diff --git a/changelog b/changelog index 4b454b8..80e94f3 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,9 @@ +20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence +20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence +20080423 tpd src/input/schaum20.input show Schaums-Axiom equivalence +20080423 tpd src/input/schaum19.input show Schaums-Axiom equivalence +20080423 tpd src/input/schaum18.input show Schaums-Axiom equivalence +20080423 tpd src/input/schaum17.input show Schaums-Axiom equivalence 20080421 tpd src/input/schaum16.input show Schaums-Axiom equivalence 20080421 tpd src/input/schaum15.input show Schaums-Axiom equivalence 20080421 tpd src/input/schaum14.input show Schaums-Axiom equivalence diff --git a/src/input/schaum17.input.pamphlet b/src/input/schaum17.input.pamphlet index fd40132..a11b623 100644 --- a/src/input/schaum17.input.pamphlet +++ b/src/input/schaum17.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 30 +--S 1 aa:=integrate(sin(a*x),x) --R --R @@ -27,6 +27,22 @@ aa:=integrate(sin(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=-cos(a*x)/a +--R +--R cos(a x) +--R (2) - -------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:339 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.340~~~~~$\displaystyle @@ -37,7 +53,7 @@ $$ <<*>>= )clear all ---S 2 of 30 +--S 4 aa:=integrate(x*sin(a*x),x) --R --R @@ -47,6 +63,23 @@ aa:=integrate(x*sin(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 5 +bb:=sin(a*x)/a^2-(x*cos(a*x))/a +--R +--R sin(a x) - a x cos(a x) +--R (2) ----------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:340 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.341~~~~~$\displaystyle @@ -57,7 +90,7 @@ $$ <<*>>= )clear all ---S 3 of 30 +--S 7 aa:=integrate(x^2*sin(a*x),x) --R --R @@ -68,6 +101,24 @@ aa:=integrate(x^2*sin(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 8 +bb:=(2*x)/a^2*sin(a*x)+(2/a^3-x^2/a)*cos(a*x) +--R +--R 2 2 +--R 2a x sin(a x) + (- a x + 2)cos(a x) +--R (2) ------------------------------------ +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:341 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.342~~~~~$\displaystyle @@ -79,7 +130,7 @@ $$ <<*>>= )clear all ---S 4 of 30 +--S 10 aa:=integrate(x^3*sin(a*x),x) --R --R @@ -90,6 +141,24 @@ aa:=integrate(x^3*sin(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 11 +bb:=((3*x^2)/a^2-6/a^4)*sin(a*x)+(6*x/a^3-x^3/a)*cos(a*x) +--R +--R 2 2 3 3 +--R (3a x - 6)sin(a x) + (- a x + 6a x)cos(a x) +--R (2) --------------------------------------------- +--R 4 +--R a +--R Type: Expression Integer +--E + +--S 12 14:342 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.343~~~~~$\displaystyle @@ -100,7 +169,7 @@ $$ <<*>>= )clear all ---S 5 of 30 +--S 13 14:343 Schaums and Axiom agree by definition aa:=integrate(sin(x)/x,x) --R --R @@ -117,7 +186,7 @@ $$ <<*>>= )clear all ---S 6 of 30 +--S 14 14:344 Axiom cannot compute this integral aa:=integrate(sin(a*x)/x^2,x) --R --R @@ -139,7 +208,7 @@ $$ <<*>>= )clear all ---S 7 of 30 +--S 15 aa:=integrate(1/sin(a*x),x) --R --R @@ -150,6 +219,35 @@ aa:=integrate(1/sin(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 16 +bb:=1/a*log(tan((a*x)/2)) +--R +--R a x +--R log(tan(---)) +--R 2 +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R a x sin(a x) +--R - log(tan(---)) + log(------------) +--R 2 cos(a x) + 1 +--R (3) ----------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 18 14:345 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.346~~~~~$\displaystyle @@ -162,7 +260,7 @@ $$ <<*>>= )clear all ---S 8 of 30 +--S 19 14:346 Axiom cannot compute this integral aa:=integrate(x/sin(a*x),x) --R --R @@ -182,7 +280,7 @@ $$ <<*>>= )clear all ---S 9 of 30 +--S 20 aa:=integrate(sin(a*x)^2,x) --R --R @@ -191,6 +289,31 @@ aa:=integrate(sin(a*x)^2,x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 21 +bb:=x/2-sin(2*a*x)/(4*a) +--R +--R - sin(2a x) + 2a x +--R (2) ------------------ +--R 4a +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R sin(2a x) - 2cos(a x)sin(a x) +--R (3) ----------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 23 14:347 Schaums and Axiom agreee +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.348~~~~~$\displaystyle @@ -201,7 +324,7 @@ $$ <<*>>= )clear all ---S 10 of 30 +--S 24 aa:=integrate(x*sin(a*x)^2,x) --R --R @@ -212,6 +335,38 @@ aa:=integrate(x*sin(a*x)^2,x) --R 4a --R Type: Union(Expression Integer,...) --E + +--S 25 +bb:=x^2/4-(x*sin(2*a*x))/(4*a)-cos(2*a*x)/(8*a^2) +--R +--R 2 2 +--R - 2a x sin(2a x) - cos(2a x) + 2a x +--R (2) ------------------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 26 +cc:=aa-bb +--R +--R 2 +--R 2a x sin(2a x) - 4a x cos(a x)sin(a x) + cos(2a x) - 2cos(a x) +--R (3) --------------------------------------------------------------- +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 27 14:348 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 1 +--R (4) - --- +--R 2 +--R 8a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.349~~~~~$\displaystyle @@ -222,7 +377,7 @@ $$ <<*>>= )clear all ---S 11 of 30 +--S 28 aa:=integrate(sin(a*x)^3,x) --R --R @@ -232,6 +387,23 @@ aa:=integrate(sin(a*x)^3,x) --R 3a --R Type: Union(Expression Integer,...) --E + +--S 29 +bb:=-cos(a*x)/a+cos(a*x)^3/(3*a) +--R +--R 3 +--R cos(a x) - 3cos(a x) +--R (2) --------------------- +--R 3a +--R Type: Expression Integer +--E + +--S 30 14:349 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.350~~~~~$\displaystyle @@ -242,7 +414,7 @@ $$ <<*>>= )clear all ---S 12 of 30 +--S 31 aa:=integrate(sin(a*x)^4,x) --R --R @@ -252,6 +424,32 @@ aa:=integrate(sin(a*x)^4,x) --R 8a --R Type: Union(Expression Integer,...) --E + +--S 32 +bb:=(3*x)/8-sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a) +--R +--R sin(4a x) - 8sin(2a x) + 12a x +--R (2) ------------------------------ +--R 32a +--R Type: Expression Integer +--E + +--S 33 +cc:=aa-bb +--R +--R 3 +--R - sin(4a x) + 8sin(2a x) + (8cos(a x) - 20cos(a x))sin(a x) +--R (3) ------------------------------------------------------------ +--R 32a +--R Type: Expression Integer +--E + +--S 34 14:350 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.351~~~~~$\displaystyle @@ -262,7 +460,7 @@ $$ <<*>>= )clear all ---S 13 of 30 +--S 35 aa:=integrate(1/sin(a*x)^2,x) --R --R @@ -271,6 +469,31 @@ aa:=integrate(1/sin(a*x)^2,x) --R a sin(a x) --R Type: Union(Expression Integer,...) --E + +--S 36 +bb:=-1/a*cot(a*x) +--R +--R cot(a x) +--R (2) - -------- +--R a +--R Type: Expression Integer +--E + +--S 37 +cc:=aa-bb +--R +--R cot(a x)sin(a x) - cos(a x) +--R (3) --------------------------- +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 38 14:351 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.352~~~~~$\displaystyle @@ -281,7 +504,7 @@ $$ <<*>>= )clear all ---S 14 of 30 +--S 39 aa:=integrate(1/sin(a*x)^3,x) --R --R @@ -293,6 +516,66 @@ aa:=integrate(1/sin(a*x)^3,x) --R 2a cos(a x) - 2a --R Type: Union(Expression Integer,...) --E + +--S 40 +bb:=-cos(a*x)/(2*a*sin(a*x)^2)+1/(2*a)*log(tan((a*x)/2)) +--R +--R 2 a x +--R sin(a x) log(tan(---)) - cos(a x) +--R 2 +--R (2) --------------------------------- +--R 2 +--R 2a sin(a x) +--R Type: Expression Integer +--E + +--S 41 +cc:=aa-bb +--R +--R (3) +--R 2 2 a x +--R (- cos(a x) + 1)sin(a x) log(tan(---)) +--R 2 +--R + +--R 2 2 sin(a x) 2 3 +--R (cos(a x) - 1)sin(a x) log(------------) + cos(a x)sin(a x) + cos(a x) +--R cos(a x) + 1 +--R + +--R - cos(a x) +--R / +--R 2 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 42 +dd:=expandLog cc +--R +--R (4) +--R 2 2 a x +--R (- cos(a x) + 1)sin(a x) log(tan(---)) +--R 2 +--R + +--R 2 2 +--R (cos(a x) - 1)sin(a x) log(sin(a x)) +--R + +--R 2 2 2 +--R (- cos(a x) + 1)sin(a x) log(cos(a x) + 1) + cos(a x)sin(a x) +--R + +--R 3 +--R cos(a x) - cos(a x) +--R / +--R 2 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 43 14:352 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.353~~~~~$\displaystyle @@ -303,7 +586,7 @@ $$ <<*>>= )clear all ---S 15 of 30 +--S 44 aa:=integrate(sin(p*x)*sin(q*x),x) --R --R @@ -312,7 +595,30 @@ aa:=integrate(sin(p*x)*sin(q*x),x) --R 2 2 --R q - p --R Type: Union(Expression Integer,...) +--E + +--S 45 +bb:=(sin(p-q)*x)/(2*(p-q))-(sin(p+q)*x)/(2*(p+q)) +--R +--R (- q + p)x sin(q + p) + (q + p)x sin(q - p) +--R (2) ------------------------------------------- +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer --E + +--S 46 14:353 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2p cos(p x)sin(q x) - 2q cos(q x)sin(p x) + (q - p)x sin(q + p) +--R + +--R (- q - p)x sin(q - p) +--R / +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.354~~~~~$\displaystyle @@ -323,7 +629,7 @@ $$ <<*>>= )clear all ---S 16 of 30 +--S 47 aa:=integrate(1/(1-sin(a*x)),x) --R --R @@ -331,7 +637,38 @@ aa:=integrate(1/(1-sin(a*x)),x) --R (1) --------------------------- --R a sin(a x) - a cos(a x) - a --R Type: Union(Expression Integer,...) +--E + +--S 48 +bb:=1/a*tan(%pi/4+(a*x)/2) +--R +--R 2a x + %pi +--R tan(----------) +--R 4 +--R (2) --------------- +--R a +--R Type: Expression Integer --E + +--S 49 +cc:=aa-bb +--R +--R 2a x + %pi +--R (- sin(a x) + cos(a x) + 1)tan(----------) - 2cos(a x) - 2 +--R 4 +--R (3) ---------------------------------------------------------- +--R a sin(a x) - a cos(a x) - a +--R Type: Expression Integer +--E + +--S 50 14:354 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 1 +--R (4) - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.355~~~~~$\displaystyle @@ -343,7 +680,7 @@ $$ <<*>>= )clear all ---S 17 of 30 +--S 51 aa:=integrate(x/(1-sin(ax)),x) --R --R @@ -353,6 +690,35 @@ aa:=integrate(x/(1-sin(ax)),x) --R 2sin(ax) - 2 --R Type: Union(Expression Integer,...) --E + +--S 52 +bb:=x/a*tan(%pi/4+(a*x)/2)+2/a^2*log(sin(%pi/4-(a*x)/2)) +--R +--R 2a x - %pi 2a x + %pi +--R 2log(- sin(----------)) + a x tan(----------) +--R 4 4 +--R (2) --------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 53 14:355 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2a x - %pi +--R (- 4sin(ax) + 4)log(- sin(----------)) +--R 4 +--R + +--R 2a x + %pi 2 2 +--R (- 2a x sin(ax) + 2a x)tan(----------) - a x +--R 4 +--R / +--R 2 2 +--R 2a sin(ax) - 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.356~~~~~$\displaystyle @@ -363,7 +729,7 @@ $$ <<*>>= )clear all ---S 18 of 30 +--S 54 aa:=integrate(1/(1+sin(ax)),x) --R --R @@ -372,6 +738,46 @@ aa:=integrate(1/(1+sin(ax)),x) --R sin(ax) + 1 --R Type: Union(Expression Integer,...) --E + +--S 55 +bb:=-1/a*tan(%pi/4-(a*x)/2) +--R +--R 2a x - %pi +--R tan(----------) +--R 4 +--R (2) --------------- +--R a +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R 2a x - %pi +--R (- sin(ax) - 1)tan(----------) + a x +--R 4 +--R (3) ------------------------------------ +--R a sin(ax) + a +--R Type: Expression Integer +--E + +--S 57 +tanrule:=rule(tan(a/b) == sin(a)/cos(b)) +--R +--R a sin(a) +--R (4) tan(-) == ------ +--R b cos(b) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 58 14:356 Axiom cannot simplify this expression +dd:=tanrule cc +--R +--R (- sin(ax) - 1)sin(2a x - %pi) + a x cos(4) +--R (5) ------------------------------------------- +--R a cos(4)sin(ax) + a cos(4) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.357~~~~~$\displaystyle @@ -383,7 +789,7 @@ $$ <<*>>= )clear all ---S 19 of 30 +--S 59 aa:=integrate(x/(1+sin(a*x)),x) --R --R @@ -402,6 +808,100 @@ aa:=integrate(x/(1+sin(a*x)),x) --R a sin(a x) + a cos(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 60 +bb:=-x/a*tan(%pi/4-(a*x)/2)+2/a^2*log(sin(%pi/4+(a*x)/2)) +--R +--R 2a x + %pi 2a x - %pi +--R 2log(sin(----------)) + a x tan(----------) +--R 4 4 +--R (2) ------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 61 +cc:=aa-bb +--R +--R (3) +--R sin(a x) + cos(a x) + 1 +--R (2sin(a x) + 2cos(a x) + 2)log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2a x + %pi +--R (- 2sin(a x) - 2cos(a x) - 2)log(sin(----------)) +--R 4 +--R + +--R 2 +--R (- sin(a x) - cos(a x) - 1)log(------------) +--R cos(a x) + 1 +--R + +--R 2a x - %pi +--R (- a x sin(a x) - a x cos(a x) - a x)tan(----------) + a x sin(a x) +--R 4 +--R + +--R - a x cos(a x) - a x +--R / +--R 2 2 2 +--R a sin(a x) + a cos(a x) + a +--R Type: Expression Integer +--E + +--S 62 +dd:=expandLog cc +--R +--R (4) +--R (2sin(a x) + 2cos(a x) + 2)log(sin(a x) + cos(a x) + 1) +--R + +--R 2a x + %pi +--R (- 2sin(a x) - 2cos(a x) - 2)log(sin(----------)) +--R 4 +--R + +--R (- sin(a x) - cos(a x) - 1)log(cos(a x) + 1) +--R + +--R 2a x - %pi +--R (- a x sin(a x) - a x cos(a x) - a x)tan(----------) +--R 4 +--R + +--R (- log(2) + a x)sin(a x) + (- log(2) - a x)cos(a x) - log(2) - a x +--R / +--R 2 2 2 +--R a sin(a x) + a cos(a x) + a +--R Type: Expression Integer +--E + +--S 63 +tanrule:=rule(tan(a/b) == sin(a)/cos(b)) +--R +--R a sin(a) +--R (5) tan(-) == ------ +--R b cos(b) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 64 14:357 Axiom cannot simplify this expression +ee:=tanrule dd +--R +--R (6) +--R (2cos(4)sin(a x) + 2cos(4)cos(a x) + 2cos(4))log(sin(a x) + cos(a x) + 1) +--R + +--R 2a x + %pi +--R (- 2cos(4)sin(a x) - 2cos(4)cos(a x) - 2cos(4))log(sin(----------)) +--R 4 +--R + +--R (- cos(4)sin(a x) - cos(4)cos(a x) - cos(4))log(cos(a x) + 1) +--R + +--R (- a x sin(a x) - a x cos(a x) - a x)sin(2a x - %pi) +--R + +--R (- cos(4)log(2) + a x cos(4))sin(a x) +--R + +--R (- cos(4)log(2) - a x cos(4))cos(a x) - cos(4)log(2) - a x cos(4) +--R / +--R 2 2 2 +--R a cos(4)sin(a x) + a cos(4)cos(a x) + a cos(4) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.358~~~~~$\displaystyle @@ -413,7 +913,7 @@ $$ <<*>>= )clear all ---S 20 of 30 +--S 65 aa:=integrate(1/(1-sin(a*x))^2,x) --R --R @@ -423,7 +923,76 @@ aa:=integrate(1/(1-sin(a*x))^2,x) --R 2 --R (3a cos(a x) + 6a)sin(a x) + 3a cos(a x) - 3a cos(a x) - 6a --R Type: Union(Expression Integer,...) +--E + +--S 66 +bb:=-1/(2*a)*tan(%pi/4+(a*x)/2)+1/(6*a)*tan(%pi/4+(a*x)/2)^3 +--R +--R 2a x + %pi 3 2a x + %pi +--R tan(----------) - 3tan(----------) +--R 4 4 +--R (2) ----------------------------------- +--R 6a +--R Type: Expression Integer --E + +--S 67 +cc:=aa-bb +--R +--R (3) +--R 2 2a x + %pi 3 +--R ((- cos(a x) - 2)sin(a x) - cos(a x) + cos(a x) + 2)tan(----------) +--R 4 +--R + +--R 2 2a x + %pi +--R ((3cos(a x) + 6)sin(a x) + 3cos(a x) - 3cos(a x) - 6)tan(----------) +--R 4 +--R + +--R 2 +--R (6cos(a x) + 6)sin(a x) + 2cos(a x) - 8cos(a x) - 10 +--R / +--R 2 +--R (6a cos(a x) + 12a)sin(a x) + 6a cos(a x) - 