diff --git a/changelog b/changelog index 9139a97..aa128b6 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080502 tpd src/input/schaum17.input post-mortem fixes 20080501 tpd src/input/schaum16.input post-mortem fixes 20080501 tpd src/input/schaum13.input post-mortem fixes 20080430 tpd src/input/schaum13.input post-mortem fixes diff --git a/src/input/schaum17.input.pamphlet b/src/input/schaum17.input.pamphlet index a11b623..6092400 100644 --- a/src/input/schaum17.input.pamphlet +++ b/src/input/schaum17.input.pamphlet @@ -598,27 +598,34 @@ aa:=integrate(sin(p*x)*sin(q*x),x) --E --S 45 -bb:=(sin(p-q)*x)/(2*(p-q))-(sin(p+q)*x)/(2*(p+q)) +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 (- q + p)sin((q + p)x) + (q + p)sin((q - p)x) +--R (2) --------------------------------------------- +--R 2 2 +--R 2q - 2p --R Type: Expression Integer --E ---S 46 14:353 Axiom cannot simplify this expression +--S 46 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 (q - p)sin((q + p)x) + 2p cos(p x)sin(q x) + (- q - p)sin((q - p)x) --R + ---R (- q - p)x sin(q - p) +--R - 2q cos(q x)sin(p x) --R / --R 2 2 --R 2q - 2p --R Type: Expression Integer --E + +--S 47 14:353 Schams and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.354~~~~~$\displaystyle @@ -629,7 +636,7 @@ $$ <<*>>= )clear all ---S 47 +--S 48 aa:=integrate(1/(1-sin(a*x)),x) --R --R @@ -639,7 +646,7 @@ aa:=integrate(1/(1-sin(a*x)),x) --R Type: Union(Expression Integer,...) --E ---S 48 +--S 49 bb:=1/a*tan(%pi/4+(a*x)/2) --R --R 2a x + %pi @@ -650,7 +657,7 @@ bb:=1/a*tan(%pi/4+(a*x)/2) --R Type: Expression Integer --E ---S 49 +--S 50 cc:=aa-bb --R --R 2a x + %pi @@ -661,7 +668,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 50 14:354 Schaums and Axiom differ by a constant +--S 51 14:354 Schaums and Axiom differ by a constant dd:=complexNormalize cc --R --R 1 @@ -680,7 +687,7 @@ $$ <<*>>= )clear all ---S 51 +--S 52 aa:=integrate(x/(1-sin(ax)),x) --R --R @@ -691,7 +698,7 @@ aa:=integrate(x/(1-sin(ax)),x) --R Type: Union(Expression Integer,...) --E ---S 52 +--S 53 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 @@ -703,7 +710,7 @@ bb:=x/a*tan(%pi/4+(a*x)/2)+2/a^2*log(sin(%pi/4-(a*x)/2)) --R Type: Expression Integer --E ---S 53 14:355 Axiom cannot simplify this expression +--S 54 14:355 Axiom cannot simplify this expression cc:=aa-bb --R --R (3) @@ -729,7 +736,7 @@ $$ <<*>>= )clear all ---S 54 +--S 55 aa:=integrate(1/(1+sin(ax)),x) --R --R @@ -739,7 +746,7 @@ aa:=integrate(1/(1+sin(ax)),x) --R Type: Union(Expression Integer,...) --E ---S 55 +--S 56 bb:=-1/a*tan(%pi/4-(a*x)/2) --R --R 2a x - %pi @@ -750,7 +757,7 @@ bb:=-1/a*tan(%pi/4-(a*x)/2) --R Type: Expression Integer --E ---S 56 +--S 57 cc:=aa-bb --R --R 2a x - %pi @@ -761,7 +768,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 57 +--S 58 tanrule:=rule(tan(a/b) == sin(a)/cos(b)) --R --R a sin(a) @@ -770,7 +777,7 @@ tanrule:=rule(tan(a/b) == sin(a)/cos(b)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 58 14:356 Axiom cannot simplify this expression +--S 59 14:356 Axiom cannot simplify this expression dd:=tanrule cc --R --R (- sin(ax) - 1)sin(2a x - %pi) + a x cos(4) @@ -789,7 +796,7 @@ $$ <<*>>= )clear all ---S 59 +--S 60 aa:=integrate(x/(1+sin(a*x)),x) --R --R @@ -809,7 +816,7 @@ aa:=integrate(x/(1+sin(a*x)),x) --R Type: Union(Expression Integer,...) --E ---S 60 +--S 61 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 @@ -821,7 +828,7 @@ bb:=-x/a*tan(%pi/4-(a*x)/2)+2/a^2*log(sin(%pi/4+(a*x)/2)) --R Type: Expression Integer --E ---S 61 +--S 62 14:257 Axiom cannot simplify this expression cc:=aa-bb --R --R (3) @@ -848,60 +855,6 @@ cc:=aa-bb --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 @@ -913,9 +866,8 @@ $$ <<*>>= )clear all ---S 65 +--S 63 aa:=integrate(1/(1-sin(a*x))^2,x) ---R --R --R 2 --R (3cos(a x) + 3)sin(a x) + cos(a x) - 4cos(a x) - 5 @@ -925,18 +877,18 @@ aa:=integrate(1/(1-sin(a*x))^2,x) --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 +--S 64 +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 tan(----------) + 3tan(----------) --R 4 4 --R (2) ----------------------------------- --R 6a --R Type: Expression Integer --E ---S 67 +--S 65 cc:=aa-bb --R --R (3) @@ -944,9 +896,9 @@ cc:=aa-bb --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 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 @@ -956,41 +908,199 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 68 -tanrule:=rule(tan(a/b) == sin(a)/cos(b)) +--S 66 +tanrule:=rule(tan(a) == sin(a)/cos(a)) --R ---R a sin(a) ---R (4) tan(-) == ------ ---R b cos(b) +--R sin(a) +--R (4) tan(a) == ------ +--R cos(a) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 69 14:358 Axiom cannot simplify this expression +--S 67 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 2a x + %pi 3 +--R (- cos(a x) - 2)sin(----------) +--R 4 +--R + +--R 2a x + %pi 2 2a x + %pi 2 2a x + %pi +--R (- 3cos(----------) cos(a x) - 6cos(----------) )sin(----------) +--R 4 4 4 +--R + +--R 2a x + %pi 3 2a x + %pi 3 +--R 6cos(----------) cos(a x) + 6cos(----------) +--R 4 4 +--R * +--R sin(a x) +--R + +--R 2 2a x + %pi 3 +--R (- cos(a x) + cos(a x) + 2)sin(----------) +--R 4 --R + ---R 2 2 2 2 ---R (3cos(4) cos(a x) + 6cos(4) )sin(a x) + 3cos(4) cos(a x) +--R 2a x + %pi 2 2 2a x + %pi 2 +--R - 3cos(----------) cos(a x) + 3cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 2 +--R 6cos(----------) +--R 4 +--R * +--R 2a x + %pi +--R sin(----------) +--R 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 2a x + %pi 3 +--R 2cos(----------) cos(a x) - 8cos(----------) cos(a x) - 10cos(----------) +--R 4 4 4 +--R / +--R 2a x + %pi 3 2a x + %pi 3 +--R (6a cos(----------) cos(a x) + 12a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 +--R 6a cos(----------) cos(a x) - 6a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 +--R - 12a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 68 +sindiffrule2:=rule(sin((a-b)/4) == sin(a/4)*cos(b/4)-cos(a/4)*sin(b/4)) +--R +--R b - a a b b a +--I (6) - %K sin(-----) == - %K cos(-)sin(-) + %K cos(-)sin(-) +--R 4 4 4 4 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 69 +ee:=sindiffrule2 dd +--R +--R (7) +--R 2a x + %pi 3 +--R (- cos(a x) - 2)sin(----------) +--R 4 --R + ---R 2 2 ---R - 3cos(4) cos(a x) - 6cos(4) +--R 2a x + %pi 2 2a x + %pi 2 2a x + %pi +--R (- 3cos(----------) cos(a x) - 6cos(----------) )sin(----------) +--R 4 4 4 +--R + +--R 2a x + %pi 3 2a x + %pi 3 +--R 6cos(----------) cos(a x) + 6cos(----------) +--R 4 4 --R * ---R sin(2a x + %pi) +--R sin(a x) +--R + +--R 2 2a x + %pi 3 +--R (- cos(a x) + cos(a x) + 2)sin(----------) +--R 4 --R + ---R 3 3 3 2 ---R (6cos(4) cos(a x) + 6cos(4) )sin(a x) + 2cos(4) cos(a x) +--R 2a x + %pi 2 2 2a x + %pi 2 +--R - 3cos(----------) cos(a x) + 3cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 2 +--R 6cos(----------) +--R 4 +--R * +--R 2a x + %pi +--R sin(----------) +--R 4 --R + ---R 