6a cos(a x) - 12a +--R Type: Expression Integer +--E + +--S 68 +tanrule:=rule(tan(a/b) == sin(a)/cos(b)) +--R +--R a sin(a) +--R (4) tan(-) == ------ +--R b cos(b) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 69 14:358 Axiom cannot simplify this expression +dd:=tanrule cc +--R +--R (5) +--R 2 3 +--R ((- cos(a x) - 2)sin(a x) - cos(a x) + cos(a x) + 2)sin(2a x + %pi) +--R + +--R 2 2 2 2 +--R (3cos(4) cos(a x) + 6cos(4) )sin(a x) + 3cos(4) cos(a x) +--R + +--R 2 2 +--R - 3cos(4) cos(a x) - 6cos(4) +--R * +--R sin(2a x + %pi) +--R + +--R 3 3 3 2 +--R (6cos(4) cos(a x) + 6cos(4) )sin(a x) + 2cos(4) cos(a x) +--R + +--R 3 3 +--R - 8cos(4) cos(a x) - 10cos(4) +--R / +--R 3 3 3 2 +--R (6a cos(4) cos(a x) + 12a cos(4) )sin(a x) + 6a cos(4) cos(a x) +--R + +--R 3 3 +--R - 6a cos(4) cos(a x) - 12a cos(4) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.359~~~~~$\displaystyle @@ -435,7 +1004,7 @@ $$ <<*>>= )clear all ---S 21 of 30 +--S 70 aa:=integrate(1/(1+sin(a*x))^2,x) --R --R @@ -445,7 +1014,38 @@ aa:=integrate(1/(1+sin(a*x))^2,x) --R 2 --R (3a cos(a x) + 6a)sin(a x) - 3a cos(a x) + 3a cos(a x) + 6a --R Type: Union(Expression Integer,...) +--E + +--S 71 +bb:=-1/(2*a)*tan(%pi/4-(a*x)/2)-1/(6*a)*tan(%pi/4-(a*x)/2)^3 +--R +--R 2a x - %pi 3 2a x - %pi +--R tan(----------) + 3tan(----------) +--R 4 4 +--R (2) ----------------------------------- +--R 6a +--R Type: Expression Integer --E + +--S 72 14:359 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2 2a x - %pi 3 +--R ((- cos(a x) - 2)sin(a x) + cos(a x) - cos(a x) - 2)tan(----------) +--R 4 +--R + +--R 2 2a x - %pi +--R ((- 3cos(a x) - 6)sin(a x) + 3cos(a x) - 3cos(a x) - 6)tan(----------) +--R 4 +--R + +--R 2 +--R (- 6cos(a x) - 6)sin(a x) + 2cos(a x) - 8cos(a x) - 10 +--R / +--R 2 +--R (6a cos(a x) + 12a)sin(a x) - 6a cos(a x) + 6a cos(a x) + 12a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.360~~~~~$\displaystyle @@ -466,7 +1066,7 @@ $$ <<*>>= )clear all ---S 22 of 30 +--S 73 aa:=integrate(1/(p+q*sin(a*x)),x) --R --R @@ -498,6 +1098,156 @@ aa:=integrate(1/(p+q*sin(a*x)),x) --R a\|- q + p --R Type: Union(List Expression Integer,...) --E + +--S 74 +bb1:=2/(a*sqrt(p^2-q^2))*atan((p*tan((a*x)/2+q))/sqrt(p^2-q^2)) +--R +--R a x + 2q +--R p tan(--------) +--R 2 +--R 2atan(---------------) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R (2) ---------------------- +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 75 +bb2:=1/(a*sqrt(q^2-p^2))*log((p*tan((a*x)/2+q-sqrt(q^2-p^2)))/(p*tan((a*x)/2+q+sqrt(q^2-p^2)))) +--R +--R +-------+ +--R | 2 2 +--R 2\|q - p - a x - 2q +--R tan(----------------------) +--R 2 +--R log(- ---------------------------) +--R +-------+ +--R | 2 2 +--R 2\|q - p + a x + 2q +--R tan(----------------------) +--R 2 +--R (3) ---------------------------------- +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 76 +cc1:=aa.1-bb1 +--R +--R (4) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R a x + 2q +--R +-------+ p tan(--------) +--R | 2 2 2 +--R - 2\|q - p atan(---------------) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 77 +cc2:=aa.2-bb1 +--R +--R (5) +--R +---------+ a x + 2q +--R | 2 2 p tan(--------) +--R (p sin(a x) + q cos(a x) + q)\|- q + p 2 +--R - 2atan(-----------------------------------------) - 2atan(---------------) +--R 2 2 2 2 +---------+ +--R (q - p )cos(a x) + q - p | 2 2 +--R \|- q + p +--R --------------------------------------------------------------------------- +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 78 +cc3:=aa.1-bb2 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R 2\|q - p - a x - 2q +--R tan(----------------------) +--R 2 +--R - log(- ---------------------------) +--R +-------+ +--R | 2 2 +--R 2\|q - p + a x + 2q +--R tan(----------------------) +--R 2 +--R + +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 79 14:360 Axiom cannot simplify these equations +cc4:=aa.2-bb2 +--R +--R (7) +--R +-------+ +--R | 2 2 +--R 2\|q - p - a x - 2q +--R +---------+ tan(----------------------) +--R | 2 2 2 +--R - \|- q + p log(- ---------------------------) +--R +-------+ +--R | 2 2 +--R 2\|q - p + a x + 2q +--R tan(----------------------) +--R 2 +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R - 2\|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.361~~~~~$\displaystyle @@ -509,7 +1259,7 @@ $$ <<*>>= )clear all ---S 23 of 30 +--S 80 aa:=integrate(1/(p+q*sin(a*x))^2,x) --R --R @@ -554,6 +1304,206 @@ aa:=integrate(1/(p+q*sin(a*x))^2,x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 81 +t1:=integrate(1/(p+q*sin(a*x)),x) +--R +--R (2) +--R [ +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a\|q - p +--R , +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R - ------------------------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 82 +bb1:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.1 +--R +--R (3) +--R 2 +--R (- p q sin(a x) - p ) +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R - q cos(a x)\|q - p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)sin(a x) + a p q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 83 +bb2:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.2 +--R +--R (4) +--R +---------+ +--R | 2 2 +--R 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R (2p q sin(a x) + 2p )atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +--R | 2 2 +--R - q cos(a x)\|- q + p +--R / +--R +---------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)sin(a x) + a p q - a p )\|- q + p +--R Type: Expression Integer +--E + +--S 84 +cc1:=aa.1-bb1 +--R +--R (5) +--R 2 +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R 2 +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R - q\|q - p +--R / +--R +-------+ +--R 2 3 | 2 2 +--R (a p q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 85 +cc2:=aa.2-bb1 +--R +--R (6) +--R +---------+ +--R 2 | 2 2 +--R p \|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R 2 | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2p \|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R - q\|- q + p \|q - p +--R / +--R +---------+ +-------+ +--R 2 3 | 2 2 | 2 2 +--R (a p q - a p )\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 86 +cc3:=aa.1-bb2 +--R +--R (7) +--R +---------+ +--R 2 | 2 2 +--R p \|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R 2 | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R - 2p \|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R - q\|- q + p \|q - p +--R / +--R +---------+ +-------+ +--R 2 3 | 2 2 | 2 2 +--R (a p q - a p )\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 87 14:361 Schaums and Axiom differ by a constant +cc4:=aa.2-bb2 +--R +--R q +--R (8) - ------------- +--R 2 3 +--R a p q - a p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.362~~~~~$\displaystyle @@ -564,7 +1514,7 @@ $$ <<*>>= )clear all ---S 24 of 30 +--S 88 aa:=integrate(1/(p^2+a^2*sin(a*x)),x) --R --R @@ -597,6 +1547,76 @@ aa:=integrate(1/(p^2+a^2*sin(a*x)),x) --R a\|p - a --R Type: Union(List Expression Integer,...) --E + +--S 89 +bb:=1/(a*p*sqrt(p^2+q^2))*atan((sqrt(p^2+q^2)*tan(a*x))/p) +--R +--R +-------+ +--R | 2 2 +--R tan(a x)\|q + p +--R atan(------------------) +--R p +--R (2) ------------------------ +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Expression Integer +--E + +--S 90 +cc1:=aa.1-bb +--R +--R (3) +--R +-------+ +--R | 2 2 +--R p\|q + p +--R * +--R log +--R +---------+ +--R 2 2 4 4 4 | 4 4 +--R (a p sin(a x) + (- p + a )cos(a x) + a )\|- p + a +--R + +--R 6 4 2 2 4 6 2 4 6 +--R (p - a p )sin(a x) + (a p - a )cos(a x) + a p - a +--R / +--R 2 2 +--R a sin(a x) + p +--R + +--R +-------+ +--R +---------+ | 2 2 +--R | 4 4 tan(a x)\|q + p +--R - \|- p + a atan(------------------) +--R p +--R / +--R +---------+ +-------+ +--R | 4 4 | 2 2 +--R a p\|- p + a \|q + p +--R Type: Expression Integer +--E + +--S 91 14:362 Axiom cannot simplify this expression +cc2:=aa.2-bb +--R +--R (4) +--R +-------+ +--R +-------+ | 2 2 +--R | 4 4 tan(a x)\|q + p +--R - \|p - a atan(------------------) +--R p +--R + +--R +-------+ +--R +-------+ 2 2 2 | 4 4 +--R | 2 2 (p sin(a x) + a cos(a x) + a )\|p - a +--R 2p\|q + p atan(----------------------------------------) +--R 4 4 4 4 +--R (p - a )cos(a x) + p - a +--R / +--R +-------+ +-------+ +--R | 4 4 | 2 2 +--R a p\|p - a \|q + p +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.363~~~~~$\displaystyle @@ -616,7 +1636,7 @@ $$ <<*>>= )clear all ---S 25 of 30 +--S 92 aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x) --R --R @@ -658,6 +1678,154 @@ aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 93 +bb1:=1/(a*p*sqrt(p^2-q^2))*atan((sqrt(p^2-q^2)*tan(a*x))/p) +--R +--R +---------+ +--R | 2 2 +--R tan(a x)\|- q + p +--R atan(--------------------) +--R p +--R (2) -------------------------- +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R Type: Expression Integer +--E + +--S 94 +bb2:=1/(2*a*p*sqrt(q^2-p^2))*log((sqrt(q^2-p^2)*tan(a*x)+p)/(sqrt(q^2-p^2)*tan(a*x)-p)) +--R +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p + p +--R log(----------------------) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p - p +--R (3) --------------------------- +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R Type: Expression Integer +--E + +--S 95 +cc1:=aa.1-bb1 +--R +--R (4) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 2 | 2 2 +--R ((- q + 2p )cos(a x) + q - p )\|q - p +--R + +--R 2 3 +--R (2p q - 2p )cos(a x)sin(a x) +--R / +--R 2 2 2 2 +--R q cos(a x) - q + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 tan(a x)\|- q + p +--R - 2\|q - p atan(--------------------) +--R p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 96 +cc2:=aa.2-bb1 +--R +--R (5) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R tan(a x)\|- q + p p sin(a x)\|- q + p +--R - atan(--------------------) - atan(-------------------------------) +--R p 2 2 2 2 +--R (2q - 2p )cos(a x) + 2q - 2p +--R + +--R 2 2 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x) +--R - atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a p\|- q + p +--R Type: Expression Integer +--E + +--S 97 +cc3:=aa.1-bb2 +--R +--R (6) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p + p +--R - log(----------------------) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p - p +--R + +--R log +--R +-------+ +--R 2 2 2 2 2 | 2 2 +--R ((- q + 2p )cos(a x) + q - p )\|q - p +--R + +--R 2 3 +--R (2p q - 2p )cos(a x)sin(a x) +--R / +--R 2 2 2 2 +--R q cos(a x) - q + p +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R Type: Expression Integer +--E + +--S 98 14:363 Axiom cannot simplify this expression +cc4:=aa.2-bb2 +--R +--R (7) +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 tan(a x)\|q - p + p +--R - \|- q + p log(----------------------) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q - p - p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 p sin(a x)\|- q + p +--R - 2\|q - p atan(-------------------------------) +--R 2 2 2 2 +--R (2q - 2p )cos(a x) + 2q - 2p +--R + +--R +-------+ 2 2 2 2 +--R | 2 2 ((2q - p )cos(a x) + 2q - 2p )sin(a x) +--R - 2\|q - p atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.364~~~~~$\displaystyle @@ -669,7 +1837,7 @@ $$ <<*>>= )clear all ---S 26 of 30 +--S 99 14:364 Axiom cannot compute this integral aa:=integrate(x^m*sin(a*x),x) --R --R @@ -689,7 +1857,7 @@ $$ <<*>>= )clear all ---S 26 of 30 +--S 100 14:365 Axiom cannot compute this integral aa:=integrate(sin(a*x)/x^n,x) --R --R @@ -710,7 +1878,7 @@ $$ <<*>>= )clear all ---S 28 of 30 +--S 101 14:366 Axiom cannot compute this integral aa:=integrate(sin(a*x)^n,x) --R --R @@ -731,7 +1899,7 @@ $$ <<*>>= )clear all ---S 29 of 30 +--S 102 14:367 Axiom cannot compute this integral aa:=integrate(1/(sin(a*x))^n,x) --R --R @@ -739,7 +1907,7 @@ aa:=integrate(1/(sin(a*x))^n,x) --R ++ 1 --I (1) | ---------- d%I --R ++ n ---i sin(%I a) +--I sin(%I a) --R Type: Union(Expression Integer,...) --E @ @@ -754,7 +1922,7 @@ $$ <<*>>= )clear all ---S 30 of 30 +--S 103 14:368 Axiom cannot compute this integral aa:=integrate(x/sin(a*x)^n,x) --R --R diff --git a/src/input/schaum18.input.pamphlet b/src/input/schaum18.input.pamphlet index 3cb4766..95fceae 100644 --- a/src/input/schaum18.input.pamphlet +++ b/src/input/schaum18.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 30 +--S 1 aa:=integrate(cos(a*x),x) --R --R @@ -27,6 +27,22 @@ aa:=integrate(cos(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=sin(a*x)/a +--R +--R sin(a x) +--R (2) -------- +--R a +--R Type: Expression Integer +--E + +--S 3 14:369 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.370~~~~~$\displaystyle @@ -37,7 +53,7 @@ $$ <<*>>= )clear all ---S 2 of 30 +--S 4 aa:=integrate(x*cos(a*x),x) --R --R @@ -47,6 +63,23 @@ aa:=integrate(x*cos(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 5 +bb:=cos(a*x)/a^2+(x*sin(a*x))/a +--R +--R a x sin(a x) + cos(a x) +--R (2) ----------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:370 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.371~~~~~$\displaystyle @@ -57,7 +90,7 @@ $$ <<*>>= )clear all ---S 3 of 30 +--S 7 aa:=integrate(x^2*cos(a*x),x) --R --R @@ -68,6 +101,24 @@ aa:=integrate(x^2*cos(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 8 +bb:=(2*x)/a^2*cos(a*x)+(x^2/a-2/a^3)*sin(a*x) +--R +--R 2 2 +--R (a x - 2)sin(a x) + 2a x cos(a x) +--R (2) ---------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 9 14:371 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.372~~~~~$\displaystyle @@ -79,7 +130,7 @@ $$ <<*>>= )clear all ---S 4 of 30 +--S 10 aa:=integrate(x^3*cos(a*x),x) --R --R @@ -90,6 +141,24 @@ aa:=integrate(x^3*cos(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 11 +bb:=((3*x^2)/a^2-6/a^4)*cos(a*x)+(x^3/a-(6*x)/a^3)*sin(a*x) +--R +--R 3 3 2 2 +--R (a x - 6a x)sin(a x) + (3a x - 6)cos(a x) +--R (2) ------------------------------------------- +--R 4 +--R a +--R Type: Expression Integer +--E + +--S 12 14:372 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.373~~~~~$\displaystyle @@ -101,7 +170,7 @@ $$ <<*>>= )clear all ---S 5 of 30 +--S 13 14:373 Schaums and Axiom agree by definition aa:=integrate(cos(x)/x,x) --R --R @@ -118,7 +187,7 @@ $$ <<*>>= )clear all ---S 6 of 30 +--S 14 14:374 Axiom cannot compute this integral aa:=integrate(cos(a*x)/x^2,x) --R --R @@ -140,7 +209,7 @@ $$ <<*>>= )clear all ---S 7 of 30 +--S 15 aa:=integrate(1/cos(a*x),x) --R --R @@ -151,6 +220,59 @@ aa:=integrate(1/cos(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 