3 3 ---R - 8cos(4) cos(a x) - 10cos(4) +--R 2a x + %pi 3 2 2a x + %pi 3 2a x + %pi 3 +--R 2cos(----------) cos(a x) - 8cos(----------) cos(a x) - 10cos(----------) +--R 4 4 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 2a x + %pi 3 2a x + %pi 3 +--R (6a cos(----------) cos(a x) + 12a cos(----------) )sin(a x) +--R 4 4 --R + ---R 3 3 ---R - 6a cos(4) cos(a x) - 12a cos(4) +--R 2a x + %pi 3 2 2a x + %pi 3 +--R 6a cos(----------) cos(a x) - 6a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 +--R - 12a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 70 +sincuberule:=rule(sin(a)^3 == 3/4*sin(a)-1/4*sin(3*a)) +--R +--R 3 - sin(3a) + 3sin(a) +--R (8) sin(a) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 71 +ff:=sincuberule ee +--R +--R (9) +--R 2 6a x + 3%pi +--R ((cos(a x) + 2)sin(a x) + cos(a x) - cos(a x) - 2)sin(-----------) +--R 4 +--R + +--R 2a x + %pi 2 2a x + %pi 2 +--R ((- 12cos(----------) - 3)cos(a x) - 24cos(----------) - 6) +--R 4 4 +--R * +--R 2a x + %pi +--R sin(----------) +--R 4 +--R + +--R 2a x + %pi 3 2a x + %pi 3 +--R 24cos(----------) cos(a x) + 24cos(----------) +--R 4 4 +--R * +--R sin(a x) +--R + +--R 2a x + %pi 2 2 +--R (- 12cos(----------) - 3)cos(a x) +--R 4 +--R + +--R 2a x + %pi 2 2a x + %pi 2 +--R (12cos(----------) + 3)cos(a x) + 24cos(----------) + 6 +--R 4 4 +--R * +--R 2a x + %pi +--R sin(----------) +--R 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 +--R 8cos(----------) cos(a x) - 32cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 +--R - 40cos(----------) +--R 4 +--R / +--R 2a x + %pi 3 2a x + %pi 3 +--R (24a cos(----------) cos(a x) + 48a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 2 2a x + %pi 3 +--R 24a cos(----------) cos(a x) - 24a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x + %pi 3 +--R - 48a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 72 14:358 Schaums and Axiom differ by a constant +complexNormalize % +--R +--R 2 +--R (10) -- +--R 3a --R Type: Expression Integer --E @ @@ -1004,9 +1114,8 @@ $$ <<*>>= )clear all ---S 70 +--S 73 aa:=integrate(1/(1+sin(a*x))^2,x) ---R --R --R 2 --R (- 3cos(a x) - 3)sin(a x) + cos(a x) - 4cos(a x) - 5 @@ -1016,7 +1125,7 @@ aa:=integrate(1/(1+sin(a*x))^2,x) --R Type: Union(Expression Integer,...) --E ---S 71 +--S 74 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 @@ -1027,7 +1136,7 @@ bb:=-1/(2*a)*tan(%pi/4-(a*x)/2)-1/(6*a)*tan(%pi/4-(a*x)/2)^3 --R Type: Expression Integer --E ---S 72 14:359 Axiom cannot simplify this expression +--S 75 cc:=aa-bb --R --R (3) @@ -1046,6 +1155,211 @@ cc:=aa-bb --R (6a cos(a x) + 12a)sin(a x) - 6a cos(a x) + 6a cos(a x) + 12a --R Type: Expression Integer --E + +--S 76 +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 77 +dd:=tanrule cc +--R +--R (5) +--R 2a x - %pi 3 +--R (- cos(a x) - 2)sin(----------) +--R 4 +--R + +--R 2a x - %pi 2 2a x - %pi 2 2a x - %pi +--R (- 3cos(----------) cos(a x) - 6cos(----------) )sin(----------) +--R 4 4 4 +--R + +--R 2a x - %pi 3 2a x - %pi 3 +--R - 6cos(----------) cos(a x) - 6cos(----------) +--R 4 4 +--R * +--R sin(a x) +--R + +--R 2 2a x - %pi 3 +--R (cos(a x) - cos(a x) - 2)sin(----------) +--R 4 +--R + +--R 2a x - %pi 2 2 2a x - %pi 2 +--R 3cos(----------) cos(a x) - 3cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x - %pi 2 +--R - 6cos(----------) +--R 4 +--R * +--R 2a x - %pi +--R sin(----------) +--R 4 +--R + +--R 2a x - %pi 3 2 2a x - %pi 3 2a x - %pi 3 +--R 2cos(----------) cos(a x) - 8cos(----------) cos(a x) - 10cos(----------) +--R 4 4 4 +--R / +--R 2a x - %pi 3 2a x - %pi 3 +--R (6a cos(----------) cos(a x) + 12a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 2 2a x - %pi 3 +--R - 6a cos(----------) cos(a x) + 6a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 +--R 12a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 78 +sindiffrule2:=rule(sin((a-b)/4) == sin(a/4)*cos(b/4)-cos(a/4)*sin(b/4)) +--R +--R +--R b - a a b b a +--I (6) - %U sin(-----) == - %U cos(-)sin(-) + %U cos(-)sin(-) +--R 4 4 4 4 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 79 +ee:=sindiffrule2 dd +--R +--R (7) +--R +-+ 2a x - %pi 2 +-+ 2a x - %pi 2 a x +--R (- 3\|2 cos(----------) cos(a x) - 6\|2 cos(----------) )sin(---) +--R 4 4 2 +--R + +--R 2a x - %pi 3 +--R (- 2cos(a x) - 4)sin(----------) +--R 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R (3\|2 cos(----------) cos(---) - 12cos(----------) )cos(a x) +--R 4 2 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R 6\|2 cos(----------) cos(---) - 12cos(----------) +--R 4 2 4 +--R * +--R sin(a x) +--R + +--R +-+ 2a x - %pi 2 +-+ 2a x - %pi 2 a x +--R (- 3\|2 cos(----------) cos(a x) - 6\|2 cos(----------) )sin(---) +--R 4 4 2 +--R + +--R 2 2a x - %pi 3 +--R (2cos(a x) - 2cos(a x) - 4)sin(----------) +--R 4 +--R + +--R 2a x - %pi 2 2 2a x - %pi 2a x - %pi 3 2 +--R 6cos(----------) cos(a x) sin(----------) + 4cos(----------) cos(a x) +--R 4 4 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R (3\|2 cos(----------) cos(---) - 16cos(----------) )cos(a x) +--R 4 2 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R 6\|2 cos(----------) cos(---) - 20cos(----------) +--R 4 2 4 +--R / +--R 2a x - %pi 3 2a x - %pi 3 +--R (12a cos(----------) cos(a x) + 24a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 2 2a x - %pi 3 +--R - 12a cos(----------) cos(a x) + 12a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 +--R 24a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 80 +sincuberule:=rule(sin(a)^3 == 3/4*sin(a)-1/4*sin(3*a)) +--R +--R 3 - sin(3a) + 3sin(a) +--R (8) sin(a) == ------------------- +--R 4 +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 81 +ff:=sincuberule ee +--R +--R (9) +--R 2 6a x - 3%pi +--R ((cos(a x) + 2)sin(a x) - cos(a x) + cos(a x) + 2)sin(-----------) +--R 4 +--R + +--R +-+ 2a x - %pi 2 +-+ 2a x - %pi 2 a x +--R (- 6\|2 cos(----------) cos(a x) - 12\|2 cos(----------) )sin(---) +--R 4 4 2 +--R + +--R 2a x - %pi +--R (- 3cos(a x) - 6)sin(----------) +--R 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R (6\|2 cos(----------) cos(---) - 24cos(----------) )cos(a x) +--R 4 2 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R 12\|2 cos(----------) cos(---) - 24cos(----------) +--R 4 2 4 +--R * +--R sin(a x) +--R + +--R +-+ 2a x - %pi 2 +-+ 2a x - %pi 2 a x +--R (- 6\|2 cos(----------) cos(a x) - 12\|2 cos(----------) )sin(---) +--R 4 4 2 +--R + +--R 2a x - %pi 2 2 2a x - %pi +--R ((12cos(----------) + 3)cos(a x) - 3cos(a x) - 6)sin(----------) +--R 4 4 +--R + +--R 2a x - %pi 3 2 +--R 8cos(----------) cos(a x) +--R 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R (6\|2 cos(----------) cos(---) - 32cos(----------) )cos(a x) +--R 4 2 4 +--R + +--R +-+ 2a x - %pi 2 a x 2a x - %pi 3 +--R 12\|2 cos(----------) cos(---) - 40cos(----------) +--R 4 2 4 +--R / +--R 2a x - %pi 3 2a x - %pi 3 +--R (24a cos(----------) cos(a x) + 48a