16 +bb1:=1/a*log(sec(a*x)+tan(a*x)) +--R +--R log(tan(a x) + sec(a x)) +--R (2) ------------------------ +--R a +--R Type: Expression Integer +--E + +--S 17 +bb2:=1/a*log(tan(%pi/4+(a*x)/2)) +--R +--R 2a x + %pi +--R log(tan(----------)) +--R 4 +--R (3) -------------------- +--R a +--R Type: Expression Integer +--E + +--S 18 +cc1:=aa-bb1 +--R +--R (4) +--R sin(a x) + cos(a x) + 1 +--R - log(tan(a x) + sec(a x)) + log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 19 14:375 Axiom cannot simplify these expressions +cc2:=aa-bb2 +--R +--R (5) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.376~~~~~$\displaystyle @@ -164,7 +286,7 @@ $$ <<*>>= )clear all ---S 8 of 30 +--S 20 14:376 Axiom cannot compute this integral aa:=integrate(x/cos(a*x),x) --R --R @@ -184,7 +306,7 @@ $$ <<*>>= )clear all ---S 9 of 30 +--S 21 aa:=integrate(cos(a*x)^2,x) --R --R @@ -193,6 +315,40 @@ aa:=integrate(cos(a*x)^2,x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 22 +bb:=x/2+sin(2*a*x)/(4*a) +--R +--R sin(2a x) + 2a x +--R (2) ---------------- +--R 4a +--R Type: Expression Integer +--E + +--S 23 +cc:=aa-bb +--R +--R - sin(2a x) + 2cos(a x)sin(a x) +--R (3) ------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 24 +cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b))) +--R +--I %S sin(b + a) - %S sin(b - a) +--I (4) %S cos(b)sin(a) == ----------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 14:377 Schaums and Axiom agree +dd:=cossinrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.378~~~~~$\displaystyle @@ -203,7 +359,7 @@ $$ <<*>>= )clear all ---S 10 of 30 +--S 26 aa:=integrate(x*cos(a*x)^2,x) --R --R @@ -214,6 +370,87 @@ aa:=integrate(x*cos(a*x)^2,x) --R 4a --R Type: Union(Expression Integer,...) --E + +--S 27 +bb:=x^2/4+(x*sin(2*a*x))/(4*a)+cos(2*a*x)/(8*a^2) +--R +--R 2 2 +--R 2a x sin(2a x) + cos(2a x) + 2a x +--R (2) ---------------------------------- +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 28 +cc:=aa-bb +--R +--R 2 +--R - 2a x sin(2a x) + 4a x cos(a x)sin(a x) - cos(2a x) + 2cos(a x) +--R (3) ----------------------------------------------------------------- +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 29 +cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b))) +--R +--I %T sin(b + a) - %T sin(b - a) +--I (4) %T cos(b)sin(a) == ----------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 30 +dd:=cossinrule cc +--R +--R 2 +--R - cos(2a x) + 2cos(a x) +--R (5) ------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 31 +coscosrule:=rule(cos(a)*cos(b) == 1/2*(cos(a-b)+cos(a+b))) +--R +--I %U cos(b + a) + %U cos(b - a) +--I (6) %U cos(a)cos(b) == ----------------------------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 32 +ee:=coscosrule dd +--R +--R 2 +--R - cos(2a x) + 2cos(a x) +--R (7) ------------------------ +--R 2 +--R 8a +--R Type: Expression Integer +--E + +--S 33 +cossqrrule1:=rule(cos(a)^2 == 1/2+1/2*cos(2*a)) +--R +--R 2 cos(2a) + 1 +--R (8) cos(a) == ----------- +--R 2 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 34 14:378 Schaums and Axiom differ by a constant +ff:=cossqrrule1 ee +--R +--R 1 +--R (9) --- +--R 2 +--R 8a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.379~~~~~$\displaystyle @@ -224,7 +461,7 @@ $$ <<*>>= )clear all ---S 11 of 30 +--S 35 aa:=integrate(cos(a*x)^3,x) --R --R @@ -233,7 +470,42 @@ aa:=integrate(cos(a*x)^3,x) --R (1) ----------------------- --R 3a --R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=sin(a*x)/a-sin(a*x)^3/(3*a) +--R +--R 3 +--R - sin(a x) + 3sin(a x) +--R (2) ----------------------- +--R 3a +--R Type: Expression Integer --E + +--S 37 +cc:=aa-bb +--R +--R 3 2 +--R sin(a x) + (cos(a x) - 1)sin(a x) +--R (3) ----------------------------------- +--R 3a +--R Type: Expression Integer +--E + +--S 38 +cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2) +--R +--R 2 2 +--R (4) cos(a) == - sin(a) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 39 14:379 Schaums and Axiom agree +dd:=cossqrrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.380~~~~~$\displaystyle @@ -244,7 +516,7 @@ $$ <<*>>= )clear all ---S 12 of 30 +--S 40 aa:=integrate(cos(a*x)^4,x) --R --R @@ -253,7 +525,26 @@ aa:=integrate(cos(a*x)^4,x) --R (1) --------------------------------------- --R 8a --R Type: Union(Expression Integer,...) +--E + +--S 41 +bb:=(3*x)/8+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a) +--R +--R sin(4a x) + 8sin(2a x) + 12a x +--R (2) ------------------------------ +--R 32a +--R Type: Expression Integer --E + +--S 42 14:380 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R 3 +--R - sin(4a x) - 8sin(2a x) + (8cos(a x) + 12cos(a x))sin(a x) +--R (3) ------------------------------------------------------------ +--R 32a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.381~~~~~$\displaystyle @@ -264,7 +555,7 @@ $$ <<*>>= )clear all ---S 13 of 30 +--S 43 aa:=integrate(1/cos(a*x)^2,x) --R --R @@ -272,7 +563,41 @@ aa:=integrate(1/cos(a*x)^2,x) --R (1) ---------- --R a cos(a x) --R Type: Union(Expression Integer,...) +--E + +--S 44 +bb:=tan(a*x)/a +--R +--R tan(a x) +--R (2) -------- +--R a +--R Type: Expression Integer --E + +--S 45 +cc:=aa-bb +--R +--R - cos(a x)tan(a x) + sin(a x) +--R (3) ----------------------------- +--R a cos(a x) +--R Type: Expression Integer +--E + +--S 46 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 47 14:381 Schaums and Axiom agree +dd:=tanrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.382~~~~~$\displaystyle @@ -284,7 +609,7 @@ $$ <<*>>= )clear all ---S 14 of 30 +--S 48 aa:=integrate(1/cos(a*x)^3,x) --R --R @@ -300,7 +625,35 @@ aa:=integrate(1/cos(a*x)^3,x) --R 2 --R 2a cos(a x) --R Type: Union(Expression Integer,...) +--E + +--S 49 +bb:=sin(a*x)/(2*a*cos(a*x)^2)+1/(2*a)*log(tan(%pi/4+(a*x)/2)) +--R +--R 2 2a x + %pi +--R cos(a x) log(tan(----------)) + sin(a x) +--R 4 +--R (2) ---------------------------------------- +--R 2 +--R 2a cos(a x) +--R Type: Expression Integer --E + +--S 50 14:382 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.383~~~~~$\displaystyle @@ -311,7 +664,7 @@ $$ <<*>>= )clear all ---S 15 of 30 +--S 51 aa:=integrate(cos(p*x)*cos(q*x),x) --R --R @@ -320,7 +673,35 @@ aa:=integrate(cos(p*x)*cos(q*x),x) --R 2 2 --R q - p --R Type: Union(Expression Integer,...) +--E + +--S 52 +bb:=(sin(a-p)*x)/(2*(a-p))+(sin(a+p)*x)/(2*(a+p)) +--R +--R (p - a)x sin(p + a) + (p + a)x sin(p - a) +--R (2) ----------------------------------------- +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer --E + +--S 53 14:383 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2 2 3 2 +--R (2p - 2a )q cos(p x)sin(q x) + (- 2p + 2a p)cos(q x)sin(p x) +--R + +--R 2 3 2 +--R ((- p + a)q + p - a p )x sin(p + a) +--R + +--R 2 3 2 +--R ((- p - a)q + p + a p )x sin(p - a) +--R / +--R 2 2 2 4 2 2 +--R (2p - 2a )q - 2p + 2a p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.384~~~~~$\displaystyle @@ -331,7 +712,7 @@ $$ <<*>>= )clear all ---S 16 of 30 +--S 54 aa:=integrate(1/(1-cos(a*x)),x) --R --R @@ -340,6 +721,24 @@ aa:=integrate(1/(1-cos(a*x)),x) --R a sin(a x) --R Type: Union(Expression Integer,...) --E + +--S 55 +bb:=-1/a*cot(a*x)/2 +--R +--R cot(a x) +--R (2) - -------- +--R 2a +--R Type: Expression Integer +--E + +--S 56 14:384 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R cot(a x)sin(a x) - 2cos(a x) - 2 +--R (3) -------------------------------- +--R 2a sin(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.385~~~~~$\displaystyle @@ -351,7 +750,7 @@ $$ <<*>>= )clear all ---S 17 of 30 +--S 57 aa:=integrate(x/(1-cos(ax)),x) --R --R @@ -361,6 +760,30 @@ aa:=integrate(x/(1-cos(ax)),x) --R 2cos(ax) - 2 --R Type: Union(Expression Integer,...) --E + +--S 58 +bb:=-x/a*cot(a*x)/2+2/a^2*log(sin((a*x)/2)) +--R +--R a x +--R 4log(sin(---)) - a x cot(a x) +--R 2 +--R (2) ----------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 59 14:385 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R a x 2 2 +--R (- 4cos(ax) + 4)log(sin(---)) + (a x cos(ax) - a x)cot(a x) - a x +--R 2 +--R (3) ------------------------------------------------------------------ +--R 2 2 +--R 2a cos(ax) - 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.386~~~~~$\displaystyle @@ -371,7 +794,7 @@ $$ <<*>>= )clear all ---S 18 of 30 +--S 60 aa:=integrate(1/(1+cos(ax)),x) --R --R @@ -379,7 +802,25 @@ aa:=integrate(1/(1+cos(ax)),x) --R (1) ----------- --R cos(ax) + 1 --R Type: Union(Expression Integer,...) ---E +--E + +--S 61 +bb:=1/a*tan(a*x)/2 +--R +--R tan(a x) +--R (2) -------- +--R 2a +--R Type: Expression Integer +--E + +--S 62 14:386 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (- cos(ax) - 1)tan(a x) + 2a x +--R (3) ------------------------------ +--R 2a cos(ax) + 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.387~~~~~$\displaystyle @@ -391,7 +832,7 @@ $$ <<*>>= )clear all ---S 19 of 30 +--S 63 aa:=integrate(x/(1+cos(a*x)),x) --R --R @@ -403,6 +844,35 @@ aa:=integrate(x/(1+cos(a*x)),x) --R a cos(a x) + a --R Type: Union(Expression Integer,...) --E + +--S 64 +bb:=x/a*tan((a*x)/2)+2/a^2*log(cos((a*x)/2)) +--R +--R a x a x +--R 2log(cos(---)) + a x tan(---) +--R 2 2 +--R (2) ----------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 65 14:387 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R a x 2 +--R (- 2cos(a x) - 2)log(cos(---)) + (- cos(a x) - 1)log(------------) +--R 2 cos(a x) + 1 +--R + +--R a x +--R (- a x cos(a x) - a x)tan(---) + a x sin(a x) +--R 2 +--R / +--R 2 2 +--R a cos(a x) + a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.388~~~~~$\displaystyle @@ -414,7 +884,7 @@ $$ <<*>>= )clear all ---S 20 of 30 +--S 66 aa:=integrate(1/(1-cos(a*x))^2,x) --R --R @@ -423,7 +893,32 @@ aa:=integrate(1/(1-cos(a*x))^2,x) --R (1) -------------------------- --R (3a cos(a x) - 3a)sin(a x) --R Type: Union(Expression Integer,...) +--E + +--S 67 +bb:=-1/(2*a)*cot((a*x)/2)-1/(6*a)*cot((a*x)/2)^3 +--R +--R a x 3 a x +--R - cot(---) - 3cot(---) +--R 2 2 +--R (2) ----------------------- +--R 6a +--R Type: Expression Integer --E + +--S 68 14:388 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R a x 3 a x 2 +--R ((cos(a x) - 1)cot(---) + (3cos(a x) - 3)cot(---))sin(a x) - 2cos(a x) +--R 2 2 +--R + +--R 2cos(a x) + 4 +--R / +--R (6a cos(a x) - 6a)sin(a x) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.389~~~~~$\displaystyle @@ -435,7 +930,7 @@ $$ <<*>>= )clear all ---S 21 of 30 +--S 69 aa:=integrate(1/(1+cos(a*x))^2,x) --R --R @@ -445,6 +940,34 @@ aa:=integrate(1/(1+cos(a*x))^2,x) --R 3a cos(a x) + 6a cos(a x) + 3a --R Type: Union(Expression Integer,...) --E + +--S 70 +bb:=1/(2*a)*tan((a*x)/2)+1/(6*a)*tan((a*x)/2)^2 +--R +--R a x 2 a x +--R tan(---) + 3tan(---) +--R 2 2 +--R (2) --------------------- +--R 6a +--R Type: Expression Integer +--E + +--S 71 14:389 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2 a x 2 +--R (- cos(a x) - 2cos(a x) - 1)tan(---) +--R 2 +--R + +--R 2 a x +--R (- 3cos(a x) - 6cos(a x) - 3)tan(---) + (2cos(a x) + 4)sin(a x) +--R 2 +--R / +--R 2 +--R 6a cos(a x) + 12a cos(a x) + 6a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.390~~~~~$\displaystyle @@ -466,7 +989,7 @@ $$ <<*>>= )clear all ---S 22 of 30 +--S 72 aa:=integrate(1/(p+q*cos(a*x)),x) --R --R @@ -491,6 +1014,128 @@ aa:=integrate(1/(p+q*cos(a*x)),x) --R a\|- q + p --R Type: Union(List Expression Integer,...) --E + +--S 73 +bb1:=2/(a*sqrt(p^2-q^2))*atan(sqrt((p-q)/(p+q)))*tan(1/2*a*x) +--R +--R +-------+ +--R a x |- q + p +--R 2tan(---)atan( |------- ) +--R 2 \| q + p +--R (2) ------------------------- +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 74 +bb2:=a/(a*sqrt(q^2-p^2))*log((tan(1/2*a*x)+sqrt((q+p)/(q-p)))/(tan(1/2*a*x)-sqrt((q+p)/(q-p)))) +--R +--R +-----+ +--R |q + p a x +--R - |----- - tan(---) +--R \|q - p 2 +--R log(---------------------) +--R +-----+ +--R |q + p a x +--R |----- - tan(---) +--R \|q - p 2 +--R (3) -------------------------- +--R +-------+ +--R | 2 2 +--R \|q - p +--R Type: Expression Integer +--E + +--S 75 +cc1:=aa.1-bb1 +--R +--R (4) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R \|- q + p log(--------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +-------+ +--R a x | 2 2 |- q + p +--R - 2tan(---)\|q - p atan( |------- ) +--R 2 \| q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 76 +cc2:=aa.2-bb1 +--R +--R +---------+ +--R +-------+ | 2 2 +--R a x |- q + p sin(a x)\|- q + p +--R - 2tan(---)atan( |------- ) + 2atan(-----------------------) +--R 2 \| q + p (q + p)cos(a x) + q + p +--R (5) ------------------------------------------------------------ +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Expression Integer +--E + +--S 77 +cc3:=aa.1-bb2 +--R +--R (6) +--R +-----+ +--R |q + p a x +--R - |----- - tan(---) +--R \|q - p 2 +--R - a log(---------------------) +--R +-----+ +--R |q + p a x +--R |----- - tan(---) +--R \|q - p 2 +--R + +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a\|q - p +--R Type: Expression Integer +--E + +--S 78 14:390 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (7) +--R +-----+ +--R |q + p a x +--R +---------+ - |----- - tan(---) +--R | 2 2 \|q - p 2 +--R - a\|- q + p log(---------------------) +--R +-----+ +--R |q + p a x +--R |----- - tan(---) +--R \|q - p 2 +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R 2\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a\|- q + p \|q - p +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.391~~~~~$\displaystyle @@ -502,7 +1147,7 @@ $$ <<*>>= )clear all ---S 23 of 30 +--S 79 aa:=integrate(1/(p+q*cos(a*x))^2,x) --R --R @@ -542,6 +1187,143 @@ aa:=integrate(1/(p+q*cos(a*x))^2,x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 80 +t1:=integrate(1/(p+q*cos(a*x)),x) +--R +--R (2) +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R [-------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R a\|q - p +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p +--R 2atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R ------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) +--E + +--S 81 +bb1:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.1 +--R +--R (3) +--R 2 +--R (- p q cos(a x) - p ) +--R * +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R q sin(a x)\|q - p +--R / +--R +-------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cos(a x) + a p q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 82 +bb2:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.2 +--R +--R (4) +--R +---------+ +--R | 2 2 +---------+ +--R 2 sin(a x)\|- q + p | 2 2 +--R (- 2p q cos(a x) - 2p )atan(-----------------------) + q sin(a x)\|- q + p +--R (q + p)cos(a x) + q + p +--R ----------------------------------------------------------------------------- +--R +---------+ +--R 3 2 2 3 | 2 2 +--R ((a q - a p q)cos(a x) + a p q - a p )\|- q + p +--R Type: Expression Integer +--E + +--S 83 +cc1:=aa.1-bb1 +--R +--R (5) +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R p log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R p log(--------------------------------------------------) +--R q cos(a x) + p +--R / +--R +-------+ +--R 2 2 | 2 2 +--R (a q - a p )\|q - p +--R Type: Expression Integer +--E + +--S 84 +cc2:=aa.2-bb1 +--R +--R (6) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R p\|- q + p log(--------------------------------------------------) +--R q cos(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R - 2p\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R 2 2 | 2 2 | 2 2 +--R (a q - a p )\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 85 +cc3:=aa.1-bb2 +--R +--R (7) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R p\|- q + p log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R 2p\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R 2 2 | 2 2 | 2 2 +--R (a q - a p )\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 86 14:391 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.392~~~~~$\displaystyle @@ -552,7 +1334,7 @@ $$ <<*>>= )clear all ---S 24 of 30 +--S 87 aa:=integrate(1/(p^2+a^2*cos(a*x)),x) --R --R @@ -579,6 +1361,63 @@ aa:=integrate(1/(p^2+a^2*cos(a*x)),x) --R a\|p - a --R Type: Union(List Expression Integer,...) --E + +--S 88 +bb:=1/(a*p*sqrt(p^2+q^2))*atan((p*tan(a*x))/sqrt(p^2+q^2)) +--R +--R p tan(a x) +--R atan(----------) +--R +-------+ +--R | 2 2 +--R \|q + p +--R (2) ---------------- +--R +-------+ +--R | 2 2 +--R a p\|q + p +--R Type: Expression Integer +--E + +--S 89 +cc1:=aa.1-bb +--R +--R (3) +--R +---------+ +--R +-------+ 2 2 | 4 4 4 4 +--R | 2 2 (- p cos(a x) - a )\|- p + a + (p - a )sin(a x) +--R p\|q + p log(---------------------------------------------------) +--R 2 2 +--R a cos(a x) + p +--R + +--R +---------+ +--R | 4 4 p tan(a x) +--R - \|- p + a atan(----------) +--R +-------+ +--R | 2 2 +--R \|q + p +--R / +--R +---------+ +-------+ +--R | 4 4 | 2 2 +--R a p\|- p + a \|q + p +--R Type: Expression Integer +--E + +--S 90 14:392 Axiom cannot simplify these expressions +cc2:=aa.2-bb +--R +--R (4) +--R +-------+ +--R +-------+ | 4 4 +-------+ +--R | 2 2 sin(a x)\|p - a | 4 4 p tan(a x) +--R 2p\|q + p atan(---------------------------) - \|p - a atan(----------) +--R 2 2 2 2 +-------+ +--R (p + a )cos(a x) + p + a | 2 2 +--R \|q + p +--R -------------------------------------------------------------------------- +--R +-------+ +-------+ +--R | 4 4 | 2 2 +--R a p\|p - a \|q + p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.393~~~~~$\displaystyle @@ -598,7 +1437,7 @@ $$ <<*>>= )clear all ---S 25 of 30 +--S 91 aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x) --R --R @@ -633,6 +1472,152 @@ aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x) --R ] --R Type: Union(List Expression Integer,...) --E + +--S 92 +bb1:=1/(a*p*sqrt(p^2-a^2))*atan((p*tan(a*x))/sqrt(p^2-q^2)) +--R +--R p tan(a x) +--R atan(------------) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R (2) ------------------ +--R +-------+ +--R | 2 2 +--R a p\|p - a +--R Type: Expression Integer +--E + +--S 93 +bb2:=1/(2*a*p*sqrt(q^2-p^2))*log((p*tan(a*x)-sqrt(q^2-p^2))/(p*tan(a*x)+sqrt(q^2-p^2))) +--R +--R +-------+ +--R | 2 2 +--R - \|q - p + p tan(a x) +--R log(-------------------------) +--R +-------+ +--R | 2 2 +--R \|q - p + p tan(a x) +--R (3) ------------------------------ +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R Type: Expression Integer +--E + +--S 94 +cc1:=aa.1-bb1 +--R +--R (4) +--R +-------+ +--R | 2 2 +--R \|p - a +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((q - 2p )cos(a x) + p )\|q - p +--R + +--R 2 3 +--R (- 2p q + 2p )cos(a x)sin(a x) +--R / +--R 2 2 2 +--R q cos(a x) - p +--R + +--R +-------+ +--R | 2 2 p tan(a x) +--R - 2\|q - p atan(------------) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R / +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|p - a \|q - p +--R Type: Expression Integer +--E + +--S 95 +cc2:=aa.2-bb1 +--R +--R (5) +--R +---------+ +--R +-------+ | 2 2 +---------+ +--R | 2 2 sin(a x)\|- q + p | 2 2 p tan(a x) +--R \|p - a atan(--------------------) - \|- q + p atan(------------) +--R 2p cos(a x) + 2p +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R +-------+ 2 2 2 +--R | 2 2 ((q + p )cos(a x) + 2p )sin(a x) +--R \|p - a atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a p\|- q + p \|p - a +--R Type: Expression Integer +--E + +--S 96 +cc3:=aa.1-bb2 +--R +--R (6) +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 2 3 +--R ((q - 2p )cos(a x) + p )\|q - p + (- 2p q + 2p )cos(a x)sin(a x) +--R ---------------------------------------------------------------------- +--R 2 2 2 +--R q cos(a x) - p +--R + +--R +-------+ +--R | 2 2 +--R - \|q - p + p tan(a x) +--R - log(-------------------------) +--R +-------+ +--R | 2 2 +--R \|q - p + p tan(a x) +--R / +--R +-------+ +--R | 2 2 +--R 2a p\|q - p +--R Type: Expression Integer +--E + +--S 97 14:393 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (7) +--R +-------+ +--R +---------+ | 2 2 +--R | 2 2 - \|q - p + p tan(a x) +--R - \|- q + p log(-------------------------) +--R +-------+ +--R | 2 2 +--R \|q - p + p tan(a x) +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R 2\|q - p atan(--------------------) +--R 2p cos(a x) + 2p +--R + +--R +-------+ 2 2 2 +--R | 2 2 ((q + p )cos(a x) + 2p )sin(a x) +--R 2\|q - p atan(-------------------------------------------) +--R +---------+ +--R 2 | 2 2 +--R (p cos(a x) + 2p cos(a x) + p)\|- q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.394~~~~~$\displaystyle @@ -644,7 +1629,7 @@ $$ <<*>>= )clear all ---S 26 of 30 +--S 98 14:394 Axiom cannot compute this integral aa:=integrate(x^m*cos(a*x),x) --R --R @@ -664,7 +1649,7 @@ $$ <<*>>= )clear all ---S 26 of 30 +--S 99 14:395 Axiom cannot compute this integral aa:=integrate(cos(a*x)/x^n,x) --R --R @@ -685,7 +1670,7 @@ $$ <<*>>= )clear all ---S 28 of 30 +--S 100 14:396 Axiom cannot compute this integral aa:=integrate(cos(a*x)^n,x) --R --R @@ -706,7 +1691,7 @@ $$ <<*>>= )clear all ---S 29 of 30 +--S 101 14:397 Axiom cannot compute this integral aa:=integrate(1/(cos(a*x))^n,x) --R --R @@ -729,7 +1714,7 @@ $$ <<*>>= )clear all ---S 30 of 30 +--S 102 14:398 Axiom cannot compute this integral aa:=integrate(x/cos(a*x)^n,x) --R --R diff --git a/src/input/schaum19.input.pamphlet b/src/input/schaum19.input.pamphlet index 37e89ee..bf580f0 100644 --- a/src/input/schaum19.input.pamphlet +++ b/src/input/schaum19.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 36 +--S 1 aa:=integrate(sin(a*x)*cos(a*x),x) --R --R @@ -28,6 +28,43 @@ aa:=integrate(sin(a*x)*cos(a*x),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=sin(a*x)^2/(2*a) +--R +--R 2 +--R sin(a x) +--R (2) --------- +--R 2a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R 2 2 +--R - sin(a x) - cos(a x) +--R (3) ----------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 4 +cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2) +--R +--R 2 2 +--R (4) cos(a) == - sin(a) + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 14:399 Schaums and Axiom differ by a constant +dd:=cossqrrule cc +--R +--R 1 +--R (5) - -- +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.400~~~~~$\displaystyle @@ -38,7 +75,7 @@ $$ <<*>>= )clear all ---S 2 of 36 +--S 6 aa:=integrate(sin(p*x)*cos(q*x),x) --R --R @@ -48,6 +85,29 @@ aa:=integrate(sin(p*x)*cos(q*x),x) --R q - p --R Type: Union(Expression Integer,...) --E + +--S 7 +bb:=-(cos(p-q)*x)/(2*(p-q))-(cos(p+q)*x)/(2*(p+q)) +--R +--R (- q + p)x cos(q + p) + (q + p)x cos(q - p) +--R (2) ------------------------------------------- +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E + +--S 8 14:400 Axiom is unable to simplify this expression +cc:=aa-bb +--R +--R (3) +--R 2q sin(p x)sin(q x) + 2p cos(p x)cos(q x) + (q - p)x cos(q + p) +--R + +--R (- q - p)x cos(q - p) +--R / +--R 2 2 +--R 2q - 2p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.401~~~~~$\displaystyle @@ -58,7 +118,7 @@ $$ <<*>>= )clear all ---S 3 of 36 +--S 9 aa:=integrate(sin(a*x)^n*cos(a*x),x) --R --R @@ -68,6 +128,51 @@ aa:=integrate(sin(a*x)^n*cos(a*x),x) --R a n + a --R Type: Union(Expression Integer,...) --E + +--S 10 +bb:=sin(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R sin(a x) +--R (2) ------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 11 +cc:=aa-bb +--R +--R n log(sin(a x)) n + 1 +--R sin(a x)%e - sin(a x) +--R (3) ----------------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 12 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 13 +dd:=explog cc +--R +--R n + 1 n +--R - sin(a x) + sin(a x)sin(a x) +--R (5) ----------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 14 14:401 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.402~~~~~$\displaystyle @@ -78,7 +183,7 @@ $$ <<*>>= )clear all ---S 4 of 36 +--S 15 aa:=integrate(cos(a*x)^n*sin(a*x),x) --R --R @@ -87,7 +192,52 @@ aa:=integrate(cos(a*x)^n*sin(a*x),x) --R (1) - ------------------------- --R a n + a --R Type: Union(Expression Integer,...) +--E + +--S 16 +bb:=-cos(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R cos(a x) +--R (2) - ------------- +--R a n + a +--R Type: Expression Integer --E + +--S 17 +cc:=aa-bb +--R +--R n log(cos(a x)) n + 1 +--R - cos(a x)%e + cos(a x) +--R (3) ------------------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 18 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 +dd:=explog cc +--R +--R n + 1 n +--R cos(a x) - cos(a x)cos(a x) +--R (5) --------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 20 14:402 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.403~~~~~$\displaystyle @@ -98,7 +248,7 @@ $$ <<*>>= )clear all ---S 5 of 36 +--S 21 aa:=integrate(sin(a*x)^2*cos(a*x)^2,x) --R --R @@ -108,6 +258,32 @@ aa:=integrate(sin(a*x)^2*cos(a*x)^2,x) --R 8a --R Type: Union(Expression Integer,...) --E + +--S 22 +bb:=x/8-sin(4*a*x)/(32*a) +--R +--R - sin(4a x) + 4a x +--R (2) ------------------ +--R 32a +--R Type: Expression Integer +--E + +--S 23 +cc:=aa-bb +--R +--R 3 +--R sin(4a x) + (- 8cos(a x) + 4cos(a x))sin(a x) +--R (3) ---------------------------------------------- +--R 32a +--R Type: Expression Integer +--E + +--S 24 14:403 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.404~~~~~$\displaystyle @@ -118,7 +294,7 @@ $$ <<*>>= )clear all ---S 6 of 36 +--S 25 aa:=integrate(1/(sin(a*x)*cos(a*x)),x) --R --R @@ -129,6 +305,55 @@ aa:=integrate(1/(sin(a*x)*cos(a*x)),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 26 +bb:=1/a*log(tan(a*x)) +--R +--R log(tan(a x)) +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 27 +cc:=aa-bb +--R +--R sin(a x) 2cos(a x) +--R - log(tan(a x)) + log(------------) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) --------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 28 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 29 +dd:=tanrule cc +--R +--R sin(a x) sin(a x) 2cos(a x) +--R - log(--------) + log(------------) - log(- ------------) +--R cos(a x) cos(a x) + 1 cos(a x) + 1 +--R (5) --------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 30 14:404 Schaums and Axiom differ by a constant +ee:=expandLog dd +--R +--R log(- 2) +--R (6) - -------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.405~~~~~$\displaystyle @@ -139,7 +364,7 @@ $$ <<*>>= )clear all ---S 7 of 36 +--S 31 aa:=integrate(1/(sin(a*x)^2*cos(a*x)),x) --R --R @@ -155,6 +380,85 @@ aa:=integrate(1/(sin(a*x)^2*cos(a*x)),x) --R a sin(a x) --R Type: Union(Expression Integer,...) --E + +--S 32 +bb:=1/a*log(tan(%pi/4+(a*x)/2))-1/(a*sin(a*x)) +--R +--R 2a x + %pi +--R sin(a x)log(tan(----------)) - 1 +--R 4 +--R (2) -------------------------------- +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 33 +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 34 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 35 +dd:=tanrule cc +--R +--R (5) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 36 +ee:=expandLog dd +--R +--R (6) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) +--R 4 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 37 14:405 Schaums and Axiom differ by a constant +ff:=complexNormalize % +--R +--R log(- 1) +--R (7) -------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.406~~~~~$\displaystyle @@ -165,7 +469,7 @@ $$ <<*>>= )clear all ---S 8 of 36 +--S 38 aa:=integrate(1/(sin(a*x)*cos(a*x)^2),x) --R --R @@ -176,6 +480,72 @@ aa:=integrate(1/(sin(a*x)*cos(a*x)^2),x) --R a cos(a x) --R Type: Union(Expression Integer,...) --E + +--S 39 +bb:=1/a*log(tan((a*x)/2))+1/(a*cos(a*x)) +--R +--R a x +--R cos(a x)log(tan(---)) + 1 +--R 2 +--R (2) ------------------------- +--R a cos(a x) +--R Type: Expression Integer +--E + +--S 40 +cc:=aa-bb +--R +--R a x sin(a x) +--R - log(tan(---)) + log(------------) + 1 +--R 2 cos(a x) + 1 +--R (3) --------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 41 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 42 +dd:=tanrule cc +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R log(------------) - log(--------) + 1 +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) ------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 43 +ee:=expandLog dd +--R +--R a x a x +--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---)) + 1 +--R 2 2 +--R (6) --------------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 44 14:406 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R 1 +--R (7) - +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.407~~~~~$\displaystyle @@ -186,7 +556,7 @@ $$ <<*>>= )clear all ---S 9 of 36 +--S 45 aa:=integrate(1/(sin(a*x)^2*cos(a*x)^2),x) --R --R @@ -196,6 +566,51 @@ aa:=integrate(1/(sin(a*x)^2*cos(a*x)^2),x) --R a cos(a x)sin(a x) --R Type: Union(Expression Integer,...) --E + +--S 46 +bb:=-(2*cot(2*a*x))/a +--R +--R 2cot(2a x) +--R (2) - ---------- +--R a +--R Type: Expression Integer +--E + +--S 47 +cc:=aa-bb +--R +--R 2 +--R 2cos(a x)cot(2a x)sin(a x) - 2cos(a x) + 1 +--R (3) ------------------------------------------- +--R a cos(a x)sin(a x) +--R Type: Expression Integer +--E + +--S 48 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 49 +dd:=cotrule cc +--R +--R 2 +--R (- 2cos(a x) + 1)sin(2a x) + 2cos(a x)cos(2a x)sin(a x) +--R (5) -------------------------------------------------------- +--R a cos(a x)sin(a x)sin(2a x) +--R Type: Expression Integer +--E + +--S 50 14:407 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.408~~~~~$\displaystyle @@ -206,7 +621,7 @@ $$ <<*>>= )clear all ---S 10 of 36 +--S 51 aa:=integrate(sin(a*x)^2/cos(a*x),x) --R --R @@ -217,6 +632,85 @@ aa:=integrate(sin(a*x)^2/cos(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 52 +bb:=-sin(a*x)/a+1/a*log(tan((a*x)/2+%pi/4)) +--R +--R 2a x + %pi +--R log(tan(----------)) - sin(a x) +--R 4 +--R (2) ------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 53 +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 54 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 55 +dd:=tanrule cc +--R +--R (5) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 56 +ee:=expandLog dd +--R +--R (6) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) +--R 4 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 57 14:408 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R log(- 1) +--R (7) -------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.409~~~~~$\displaystyle @@ -227,7 +721,7 @@ $$ <<*>>= )clear all ---S 11 of 36 +--S 58 aa:=integrate(cos(a*x)^2/sin(a*x),x) --R --R @@ -238,6 +732,70 @@ aa:=integrate(cos(a*x)^2/sin(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 59 +bb:=cos(a*x)/a+1/a*log(tan((a*x)/2)) +--R +--R a x +--R log(tan(---)) + cos(a x) +--R 2 +--R (2) ------------------------ +--R a +--R Type: Expression Integer +--E + +--S 60 +cc:=aa-bb +--R +--R a x sin(a x) +--R - log(tan(---)) + log(------------) +--R 2 cos(a x) + 1 +--R (3) ----------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 61 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 62 +dd:=tanrule cc +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R log(------------) - log(--------) +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) --------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 63 +ee:=expandLog dd +--R +--R a x a x +--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---)) +--R 2 2 +--R (6) ----------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 64 14:409 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.410~~~~~$\displaystyle @@ -249,7 +807,7 @@ $$ <<*>>= )clear all ---S 12 of 36 +--S 65 aa:=integrate(1/(cos(a*x)*(1+sin(a*x))),x) --R --R @@ -266,9 +824,88 @@ aa:=integrate(1/(cos(a*x)*(1+sin(a*x))),x) --R Type: Union(Expression Integer,...) --E +--S 66 +bb:=-1/(2*a*(1+sin(a*x)))+1/(2*a)*log(tan((a*x)/2+%pi/4)) +--R +--R 2a x + %pi +--R (sin(a x) + 1)log(tan(----------)) - 1 +--R 4 +--R (2) -------------------------------------- +--R 2a sin(a x) + 2a +--R Type: Expression Integer +--E + +--S 67 +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) + 1 +--R cos(a x) + 1 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 68 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 69 +dd:=tanrule cc +--R +--R (5) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) + 1 +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 70 +ee:=expandLog dd +--R +--R (6) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) + 1 +--R 4 4 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 71 +ff:=complexNormalize ee +--R +--R log(- 1) + 1 +--R (7) ------------ +--R 2a +--R Type: Expression Integer +--E + )clear all ---S 13 of 36 +--S 72 aa:=integrate(1/(cos(a*x)*(1-sin(a*x))),x) --R --R @@ -284,6 +921,86 @@ aa:=integrate(1/(cos(a*x)*(1-sin(a*x))),x) --R 2a sin(a x) - 2a --R Type: Union(Expression Integer,...) --E + +--S 73 +bb:=1/(2*a*(1-sin(a*x)))+1/(2*a)*log(tan((a*x)/2+%pi/4)) +--R +--R 2a x + %pi +--R (sin(a x) - 1)log(tan(----------)) - 1 +--R 4 +--R (2) -------------------------------------- +--R 2a sin(a x) - 2a +--R Type: Expression Integer +--E + +--S 74 +cc:=aa-bb +--R +--R (3) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) - 1 +--R cos(a x) + 1 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 75 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 76 +dd:=tanrule cc +--R +--R (5) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) - 1 +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 77 +ee:=expandLog dd +--R +--R (6) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) - 1 +--R 4 4 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 78 14:410 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R log(- 1) - 1 +--R (7) ------------ +--R 2a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.411~~~~~$\displaystyle @@ -294,7 +1011,7 @@ $$ <<*>>= )clear all ---S 14 of 36 +--S 79 aa:=integrate(1/(sin(a*x)*(1+cos(a*x))),x) --R --R @@ -306,9 +1023,76 @@ aa:=integrate(1/(sin(a*x)*(1+cos(a*x))),x) --R Type: Union(Expression Integer,...) --E +--S 80 +bb:=1/(2*a*(1+cos(a*x)))+1/(2*a)*log(tan((a*x)/2)) +--R +--R a x +--R (cos(a x) + 1)log(tan(---)) + 1 +--R 2 +--R (2) ------------------------------- +--R 2a cos(a x) + 2a +--R Type: Expression Integer +--E + +--S 81 +cc:=aa-bb +--R +--R a x sin(a x) +--R - 2log(tan(---)) + 2log(------------) - 1 +--R 2 cos(a x) + 1 +--R (3) ----------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 82 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 83 +dd:=tanrule cc +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R 2log(------------) - 2log(--------) - 1 +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) --------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 84 +ee:=expandLog dd +--R +--R (6) +--R a x a x +--R 2log(sin(a x)) - 2log(sin(---)) - 2log(cos(a x) + 1) + 2log(cos(---)) - 1 +--R 2 2 +--R ------------------------------------------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 85 +ff:=complexNormalize ee +--R +--R 1 +--R (7) - -- +--R 4a +--R Type: Expression Integer +--E + )clear all ---S 15 of 36 +--S 86 aa:=integrate(1/(sin(a*x)*(1-cos(a*x))),x) --R --R @@ -319,6 +1103,73 @@ aa:=integrate(1/(sin(a*x)*(1-cos(a*x))),x) --R 4a cos(a x) - 4a --R Type: Union(Expression Integer,...) --E + +--S 87 +bb:=-1/(2*a*(1-cos(a*x)))+1/(2*a)*log(tan((a*x)/2)) +--R +--R a x +--R (cos(a x) - 1)log(tan(---)) + 1 +--R 2 +--R (2) ------------------------------- +--R 2a cos(a x) - 2a +--R Type: Expression Integer +--E + +--S 88 +cc:=aa-bb +--R +--R a x sin(a x) +--R - 2log(tan(---)) + 2log(------------) + 1 +--R 2 cos(a x) + 1 +--R (3) ----------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 89 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 90 +dd:=tanrule cc +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R 2log(------------) - 2log(--------) + 1 +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) --------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 91 +ee:=expandLog dd +--R +--R (6) +--R a x a x +--R 2log(sin(a x)) - 2log(sin(---)) - 2log(cos(a x) + 1) + 2log(cos(---)) + 1 +--R 2 2 +--R ------------------------------------------------------------------------- +--R 4a +--R Type: Expression Integer +--E + +--S 92 14:411 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R 1 +--R (7) -- +--R 4a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.412~~~~~$\displaystyle @@ -329,7 +1180,7 @@ $$ <<*>>= )clear all ---S 16 of 36 +--S 93 aa:=integrate(1/(sin(a*x)+cos(a*x)),x) --R --R @@ -342,9 +1193,37 @@ aa:=integrate(1/(sin(a*x)+cos(a*x)),x) --R Type: Union(Expression Integer,...) --E +--S 94 +bb:=1/(a*sqrt(2))*log(tan((a*x)/2+%pi/8)) +--R +--R +-+ 4a x + %pi +--R \|2 log(tan(----------)) +--R 8 +--R (2) ------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 95 +cc:=aa-bb +--R +--R (3) +--R +-+ 4a x + %pi +--R - \|2 log(tan(----------)) +--R 8 +--R + +--R +-+ +-+ +-+ +--R +-+ (- \|2 + 1)sin(a x) + (\|2 - 1)cos(a x) + \|2 - 2 +--R \|2 log(----------------------------------------------------) +--R sin(a x) + cos(a x) +--R / +--R 2a +--R Type: Expression Integer +--E + )clear all ---S 17 of 36 +--S 96 aa:=integrate(1/(sin(a*x)-cos(a*x)),x) --R --R @@ -356,6 +1235,35 @@ aa:=integrate(1/(sin(a*x)-cos(a*x)),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 97 +bb:=1/(a*sqrt(2))*log(tan((a*x)/2-%pi/8)) +--R +--R +-+ 4a x - %pi +--R \|2 log(tan(----------)) +--R 8 +--R (2) ------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 98 14:412 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R +-+ 4a x - %pi +--R - \|2 log(tan(----------)) +--R 8 +--R + +--R +-+ +-+ +-+ +--R +-+ (- \|2 + 1)sin(a x) + (- \|2 + 1)cos(a x) - \|2 + 2 +--R \|2 log(------------------------------------------------------) +--R sin(a x) - cos(a x) +--R / +--R 2a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.413~~~~~$\displaystyle @@ -366,7 +1274,7 @@ $$ <<*>>= )clear all ---S 18 of 36 +--S 99 aa:=integrate(sin(a*x)/(sin(a*x)+cos(a*x)),x) --R --R @@ -378,9 +1286,48 @@ aa:=integrate(sin(a*x)/(sin(a*x)+cos(a*x)),x) --R Type: Union(Expression Integer,...) --E +--S 100 +bb:=x/2-1/(2*a)*log(sin(a*x)+cos(a*x)) +--R +--R - log(sin(a x) + cos(a x)) + a x +--R (2) -------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 101 +cc:=aa-bb +--R +--R (3) +--R 2 - 2sin(a x) - 2cos(a x) +--R log(sin(a x) + cos(a x)) + log(------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R --------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 102 +dd:=expandLog cc +--R +--R log(sin(a x) + cos(a x)) - log(- sin(a x) - cos(a x)) +--R (4) ----------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 103 +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) -------- +--R 2a +--R Type: Expression Integer +--E + )clear all ---S 19 of 36 +--S 104 aa:=integrate(sin(a*x)/(sin(a*x)-cos(a*x)),x) --R --R @@ -391,6 +1338,34 @@ aa:=integrate(sin(a*x)/(sin(a*x)-cos(a*x)),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 105 +bb:=x/2+1/(2*a)*log(sin(a*x)-cos(a*x)) +--R +--R log(sin(a x) - cos(a x)) + a x +--R (2) ------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 106 +cc:=aa-bb +--R +--R (3) +--R 2sin(a x) - 2cos(a x) 2 +--R - log(sin(a x) - cos(a x)) + log(---------------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R --------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 107 14:413 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.414~~~~~$\displaystyle @@ -401,7 +1376,7 @@ $$ <<*>>= )clear all ---S 20 of 36 +--S 108 aa:=integrate(cos(a*x)/(sin(a*x)+cos(a*x)),x) --R --R @@ -413,9 +1388,48 @@ aa:=integrate(cos(a*x)/(sin(a*x)+cos(a*x)),x) --R Type: Union(Expression Integer,...) --E +--S 109 +bb:=x/2+1/(2*a)*log(sin(a*x)+cos(a*x)) +--R +--R log(sin(a x) + cos(a x)) + a x +--R (2) ------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 110 +cc:=aa-bb +--R +--R (3) +--R 2 - 2sin(a x) - 2cos(a x) +--R - log(sin(a x) + cos(a x)) - log(------------) + log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R ----------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 111 +dd:=expandLog cc +--R +--R - log(sin(a x) + cos(a x)) + log(- sin(a x) - cos(a x)) +--R (4) ------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 112 +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R 2a +--R Type: Expression Integer +--E + )clear all ---S 21 of 36 +--S 113 aa:=integrate(cos(a*x)/(sin(a*x)-cos(a*x)),x) --R --R @@ -426,6 +1440,34 @@ aa:=integrate(cos(a*x)/(sin(a*x)-cos(a*x)),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 114 +bb:=-x/2+1/(2*a)*log(sin(a*x)-cos(a*x)) +--R +--R log(sin(a x) - cos(a x)) - a x +--R (2) ------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 115 +cc:=aa-bb +--R +--R (3) +--R 2sin(a x) - 2cos(a x) 2 +--R - log(sin(a x) - cos(a x)) + log(---------------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R --------------------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 116 14:414 Schaums and Axiom agree +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.415~~~~~$\displaystyle @@ -436,7 +1478,7 @@ $$ <<*>>= )clear all ---S 22 of 36 +--S 117 aa:=integrate(sin(a*x)/(p+q*cos(a*x)),x) --R --R @@ -447,6 +1489,44 @@ aa:=integrate(sin(a*x)/(p+q*cos(a*x)),x) --R a q --R Type: Union(Expression Integer,...) --E + +--S 118 +bb:=-1/(a*q)*log(p+q*cos(a*x)) +--R +--R log(q cos(a x) + p) +--R (2) - ------------------- +--R a q +--R Type: Expression Integer +--E + +--S 119 +cc:=aa-bb +--R +--R 2 - 2q cos(a x) - 2p +--R log(q cos(a x) + p) + log(------------) - log(------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) ----------------------------------------------------------------- +--R a q +--R Type: Expression Integer +--E + +--S 120 +dd:=expandLog cc +--R +--R log(q cos(a x) + p) - log(- q cos(a x) - p) +--R (4) ------------------------------------------- +--R a q +--R Type: Expression Integer +--E + +--S 121 14:415 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) -------- +--R a q +--R Type: Expression Integer +--E @ \section{\cite{1}:14.416~~~~~$\displaystyle @@ -457,7 +1537,7 @@ $$ <<*>>= )clear all ---S 23 of 36 +--S 122 aa:=integrate(cos(a*x)/(p+q*sin(a*x)),x) --R --R @@ -468,6 +1548,33 @@ aa:=integrate(cos(a*x)/(p+q*sin(a*x)),x) --R a q --R Type: Union(Expression Integer,...) --E + +--S 123 +bb:=1/(a*q)*log(p+q*sin(a*x)) +--R +--R log(q sin(a x) + p) +--R (2) ------------------- +--R a q +--R Type: Expression Integer +--E + +--S 124 +cc:=aa-bb +--R +--R 2q sin(a x) + 2p 2 +--R - log(q sin(a x) + p) + log(----------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) ----------------------------------------------------------------- +--R a q +--R Type: Expression Integer +--E + +--S 125 14:416 Schaums and Axiom differ by a constant +dd:=expandLog cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.417~~~~~$\displaystyle @@ -478,7 +1585,7 @@ $$ <<*>>= )clear all ---S 24 of 36 +--S 126 aa:=integrate(sin(a*x)/(p+q*cos(a*x))^n,x) --R --R @@ -488,6 +1595,53 @@ aa:=integrate(sin(a*x)/(p+q*cos(a*x))^n,x) --R (a n - a)q %e --R Type: Union(Expression Integer,...) --E + +--S 127 +bb:=1/(a*q*(n-1)*(p+q*cos(a*x))^(n-1)) +--R +--R 1 +--R (2) -------------------------------- +--R n - 1 +--R (a n - a)q (q cos(a x) + p) +--R Type: Expression Integer +--E + +--S 128 +cc:=aa-bb +--R +--R n log(q cos(a x) + p) n - 1 +--R - %e + (q cos(a x) + p)(q cos(a x) + p) +--R (3) ----------------------------------------------------------------- +--R n - 1 n log(q cos(a x) + p) +--R (a n - a)q (q cos(a x) + p) %e +--R Type: Expression Integer +--E + +--S 129 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 130 +dd:=explog cc +--R +--R n n - 1 +--R - (q cos(a x) + p) + (q cos(a x) + p)(q cos(a x) + p) +--R (5) ----------------------------------------------------------- +--R n - 1 n +--R (a n - a)q (q cos(a x) + p) (q cos(a x) + p) +--R Type: Expression Integer +--E + +--S 131 14:417 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.418~~~~~$\displaystyle @@ -498,7 +1652,7 @@ $$ <<*>>= )clear all ---S 25 of 36 +--S 132 aa:=integrate(cos(a*x)/(p+q*sin(a*x))^n,x) --R --R @@ -508,6 +1662,53 @@ aa:=integrate(cos(a*x)/(p+q*sin(a*x))^n,x) --R (a n - a)q %e --R Type: Union(Expression Integer,...) --E + +--S 133 +bb:=-1/(a*q*(n-1)*(p+q*sin(a*x))^(n-1)) +--R +--R 1 +--R (2) - -------------------------------- +--R n - 1 +--R (a n - a)q (q sin(a x) + p) +--R Type: Expression Integer +--E + +--S 134 +cc:=aa-bb +--R +--R n log(q sin(a x) + p) n - 1 +--R %e + (- q sin(a x) - p)(q sin(a x) + p) +--R (3) ----------------------------------------------------------------- +--R n - 1 n log(q sin(a x) + p) +--R (a n - a)q (q sin(a x) + p) %e +--R Type: Expression Integer +--E + +--S 135 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 136 +dd:=explog cc +--R +--R n n - 1 +--R (q sin(a x) + p) + (- q sin(a x) - p)(q sin(a x) + p) +--R (5) ----------------------------------------------------------- +--R n - 1 n +--R (a n - a)q (q sin(a x) + p) (q sin(a x) + p) +--R Type: Expression Integer +--E + +--S 137 14:418 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.419~~~~~$\displaystyle @@ -518,7 +1719,7 @@ $$ <<*>>= )clear all ---S 26 of 36 +--S 138 aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)),x) --R --R @@ -538,6 +1739,47 @@ aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)),x) --R a\|q + p --R Type: Union(Expression Integer,...) --E + +--S 139 +bb:=1/(a*sqrt(p^2+q^2))*log(tan((a*x+atan(q/p))/2)) +--R +--R q +--R atan(-) + a x +--R p +--R log(tan(-------------)) +--R 2 +--R (2) ----------------------- +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 140 14:419 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R q +--R atan(-) + a x +--R p +--R - log(tan(-------------)) +--R 2 +--R + +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) - p cos(a x) - q - p )\|q + p +--R + +--R 3 2 2 3 2 3 +--R (- q - p q)sin(a x) + (p q + p )cos(a x) + p q + p +--R / +--R p sin(a x) + q cos(a x) +--R / +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.420~~~~~$\displaystyle @@ -559,7 +1801,7 @@ $$ <<*>>= )clear all ---S 27 of 36 +--S 141 aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+r),x) --R --R @@ -600,6 +1842,174 @@ aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+r),x) --R a\|r - q - p --R Type: Union(List Expression Integer,...) --E + +--S 142 +bb1:=2/(a*sqrt(r^2-p^2-q^2))*atan((p+(r-q)*tan((a*x)/2))/sqrt(r^2-p^2-q^2)) +--R +--R a x +--R (r - q)tan(---) + p +--R 2 +--R 2atan(-------------------) +--R +------------+ +--R | 2 2 2 +--R \|r - q - p +--R (2) -------------------------- +--R +------------+ +--R | 2 2 2 +--R a\|r - q - p +--R Type: Expression Integer +--E + +--S 143 +bb2:=1/(a*sqrt(p^2+q^2-r^2))*log((p-sqrt(p^2+q^2-r^2)+(r-q)*tan((a*x)/2))/(p+sqrt(p^2+q^2-r^2)+(r-q)*tan((a*x)/2))) +--R +--R +--------------+ +--R | 2 2 2 a x +--R - \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R log(-----------------------------------------) +--R +--------------+ +--R | 2 2 2 a x +--R \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R (3) ---------------------------------------------- +--R +--------------+ +--R | 2 2 2 +--R a\|- r + q + p +--R Type: Expression Integer +--E + +--S 144 +cc1:=aa.1-bb1 +--R +--R (4) +--R +------------+ +--R | 2 2 2 +--R \|r - q - p +--R * +--R log +--R 2 2 2 +--R (p r - p q)sin(a x) + (- r + q r + p )cos(a x) - q r + q +--R + +--R 2 +--R p +--R * +--R +--------------+ +--R | 2 2 2 +--R \|- r + q + p +--R + +--R 3 2 2 2 3 2 +--R (r - q r + (- q - p )r + q + p q)sin(a x) +--R + +--R 2 2 3 2 2 3 +--R (p r - p q - p )cos(a x) + p r - p q - p +--R / +--R p sin(a x) + q cos(a x) + r +--R + +--R a x +--R +--------------+ (r - q)tan(---) + p +--R | 2 2 2 2 +--R - 2\|- r + q + p atan(-------------------) +--R +------------+ +--R | 2 2 2 +--R \|r - q - p +--R / +--R +--------------+ +------------+ +--R | 2 2 2 | 2 2 2 +--R a\|- r + q + p \|r - q - p +--R Type: Expression Integer +--E + +--S 145 +cc2:=aa.2-bb1 +--R +--R (5) +--R +------------+ +--R | 2 2 2 +--R ((r - q)sin(a x) + p cos(a x) + p)\|r - q - p +--R 2atan(-------------------------------------------------) +--R 2 2 2 2 2 2 +--R (r - q - p )cos(a x) + r - q - p +--R + +--R a x +--R (r - q)tan(---) + p +--R 2 +--R - 2atan(-------------------) +--R +------------+ +--R | 2 2 2 +--R \|r - q - p +--R / +--R +------------+ +--R | 2 2 2 +--R a\|r - q - p +--R Type: Expression Integer +--E + +--S 146 +cc3:=aa.1-bb2 +--R +--R (6) +--R log +--R 2 2 2 +--R (p r - p q)sin(a x) + (- r + q r + p )cos(a x) - q r + q +--R + +--R 2 +--R p +--R * +--R +--------------+ +--R | 2 2 2 +--R \|- r + q + p +--R + +--R 3 2 2 2 3 2 +--R (r - q r + (- q - p )r + q + p q)sin(a x) +--R + +--R 2 2 3 2 2 3 +--R (p r - p q - p )cos(a x) + p r - p q - p +--R / +--R p sin(a x) + q cos(a x) + r +--R + +--R +--------------+ +--R | 2 2 2 a x +--R - \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R - log(-----------------------------------------) +--R +--------------+ +--R | 2 2 2 a x +--R \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R / +--R +--------------+ +--R | 2 2 2 +--R a\|- r + q + p +--R Type: Expression Integer +--E + +--S 147 14:420 Axiom cannot simplify these expressions +cc4:=aa.2-bb2 +--R +--R (7) +--R +--------------+ +--R | 2 2 2 a x +--R +------------+ - \|- r + q + p + (r - q)tan(---) + p +--R | 2 2 2 2 +--R - \|r - q - p log(-----------------------------------------) +--R +--------------+ +--R | 2 2 2 a x +--R \|- r + q + p + (r - q)tan(---) + p +--R 2 +--R + +--R +------------+ +--R +--------------+ | 2 2 2 +--R | 2 2 2 ((r - q)sin(a x) + p cos(a x) + p)\|r - q - p +--R 2\|- r + q + p atan(-------------------------------------------------) +--R 2 2 2 2 2 2 +--R (r - q - p )cos(a x) + r - q - p +--R / +--R +--------------+ +------------+ +--R | 2 2 2 | 2 2 2 +--R a\|- r + q + p \|r - q - p +--R Type: Expression Integer +--E @ \section{\cite{1}:14.421~~~~~$\displaystyle @@ -610,7 +2020,7 @@ $$ <<*>>= )clear all ---S 28 of 36 +--S 148 aa:=integrate(1/(p*sin(a*x)+q*(1+cos(a*x))),x) --R --R @@ -620,7 +2030,76 @@ aa:=integrate(1/(p*sin(a*x)+q*(1+cos(a*x))),x) --R (1) -------------------------------- --R a p --R Type: Union(Expression Integer,...) +--E + +--S 149 +bb:=1/(a*p)*log(q+p*tan((a*x)/2)) +--R +--R a x +--R log(p tan(---) + q) +--R 2 +--R (2) ------------------- +--R a p +--R Type: Expression Integer --E + +--S 150 +cc:=aa-bb +--R +--R a x p sin(a x) + q cos(a x) + q +--R - log(p tan(---) + q) + log(---------------------------) +--R 2 cos(a x) + 1 +--R (3) -------------------------------------------------------- +--R a p +--R Type: Expression Integer +--E + +--S 151 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 152 +dd:=tanrule cc +--R +--R a x a x +--R p sin(---) + q cos(---) +--R p sin(a x) + q cos(a x) + q 2 2 +--R log(---------------------------) - log(-----------------------) +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (5) --------------------------------------------------------------- +--R a p +--R Type: Expression Integer +--E + +--S 153 +ee:=expandLog dd +--R +--R (6) +--R a x a x +--R log(p sin(a x) + q cos(a x) + q) - log(p sin(---) + q cos(---)) +--R 2 2 +--R + +--R a x +--R - log(cos(a x) + 1) + log(cos(---)) +--R 2 +--R / +--R a p +--R Type: Expression Integer +--E + +--S 154 14:421 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.422~~~~~$\displaystyle @@ -632,7 +2111,7 @@ $$ <<*>>= )clear all ---S 29 of 36 +--S 155 aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+sqrt(p^2+q^2)),x) --R --R @@ -665,9 +2144,88 @@ aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)+sqrt(p^2+q^2)),x) --R Type: Union(Expression Integer,...) --E +--S 156 +bb:=-1/(a*sqrt(p^2+q^2))*tan(%pi/4-(a*x+atan(q/p))/2) +--R +--R q +--R 2atan(-) + 2a x - %pi +--R p +--R tan(---------------------) +--R 4 +--R (2) -------------------------- +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 157 +cc:=aa-bb +--R +--R (3) +--R 6 2 4 4 2 6 +--R (64q + 80p q + 24p q + p )sin(a x) +--R + +--R 5 3 3 5 5 3 3 5 +--R (- 32p q - 32p q - 6p q)cos(a x) - 32p q - 32p q - 6p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 +--R (- 64q - 112p q - 56p q - 7p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 3 4 5 2 7 +--R (32p q + 48p q + 18p q + p )cos(a x) + 32p q + 48p q + 18p q + p +--R * +--R q +--R 2atan(-) + 2a x - %pi +--R p +--R tan(---------------------) +--R 4 +--R + +--R 6 2 4 4 2 6 6 2 4 4 2 6 +--R ((64q + 96p q + 36p q + 2p )cos(a x) + 64q + 96p q + 36p q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 7 2 5 4 3 +--R (- 64q - 128p q - 76p q - 12p q)cos(a x) - 64q - 128p q - 76p q +--R + +--R 6 +--R - 12p q +--R / +--R 7 2 5 4 3 6 +--R (64a q + 112a p q + 56a p q + 7a p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 +--R (- 32a p q - 48a p q - 18a p q - a p )cos(a x) - 32a p q +--R + +--R 3 4 5 2 7 +--R - 48a p q - 18a p q - a p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 8 2 6 4 4 6 2 8 +--R (- 64a q - 144a p q - 104a p q - 25a p q - a p )sin(a x) +--R + +--R 7 3 5 5 3 7 7 3 5 +--R (32a p q + 64a p q + 38a p q + 6a p q)cos(a x) + 32a p q + 64a p q +--R + +--R 5 3 7 +--R 38a p q + 6a p q +--R Type: Expression Integer +--E + + )clear all ---S 30 of 36 +--S 158 aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)-sqrt(p^2+q^2)),x) --R --R @@ -699,6 +2257,84 @@ aa:=integrate(1/(p*sin(a*x)+q*cos(a*x)-sqrt(p^2+q^2)),x) --R - 18a p q - a p --R Type: Union(Expression Integer,...) --E + +--S 159 +bb:=-1/(a*sqrt(p^2+q^2))*tan(%pi/4+(a*x+atan(q/p))/2) +--R +--R q +--R 2atan(-) + 2a x + %pi +--R p +--R tan(---------------------) +--R 4 +--R (2) - -------------------------- +--R +-------+ +--R | 2 2 +--R a\|q + p +--R Type: Expression Integer +--E + +--S 160 14:422 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R 6 2 4 4 2 6 +--R (64q + 80p q + 24p q + p )sin(a x) +--R + +--R 5 3 3 5 5 3 3 5 +--R (- 32p q - 32p q - 6p q)cos(a x) - 32p q - 32p q - 6p q +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 +--R (64q + 112p q + 56p q + 7p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 3 4 5 2 +--R (- 32p q - 48p q - 18p q - p )cos(a x) - 32p q - 48p q - 18p q +--R + +--R 7 +--R - p +--R * +--R q +--R 2atan(-) + 2a x + %pi +--R p +--R tan(---------------------) +--R 4 +--R + +--R 6 2 4 4 2 6 6 2 4 4 2 6 +--R ((64q + 96p q + 36p q + 2p )cos(a x) + 64q + 96p q + 36p q + 2p ) +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 7 2 5 4 3 6 7 2 5 4 3 6 +--R (64q + 128p q + 76p q + 12p q)cos(a x) + 64q + 128p q + 76p q + 12p q +--R / +--R 7 2 5 4 3 6 +--R (64a q + 112a p q + 56a p q + 7a p q)sin(a x) +--R + +--R 6 3 4 5 2 7 6 +--R (- 32a p q - 48a p q - 18a p q - a p )cos(a x) - 32a p q +--R + +--R 3 4 5 2 7 +--R - 48a p q - 18a p q - a p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 8 2 6 4 4 6 2 8 +--R (64a q + 144a p q + 104a p q + 25a p q + a p )sin(a x) +--R + +--R 7 3 5 5 3 7 7 3 5 +--R (- 32a p q - 64a p q - 38a p q - 6a p q)cos(a x) - 32a p q - 64a p q +--R + +--R 5 3 7 +--R - 38a p q - 6a p q +--R Type: Expression Integer +--E @ \section{\cite{1}:14.423~~~~~$\displaystyle @@ -709,7 +2345,7 @@ $$ <<*>>= )clear all ---S 31 of 36 +--S 161 aa:=integrate(1/(p^2*sin(a*x)^2+q^2*cos(a*x)^2),x) --R --R @@ -722,6 +2358,37 @@ aa:=integrate(1/(p^2*sin(a*x)^2+q^2*cos(a*x)^2),x) --R a p q --R Type: Union(Expression Integer,...) --E + +--S 162 +bb:=1/(a*p*q)*atan((p*tan(a*x))/q) +--R +--R p tan(a x) +--R atan(----------) +--R q +--R (2) ---------------- +--R a p q +--R Type: Expression Integer +--E + +--S 163 14:423 Axiom cannot simplify this expressions +cc:=aa-bb +--R +--R (3) +--R 2 2 2 +--R p tan(a x) ((q - 2p )cos(a x) - 2p )sin(a x) +--R - atan(----------) - atan(-----------------------------------) +--R q 2 +--R p q cos(a x) + 2p q cos(a x) + p q +--R + +--R q sin(a x) +--R atan(----------------) +--R 2p cos(a x) + 2p +--R / +--R a p q +--R Type: Expression Integer +--E + + @ \section{\cite{1}:14.424~~~~~$\displaystyle @@ -732,7 +2399,7 @@ $$ <<*>>= )clear all ---S 32 of 36 +--S 164 aa:=integrate(1/(p^2*sin(a*x)^2-q^2*cos(a*x)^2),x) --R --R @@ -742,7 +2409,77 @@ aa:=integrate(1/(p^2*sin(a*x)^2-q^2*cos(a*x)^2),x) --R (1) ----------------------------------------------------------------- --R 2a p q --R Type: Union(Expression Integer,...) +--E + +--S 165 +bb:=1/(2*a*p*q)*log((p*tan(a*x)-q)/(p*tan(a*x)+q)) +--R +--R p tan(a x) - q +--R log(--------------) +--R p tan(a x) + q +--R (2) ------------------- +--R 2a p q +--R Type: Expression Integer --E + +--S 166 +cc:=aa-bb +--R +--R (3) +--R 2p sin(a x) - 2q cos(a x) p tan(a x) - q +--R log(-------------------------) - log(--------------) +--R cos(a x) + 1 p tan(a x) + q +--R + +--R - 2p sin(a x) - 2q cos(a x) +--R - log(---------------------------) +--R cos(a x) + 1 +--R / +--R 2a p q +--R Type: Expression Integer +--E + +--S 167 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 168 +dd:=tanrule cc +--R +--R (5) +--R 2p sin(a x) - 2q cos(a x) p sin(a x) - q cos(a x) +--R log(-------------------------) - log(-----------------------) +--R cos(a x) + 1 p sin(a x) + q cos(a x) +--R + +--R - 2p sin(a x) - 2q cos(a x) +--R - log(---------------------------) +--R cos(a x) + 1 +--R / +--R 2a p q +--R Type: Expression Integer +--E + +--S 169 +ee:=expandLog dd +--R +--R log(p sin(a x) + q cos(a x)) - log(- p sin(a x) - q cos(a x)) +--R (6) ------------------------------------------------------------- +--R 2a p q +--R Type: Expression Integer +--E + +--S 170 14:424 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R log(- 1) +--R (7) -------- +--R 2a p q +--R Type: Expression Integer +--E @ \section{\cite{1}:14.425~~~~~$\displaystyle @@ -763,7 +2500,7 @@ $$ <<*>>= )clear all ---S 33 of 36 +--S 171 14:425 Axiom cannot compute this integral aa:=integrate(sin(a*x)^m*cos(a*x)^n,x) --R --R @@ -797,7 +2534,7 @@ $$ <<*>>= )clear all ---S 34 of 36 +--S 172 14:426 Axiom cannot compute this integral aa:=integrate(sin(a*x)^m/cos(a*x)^n,x) --R --R @@ -832,7 +2569,7 @@ $$ <<*>>= )clear all ---S 35 of 36 +--S 173 14:427 Axiom cannot compute this integral aa:=integrate(cos(a*x)^m/sin(a*x)^n,x) --R --R @@ -863,7 +2600,7 @@ $$ <<*>>= )clear all ---S 36 of 36 +--S 174 14:428 Axiom cannot compute this integral aa:=integrate(1/(sin(a*x)^m*cos(a*x)^n),x) --R --R diff --git a/src/input/schaum20.input.pamphlet b/src/input/schaum20.input.pamphlet index a25209e..e6981a6 100644 --- a/src/input/schaum20.input.pamphlet +++ b/src/input/schaum20.input.pamphlet @@ -19,7 +19,7 @@ $$ )set message auto off )clear all ---S 1 of 11 +--S 1 aa:=integrate(tan(a*x),x) --R --R @@ -29,6 +29,81 @@ aa:=integrate(tan(a*x),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb1:=-1/a*log(cos(a*x)) +--R +--R log(cos(a x)) +--R (2) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +bb2:=1/a*log(sec(a*x)) +--R +--R log(sec(a x)) +--R (3) ------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +cc1:=aa-bb1 +--R +--R 2 +--R log(tan(a x) + 1) + 2log(cos(a x)) +--R (4) ----------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 5 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (5) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 6 +dd1:=tanrule cc1 +--R +--R 2 2 +--R sin(a x) + cos(a x) +--R log(---------------------) + 2log(cos(a x)) +--R 2 +--R cos(a x) +--R (6) ------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 7 +ee1:=expandLog dd1 +--R +--R 2 2 +--R log(sin(a x) + cos(a x) ) +--R (7) -------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 8 +sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1) +--R +--R 2 2 +--I (8) sin(a) + cos(a) + %K == %K + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 9 14:429 Schaums and Axiom agree +ff1:=sincossqrrule ee1 +--R +--R (9) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.430~~~~~$\displaystyle @@ -39,7 +114,7 @@ $$ <<*>>= )clear all ---S 2 of 11 +--S 10 aa:=integrate(tan(a*x)^2,x) --R --R @@ -48,6 +123,22 @@ aa:=integrate(tan(a*x)^2,x) --R a --R Type: Union(Expression Integer,...) --E + +--S 11 +bb:=tan(a*x)/a-x +--R +--R tan(a x) - a x +--R (2) -------------- +--R a +--R Type: Expression Integer +--E + +--S 12 14:430 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.431~~~~~$\displaystyle @@ -58,7 +149,7 @@ $$ <<*>>= )clear all ---S 3 of 11 +--S 13 aa:=integrate(tan(a*x)^3,x) --R --R @@ -68,6 +159,73 @@ aa:=integrate(tan(a*x)^3,x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 14 +bb:=tan(a*x)^2/(2*a)+1/a*log(cos(a*x)) +--R +--R 2 +--R 2log(cos(a x)) + tan(a x) +--R (2) -------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R 2 +--R - log(tan(a x) + 1) - 2log(cos(a x)) +--R (3) ------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 16 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 17 +dd:=tanrule cc +--R +--R 2 2 +--R sin(a x) + cos(a x) +--R - log(---------------------) - 2log(cos(a x)) +--R 2 +--R cos(a x) +--R (5) --------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 18 +ee:=expandLog dd +--R +--R 2 2 +--R log(sin(a x) + cos(a x) ) +--R (6) - -------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 19 +sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1) +--R +--R 2 2 +--I (7) sin(a) + cos(a) + %L == %L + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 14:431 Schaums and Axiom agree +ff:=sincossqrrule ee +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.432~~~~~$\displaystyle @@ -78,7 +236,7 @@ $$ <<*>>= )clear all ---S 4 of 11 +--S 21 aa:=integrate(tan(a*x)^n*sec(a*x)^2,x) --R --R @@ -90,6 +248,74 @@ aa:=integrate(tan(a*x)^n*sec(a*x)^2,x) --R (a n + a)cos(a x) --R Type: Union(Expression Integer,...) --E + +--S 22 +bb:=tan(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R tan(a x) +--R (2) ------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 23 +cc:=aa-bb +--R +--R sin(a x) +--R n log(--------) +--R cos(a x) n + 1 +--R sin(a x)%e - cos(a x)tan(a x) +--R (3) ------------------------------------------------- +--R (a n + a)cos(a x) +--R Type: Expression Integer +--E + +--S 24 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +dd:=explog cc +--R +--R n + 1 sin(a x) n +--R - cos(a x)tan(a x) + sin(a x)(--------) +--R cos(a x) +--R (5) --------------------------------------------- +--R (a n + a)cos(a x) +--R Type: Expression Integer +--E + +--S 26 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (6) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 +ee:=tanrule dd +--R +--R sin(a x) n + 1 sin(a x) n +--R - cos(a x)(--------) + sin(a x)(--------) +--R cos(a x) cos(a x) +--R (7) ----------------------------------------------- +--R (a n + a)cos(a x) +--R Type: Expression Integer +--E + +--S 28 14:432 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.433~~~~~$\displaystyle @@ -100,7 +326,7 @@ $$ <<*>>= )clear all ---S 5 of 11 +--S 29 aa:=integrate(sec(a*x)^2/tan(a*x),x) --R --R @@ -111,6 +337,44 @@ aa:=integrate(sec(a*x)^2/tan(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 30 +bb:=1/a*log(tan(a*x)) +--R +--R log(tan(a x)) +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R sin(a x) 2cos(a x) +--R - log(tan(a x)) + log(------------) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) --------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 32 +dd:=expandLog cc +--R +--R - log(tan(a x)) + log(sin(a x)) - log(cos(a x)) - log(- 2) +--R (4) ---------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 33 14:433 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 2) +--R (5) - -------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.434~~~~~$\displaystyle @@ -121,7 +385,7 @@ $$ <<*>>= )clear all ---S 6 of 11 +--S 34 aa:=integrate(1/tan(a*x),x) --R --R @@ -131,6 +395,32 @@ aa:=integrate(1/tan(a*x),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 35 +bb:=1/a*log(sin(a*x)) +--R +--R log(sin(a x)) +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 36 +cc:=aa-bb +--R +--R 2 +--R - log(tan(a x) + 1) + 2log(tan(a x)) - 2log(sin(a x)) +--R (3) ------------------------------------------------------ +--R 2a +--R Type: Expression Integer +--E + +--S 37 +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.435~~~~~$\displaystyle @@ -142,7 +432,7 @@ $$ <<*>>= )clear all ---S 7 of 11 +--S 38 14:435 Axiom cannot compute this integral aa:=integrate(x*tan(a*x),x) --R --R @@ -163,7 +453,7 @@ $$ <<*>>= )clear all ---S 8 of 11 +--S 39 14:436 Axiom cannot compute this integral aa:=integrate(tan(a*x)/x,x) --R --R @@ -183,7 +473,7 @@ $$ <<*>>= )clear all ---S 9 of 11 +--S 40 aa:=integrate(x*tan(a*x)^2,x) --R --R @@ -194,6 +484,77 @@ aa:=integrate(x*tan(a*x)^2,x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 41 +bb:=(x*tan(a*x))/a+1/a^2*log(cos(a*x))-x^2/2 +--R +--R 2 2 +--R 2log(cos(a x)) + 2a x tan(a x) - a x +--R (2) ------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 42 +cc:=aa-bb +--R +--R 2 +--R - log(tan(a x) + 1) - 2log(cos(a x)) +--R (3) ------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 43 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 44 +dd:=tanrule cc +--R +--R 2 2 +--R sin(a x) + cos(a x) +--R - log(---------------------) - 2log(cos(a x)) +--R 2 +--R cos(a x) +--R (5) --------------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 45 +ee:=expandLog dd +--R +--R 2 2 +--R log(sin(a x) + cos(a x) ) +--R (6) - -------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 46 +sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1) +--R +--R 2 2 +--I (7) sin(a) + cos(a) + %R == %R + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 47 14:437 Schaums and Axiom agree +ff:=sincossqrrule ee +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.438~~~~~$\displaystyle @@ -204,7 +565,7 @@ $$ <<*>>= )clear all ---S 10 of 11 +--S 48 aa:=integrate(1/(p+q*tan(a*x)),x) --R --R @@ -215,6 +576,84 @@ aa:=integrate(1/(p+q*tan(a*x)),x) --R 2a q + 2a p --R Type: Union(Expression Integer,...) --E + +--S 49 +bb:=(p*x)/(p^2+q^2)+q/(a*(p^2+q^2))*log(q*sin(a*x)+p*cos(a*x)) +--R +--R q log(q sin(a x) + p cos(a x)) + a p x +--R (2) -------------------------------------- +--R 2 2 +--R a q + a p +--R Type: Expression Integer +--E + +--S 50 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - q log(tan(a x) + 1) + 2q log(q tan(a x) + p) +--R + +--R - 2q log(q sin(a x) + p cos(a x)) +--R / +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + +--S 51 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 52 +dd:=tanrule cc +--R +--R (5) +--R 2 2 +--R sin(a x) + cos(a x) +--R - q log(---------------------) - 2q log(q sin(a x) + p cos(a x)) +--R 2 +--R cos(a x) +--R + +--R q sin(a x) + p cos(a x) +--R 2q log(-----------------------) +--R cos(a x) +--R / +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + +--S 53 +ee:=expandLog dd +--R +--R 2 2 +--R q log(sin(a x) + cos(a x) ) +--R (6) - ---------------------------- +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + +--S 54 +sincossqrrule:=rule(sin(a)^2+cos(a)^2 == 1) +--R +--R 2 2 +--I (7) sin(a) + cos(a) + %S == %S + 1 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 55 14:438 Schaums and Axiom agree +ff:=sincossqrrule ee +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.439~~~~~$\displaystyle @@ -225,7 +664,7 @@ $$ <<*>>= )clear all ---S 11 of 11 +--S 56 14:439 Axiom cannot compute this integral aa:=integrate(tan(a*x)^n,x) --R --R diff --git a/src/input/schaum21.input.pamphlet b/src/input/schaum21.input.pamphlet index 6e02ffe..4d6a8be 100644 --- a/src/input/schaum21.input.pamphlet +++ b/src/input/schaum21.input.pamphlet @@ -18,7 +18,7 @@ $$ )set message auto off )clear all ---S 1 of 11 +--S 1 aa:=integrate(cot(a*x),x) --R --R @@ -29,6 +29,42 @@ aa:=integrate(cot(a*x),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=1/a*log(sin(a*x)) +--R +--R log(sin(a x)) +--R (2) ------------- +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R sin(2a x) 2 +--R 2log(-------------) - 2log(sin(a x)) - log(-------------) +--R cos(2a x) + 1 cos(2a x) + 1 +--R (3) --------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 4 +dd:=expandLog cc +--R +--R 2log(sin(2a x)) - 2log(sin(a x)) - log(cos(2a x) + 1) - log(2) +--R (4) -------------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 5 14:440 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.441~~~~~$\displaystyle @@ -39,7 +75,7 @@ $$ <<*>>= )clear all ---S 2 of 11 +--S 6 aa:=integrate(cot(a*x)^2,x) --R --R @@ -48,6 +84,49 @@ aa:=integrate(cot(a*x)^2,x) --R a sin(2a x) --R Type: Union(Expression Integer,...) --E + +--S 7 +bb:=-cot(a*x)/a-x +--R +--R - cot(a x) - a x +--R (2) ---------------- +--R a +--R Type: Expression Integer +--E + +--S 8 +cc:=aa-bb +--R +--R cot(a x)sin(2a x) - cos(2a x) - 1 +--R (3) --------------------------------- +--R a sin(2a x) +--R Type: Expression Integer +--E + +--S 9 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 10 +dd:=cotrule cc +--R +--R cos(a x)sin(2a x) + (- cos(2a x) - 1)sin(a x) +--R (5) --------------------------------------------- +--R a sin(a x)sin(2a x) +--R Type: Expression Integer +--E + +--S 11 14:441 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.442~~~~~$\displaystyle @@ -58,7 +137,7 @@ $$ <<*>>= )clear all ---S 3 of 11 +--S 12 aa:=integrate(cot(a*x)^3,x) --R --R @@ -72,6 +151,97 @@ aa:=integrate(cot(a*x)^3,x) --R 2a cos(2a x) - 2a --R Type: Union(Expression Integer,...) --E + +--S 13 +bb:=-cot(a*x)^2/(2*a)-1/a*log(sin(a*x)) +--R +--R 2 +--R - 2log(sin(a x)) - cot(a x) +--R (2) ---------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 14 +cc:=aa-bb +--R +--R (3) +--R sin(2a x) +--R (- 2cos(2a x) + 2)log(-------------) + (2cos(2a x) - 2)log(sin(a x)) +--R cos(2a x) + 1 +--R + +--R 2 2 +--R (cos(2a x) - 1)log(-------------) + (cos(2a x) - 1)cot(a x) + cos(2a x) +--R cos(2a x) + 1 +--R + +--R 1 +--R / +--R 2a cos(2a x) - 2a +--R Type: Expression Integer +--E + +--S 15 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 16 +dd:=cotrule cc +--R +--R (5) +--R 2 sin(2a x) +--R (- 2cos(2a x) + 2)sin(a x) log(-------------) +--R cos(2a x) + 1 +--R + +--R 2 +--R (2cos(2a x) - 2)sin(a x) log(sin(a x)) +--R + +--R 2 2 2 +--R (cos(2a x) - 1)sin(a x) log(-------------) + (cos(2a x) + 1)sin(a x) +--R cos(2a x) + 1 +--R + +--R 2 2 +--R cos(a x) cos(2a x) - cos(a x) +--R / +--R 2 +--R (2a cos(2a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 17 +ee:=expandLog dd +--R +--R (6) +--R 2 +--R (- 2cos(2a x) + 2)sin(a x) log(sin(2a x)) +--R + +--R 2 +--R (2cos(2a x) - 2)sin(a x) log(sin(a x)) +--R + +--R 2 +--R (cos(2a x) - 1)sin(a x) log(cos(2a x) + 1) +--R + +--R 2 2 +--R ((log(2) + 1)cos(2a x) - log(2) + 1)sin(a x) + cos(a x) cos(2a x) +--R + +--R 2 +--R - cos(a x) +--R / +--R 2 +--R (2a cos(2a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 18 14:442 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (7) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.443~~~~~$\displaystyle @@ -82,7 +252,7 @@ $$ <<*>>= )clear all ---S 4 of 11 +--S 19 aa:=integrate(cot(a*x)^n*csc(a*x)^2,x) --R --R @@ -94,6 +264,74 @@ aa:=integrate(cot(a*x)^n*csc(a*x)^2,x) --R (a n + a)sin(a x) --R Type: Union(Expression Integer,...) --E + +--S 20 +bb:=-cot(a*x)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R cot(a x) +--R (2) - ------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 21 +cc:=aa-bb +--R +--R cos(a x) +--R n log(--------) +--R sin(a x) n + 1 +--R - cos(a x)%e + sin(a x)cot(a x) +--R (3) --------------------------------------------------- +--R (a n + a)sin(a x) +--R Type: Expression Integer +--E + +--S 22 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 23 +dd:=explog cc +--R +--R n + 1 cos(a x) n +--R sin(a x)cot(a x) - cos(a x)(--------) +--R sin(a x) +--R (5) ------------------------------------------- +--R (a n + a)sin(a x) +--R Type: Expression Integer +--E + +--S 24 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (6) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +ee:=cotrule dd +--R +--R cos(a x) n + 1 cos(a x) n +--R sin(a x)(--------) - cos(a x)(--------) +--R sin(a x) sin(a x) +--R (7) --------------------------------------------- +--R (a n + a)sin(a x) +--R Type: Expression Integer +--E + +--S 26 14:443 Schaums and Axiom agree +ff:=complexNormalize ee +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.444~~~~~$\displaystyle @@ -104,7 +342,7 @@ $$ <<*>>= )clear all ---S 5 of 11 +--S 27 aa:=integrate(csc(a*x)^2/cot(a*x),x) --R --R @@ -115,6 +353,55 @@ aa:=integrate(csc(a*x)^2/cot(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 28 +bb:=-1/a*log(cot(a*x)) +--R +--R log(cot(a x)) +--R (2) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 29 +cc:=aa-bb +--R +--R sin(a x) 2cos(a x) +--R log(------------) + log(cot(a x)) - log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (3) ------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 30 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 31 +dd:=cotrule cc +--R +--R sin(a x) cos(a x) 2cos(a x) +--R log(------------) + log(--------) - log(- ------------) +--R cos(a x) + 1 sin(a x) cos(a x) + 1 +--R (5) ------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 32 14:444 Schaums and Axiom differ by a constant +ee:=expandLog dd +--R +--R log(- 2) +--R (6) - -------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.445~~~~~$\displaystyle @@ -125,7 +412,7 @@ $$ <<*>>= )clear all ---S 6 of 11 +--S 33 aa:=integrate(1/cot(a*x),x) --R --R @@ -136,6 +423,42 @@ aa:=integrate(1/cot(a*x),x) --R 2a --R Type: Union(Expression Integer,...) --E + +--S 34 +bb:=-1/a*log(cos(a*x)) +--R +--R log(cos(a x)) +--R (2) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 35 +cc:=aa-bb +--R +--R 2 +--R 2log(cos(a x)) + log(-------------) +--R cos(2a x) + 1 +--R (3) ----------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 36 +dd:=expandLog cc +--R +--R - log(cos(2a x) + 1) + 2log(cos(a x)) + log(2) +--R (4) ---------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 37 14:445 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.446~~~~~$\displaystyle @@ -148,7 +471,7 @@ $$ <<*>>= )clear all ---S 7 of 11 +--S 38 14:446 Axiom cannot compute this integral aa:=integrate(x*cot(a*x),x) --R --R @@ -169,7 +492,7 @@ $$ <<*>>= )clear all ---S 8 of 11 +--S 39 14:447 Axiom cannot compute this integral aa:=integrate(cot(a*x)/x,x) --R --R @@ -189,7 +512,7 @@ $$ <<*>>= )clear all ---S 9 of 11 +--S 40 aa:=integrate(x*cot(a*x)^2,x) --R --R @@ -205,6 +528,58 @@ aa:=integrate(x*cot(a*x)^2,x) --R 2a sin(2a x) --R Type: Union(Expression Integer,...) --E + +--S 41 +bb:=-(x*cot(a*x))/a+1/a^2*log(sin(a*x))-x^2/2 +--R +--R 2 2 +--R 2log(sin(a x)) - 2a x cot(a x) - a x +--R (2) ------------------------------------- +--R 2 +--R 2a +--R Type: Expression Integer +--E + +--S 42 +cc:=aa-bb +--R +--R (3) +--R sin(2a x) +--R 2sin(2a x)log(-------------) - 2sin(2a x)log(sin(a x)) +--R cos(2a x) + 1 +--R + +--R 2 +--R - sin(2a x)log(-------------) + 2a x cot(a x)sin(2a x) - 2a x cos(2a x) +--R cos(2a x) + 1 +--R + +--R - 2a x +--R / +--R 2 +--R 2a sin(2a x) +--R Type: Expression Integer +--E + +--S 43 +dd:=expandLog cc +--R +--R (4) +--R 2sin(2a x)log(sin(2a x)) - 2sin(2a x)log(sin(a x)) +--R + +--R - sin(2a x)log(cos(2a x) + 1) + (2a x cot(a x) - log(2))sin(2a x) +--R + +--R - 2a x cos(2a x) - 2a x +--R / +--R 2 +--R 2a sin(2a x) +--R Type: Expression Integer +--E + +--S 44 14:448 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.449~~~~~$\displaystyle @@ -215,7 +590,7 @@ $$ <<*>>= )clear all ---S 10 of 11 +--S 45 aa:=integrate(1/(p+q*cot(a*x)),x) --R --R @@ -228,6 +603,34 @@ aa:=integrate(1/(p+q*cot(a*x)),x) --R 2a q + 2a p --R Type: Union(Expression Integer,...) --E + +--S 46 +bb:=(p*x)/(p^2+q^2)-q/(a*(p^2+q^2))*log(p*sin(a*x)+q*cos(a*x)) +--R +--R - q log(p sin(a x) + q cos(a x)) + a p x +--R (2) ---------------------------------------- +--R 2 2 +--R a q + a p +--R Type: Expression Integer +--E + +--S 47 14:449 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R (3) +--R p sin(2a x) + q cos(2a x) + q +--R - 2q log(-----------------------------) + 2q log(p sin(a x) + q cos(a x)) +--R cos(2a x) + 1 +--R + +--R 2 +--R q log(-------------) +--R cos(2a x) + 1 +--R / +--R 2 2 +--R 2a q + 2a p +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.450~~~~~$\displaystyle @@ -238,7 +641,7 @@ $$ <<*>>= )clear all ---S 11 of 11 +--S 48 14:450 Axiom cannot compute this integral aa:=integrate(cot(a*x)^n,x) --R --R diff --git a/src/input/schaum22.input.pamphlet b/src/input/schaum22.input.pamphlet index 66befb9..173338a 100644 --- a/src/input/schaum22.input.pamphlet +++ b/src/input/schaum22.input.pamphlet @@ -19,7 +19,7 @@ $$ )set message auto off )clear all ---S 1 of 10 +--S 1 aa:=integrate(sec(a*x),x) --R --R @@ -30,6 +30,172 @@ aa:=integrate(sec(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb1:=1/a*log(sec(a*x)+tan(a*x)) +--R +--R log(tan(a x) + sec(a x)) +--R (2) ------------------------ +--R a +--R Type: Expression Integer +--E + +--S 3 +bb2:=1/a*log(tan((a*x)/2+%pi/4)) +--R +--R 2a x + %pi +--R log(tan(----------)) +--R 4 +--R (3) -------------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +cc1:=aa-bb1 +--R +--R (4) +--R sin(a x) + cos(a x) + 1 +--R - log(tan(a x) + sec(a x)) + log(-----------------------) +--R cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 5 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (5) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 6 +dd1:=tanrule cc1 +--R +--R (6) +--R sin(a x) + cos(a x)sec(a x) sin(a x) + cos(a x) + 1 +--R - log(---------------------------) + log(-----------------------) +--R cos(a x) cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 7 +secrule:=rule(sec(a) == 1/cos(a)) +--R +--R 1 +--R (7) sec(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 8 +ee1:=secrule dd1 +--R +--R (8) +--R sin(a x) + 1 sin(a x) + cos(a x) + 1 +--R - log(------------) + log(-----------------------) +--R cos(a x) cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 9 +ff1:=expandLog ee1 +--R +--R (9) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) + 1) +--R + +--R - log(sin(a x) - cos(a x) - 1) + log(cos(a x)) +--R / +--R a +--R Type: Expression Integer +--E + +--S 10 +gg1:=complexNormalize ff1 +--R +--R log(- 1) +--R (10) -------- +--R a +--R Type: Expression Integer +--E + +--S 11 +cc2:=aa-bb2 +--R +--R (11) +--R 2a x + %pi sin(a x) + cos(a x) + 1 +--R - log(tan(----------)) + log(-----------------------) +--R 4 cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R a +--R Type: Expression Integer +--E + +--S 12 +dd2:=tanrule cc2 +--R +--R (12) +--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1 +--R log(-----------------------) - log(-----------------------) +--R cos(a x) + 1 cos(a x) + 1 +--R + +--R 2a x + %pi +--R sin(----------) +--R 4 +--R - log(---------------) +--R 2a x + %pi +--R cos(----------) +--R 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 13 +ee2:=expandLog dd2 +--R +--R (13) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) - cos(a x) - 1) +--R + +--R 2a x + %pi 2a x + %pi +--R - log(sin(----------)) + log(cos(----------)) +--R 4 4 +--R / +--R a +--R Type: Expression Integer +--E + +--S 14 14:451 Schaums and Axiom differ by a constant +ff2:=complexNormalize ee2 +--R +--R log(- 1) +--R (14) -------- +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.452~~~~~$\displaystyle @@ -40,7 +206,7 @@ $$ <<*>>= )clear all ---S 2 of 10 +--S 15 aa:=integrate(sec(a*x)^2,x) --R --R @@ -49,6 +215,40 @@ aa:=integrate(sec(a*x)^2,x) --R a cos(a x) --R Type: Union(Expression Integer,...) --E + +--S 16 +bb:=tan(a*x)/a +--R +--R tan(a x) +--R (2) -------- +--R a +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R - cos(a x)tan(a x) + sin(a x) +--R (3) ----------------------------- +--R a cos(a x) +--R Type: Expression Integer +--E + +--S 18 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 19 14:452 Schaums and Axiom agree +dd:=tanrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.453~~~~~$\displaystyle @@ -59,7 +259,7 @@ $$ <<*>>= )clear all ---S 3 of 10 +--S 20 aa:=integrate(sec(a*x)^3,x) --R --R @@ -76,6 +276,113 @@ aa:=integrate(sec(a*x)^3,x) --R 2a cos(a x) --R Type: Union(Expression Integer,...) --E + +--S 21 +bb:=(sec(a*x)*tan(a*x))/(2*a)+1/(2*a)*log(sec(a*x)+tan(a*x)) +--R +--R log(tan(a x) + sec(a x)) + sec(a x)tan(a x) +--R (2) ------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - cos(a x) log(tan(a x) + sec(a x)) +--R + +--R 2 sin(a x) + cos(a x) + 1 +--R cos(a x) log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2 sin(a x) - cos(a x) - 1 2 +--R - cos(a x) log(-----------------------) - cos(a x) sec(a x)tan(a x) +--R cos(a x) + 1 +--R + +--R sin(a x) +--R / +--R 2 +--R 2a cos(a x) +--R Type: Expression Integer +--E + +--S 23 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 24 +dd:=tanrule cc +--R +--R (5) +--R 2 sin(a x) + cos(a x)sec(a x) +--R - cos(a x) log(---------------------------) +--R cos(a x) +--R + +--R 2 sin(a x) + cos(a x) + 1 +--R cos(a x) log(-----------------------) +--R cos(a x) + 1 +--R + +--R 2 sin(a x) - cos(a x) - 1 +--R - cos(a x) log(-----------------------) + (- cos(a x)sec(a x) + 1)sin(a x) +--R cos(a x) + 1 +--R / +--R 2 +--R 2a cos(a x) +--R Type: Expression Integer +--E + +--S 25 +secrule:=rule(sec(a) == 1/cos(a)) +--R +--R 1 +--R (6) sec(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 26 +ee:=secrule dd +--R +--R (7) +--R sin(a x) + 1 sin(a x) + cos(a x) + 1 +--R - log(------------) + log(-----------------------) +--R cos(a x) cos(a x) + 1 +--R + +--R sin(a x) - cos(a x) - 1 +--R - log(-----------------------) +--R cos(a x) + 1 +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 27 +ff:=expandLog ee +--R +--R (8) +--R log(sin(a x) + cos(a x) + 1) - log(sin(a x) + 1) +--R + +--R - log(sin(a x) - cos(a x) - 1) + log(cos(a x)) +--R / +--R 2a +--R Type: Expression Integer +--E + +--S 28 14:453 Schaums and Axiom differ by a constant +gg:=complexNormalize ff +--R +--R log(- 1) +--R (9) -------- +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.454~~~~~$\displaystyle @@ -86,7 +393,7 @@ $$ <<*>>= )clear all ---S 4 of 10 +--S 29 aa:=integrate(sec(a*x)^n*tan(a*x),x) --R --R 1 @@ -100,6 +407,31 @@ aa:=integrate(sec(a*x)^n*tan(a*x),x) --R a n --R Type: Union(Expression Integer,...) --E + +--S 30 +bb:=sec(a*x)^n/(n*a) +--R +--R n +--R sec(a x) +--R (2) --------- +--R a n +--R Type: Expression Integer +--E + +--S 31 14:454 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R 1 +--R n log(---------) +--R 2 +--R cos(a x) +--R ---------------- +--R 2 n +--R %e - sec(a x) +--R (3) ------------------------------ +--R a n +--R Type: Expression Integer +--E @ \section{\cite{1}:14.455~~~~~$\displaystyle @@ -110,7 +442,7 @@ $$ <<*>>= )clear all ---S 5 of 10 +--S 32 aa:=integrate(1/sec(a*x),x) --R --R @@ -118,7 +450,23 @@ aa:=integrate(1/sec(a*x),x) --R (1) -------- --R a --R Type: Union(Expression Integer,...) +--E + +--S 33 +bb:=sin(a*x)/a +--R +--R sin(a x) +--R (2) -------- +--R a +--R Type: Expression Integer --E + +--S 34 14:455 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.456~~~~~$\displaystyle @@ -130,7 +478,7 @@ $$ <<*>>= )clear all ---S 6 of 10 +--S 35 14:456 Axiom cannot compute this integral aa:=integrate(x*sec(a*x),x) --R --R @@ -151,7 +499,7 @@ $$ <<*>>= )clear all ---S 7 of 10 +--S 36 14:457 Axiom cannot compute this integral aa:=integrate(sec(a*x)/x,x) --R --R @@ -171,7 +519,7 @@ $$ <<*>>= )clear all ---S 8 of 10 +--S 37 aa:=integrate(x*sec(a*x)^2,x) --R --R @@ -184,6 +532,64 @@ aa:=integrate(x*sec(a*x)^2,x) --R a cos(a x) --R Type: Union(Expression Integer,...) --E + +--S 38 +bb:=x/a*tan(a*x)+1/a^2*log(cos(a*x)) +--R +--R log(cos(a x)) + a x tan(a x) +--R (2) ---------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 39 +cc:=aa-bb +--R +--R (3) +--R 2 +--R - cos(a x)log(cos(a x)) - cos(a x)log(------------) +--R cos(a x) + 1 +--R + +--R 2cos(a x) +--R cos(a x)log(- ------------) - a x cos(a x)tan(a x) + a x sin(a x) +--R cos(a x) + 1 +--R / +--R 2 +--R a cos(a x) +--R Type: Expression Integer +--E + +--S 40 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 41 +dd:=tanrule cc +--R +--R 2 2cos(a x) +--R - log(cos(a x)) - log(------------) + log(- ------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (5) --------------------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 42 14:458 Schaums and Axiom differ by a constant +ee:=expandLog dd +--R +--R - log(2) + log(- 2) +--R (6) ------------------- +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.459~~~~~$\displaystyle @@ -194,7 +600,7 @@ $$ <<*>>= )clear all ---S 9 of 10 +--S 43 aa:=integrate(1/(q+p*sec(a*x)),x) --R --R @@ -218,7 +624,136 @@ aa:=integrate(1/(q+p*sec(a*x)),x) --R | 2 2 --R a q\|- q + p --R Type: Union(List Expression Integer,...) +--E + +--S 44 +t1:=integrate(1/(p+q*cos(a*x)),x) +--R +--R (2) +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R log(--------------------------------------------------) +--R q cos(a x) + p +--R [-------------------------------------------------------, +--R +-------+ +--R | 2 2 +--R a\|q - p +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p +--R 2atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R ------------------------------] +--R +---------+ +--R | 2 2 +--R a\|- q + p +--R Type: Union(List Expression Integer,...) --E + +--S 45 +bb1:=x/q-p/q*t1.1 +--R +--R (3) +--R +-------+ +--R | 2 2 2 2 +-------+ +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) | 2 2 +--R - p log(--------------------------------------------------) + a x\|q - p +--R q cos(a x) + p +--R --------------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 46 +bb2:=x/q-p/q*t1.2 +--R +--R +---------+ +--R | 2 2 +---------+ +--R sin(a x)\|- q + p | 2 2 +--R - 2p atan(-----------------------) + a x\|- q + p +--R (q + p)cos(a x) + q + p +--R (4) ---------------------------------------------------- +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 47 +cc1:=aa.1-bb1 +--R +--R (5) +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R p log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +-------+ +--R | 2 2 2 2 +--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R p log(--------------------------------------------------) +--R q cos(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 48 +cc2:=aa.1-bb2 +--R +--R (6) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (q - p )sin(a x) +--R p\|- q + p log(------------------------------------------------) +--R q cos(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R 2p\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 49 +cc3:=aa.2-bb1 +--R +--R (7) +--R +-------+ +--R +---------+ | 2 2 2 2 +--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x) +--R p\|- q + p log(--------------------------------------------------) +--R q cos(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R - 2p\|q - p atan(-----------------------) +--R (q + p)cos(a x) + q + p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 50 14:459 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.460~~~~~$\displaystyle @@ -230,7 +765,7 @@ $$ <<*>>= )clear all ---S 10 of 10 +--S 51 14:460 Axiom cannot compute this integral aa:=integrate(sec(a*x)^n,x) --R --R