cos(----------) )sin(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 2 2a x - %pi 3 +--R - 24a cos(----------) cos(a x) + 24a cos(----------) cos(a x) +--R 4 4 +--R + +--R 2a x - %pi 3 +--R 48a cos(----------) +--R 4 +--R Type: Expression Integer +--E + +--S 82 14:359 Schaums and Axiom differ by a constant +complexNormalize % +--R +--R 2 +--R (10) - -- +--R 3a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.360~~~~~$\displaystyle @@ -1066,9 +1380,8 @@ $$ <<*>>= )clear all ---S 73 +--S 83 aa:=integrate(1/(p+q*sin(a*x)),x) ---R --R --R (1) --R [ @@ -1099,37 +1412,35 @@ aa:=integrate(1/(p+q*sin(a*x)),x) --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 +--S 84 +bb1:=2/(a*sqrt(p^2-q^2))*atan((p*tan(a*x/2)+q)/sqrt(p^2-q^2)) +--R +--R a x +--R p tan(---) + q +--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 +--S 85 +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 a x +--R - \|q - p + p tan(---) + q +--R 2 +--R log(-----------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q - p + p tan(---) + q +--R 2 --R (3) ---------------------------------- --R +-------+ --R | 2 2 @@ -1137,7 +1448,7 @@ 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+s --R Type: Expression Integer --E ---S 76 +--S 86 cc1:=aa.1-bb1 --R --R (4) @@ -1155,13 +1466,13 @@ cc1:=aa.1-bb1 --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 a x +--R +-------+ p tan(---) + q +--R | 2 2 2 +--R - 2\|q - p atan(--------------) +--R +---------+ +--R | 2 2 +--R \|- q + p --R / --R +---------+ +-------+ --R | 2 2 | 2 2 @@ -1169,40 +1480,28 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 77 +--S 87 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 +---------+ a x +--R | 2 2 p tan(---) + q +--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 +--S 88 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 @@ -1212,6 +1511,16 @@ cc3:=aa.1-bb2 --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 a x +--R - \|q - p + p tan(---) + q +--R 2 +--R - log(-----------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q - p + p tan(---) + q +--R 2 --R / --R +-------+ --R | 2 2 @@ -1219,21 +1528,19 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 79 14:360 Axiom cannot simplify these equations +--S 89 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 | 2 2 a x +--R +---------+ - \|q - p + p tan(---) + q +--R | 2 2 2 +--R - \|- q + p log(-----------------------------) +--R +-------+ +--R | 2 2 a x +--R \|q - p + p tan(---) + q +--R 2 --R + --R +---------+ --R +-------+ | 2 2 @@ -1248,6 +1555,233 @@ cc4:=aa.2-bb2 --R Type: Expression Integer --E +--S 90 +dd2:=ratDenom cc2 +--R +--R (8) +--R +---------+ +--R a x | 2 2 +--R +---------+ (p tan(---) + q)\|- q + p +--R | 2 2 2 +--R - 2\|- q + p atan(----------------------------) +--R 2 2 +--R q - p +--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 2 2 +--R a q - a p +--R Type: Expression Integer +--E + +--S 91 +atanrule2:=rule(atan(x) == 1/2*%i*(log(1-%i*x)-log(1+%i*x))) +--R +--R 1 1 +--R (9) atan(x) == - - %i log(%i x + 1) + - %i log(- %i x + 1) +--R 2 2 +--RType: RewriteRule(Integer,Complex Fraction Integer,Expression Complex Fraction Integer) +--E + +--S 92 +ee2:=atanrule2 dd2 +--R +--R (10) +--R +---------+ +--R 1 | 2 2 2 2 +--R +---------+ (%i p tan(- a x) + %i q)\|- q + p + q - p +--R | 2 2 2 +--R %i\|- q + p log(----------------------------------------------) +--R 2 2 +--R q - p +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (%i p sin(a x) + %i q cos(a x) + %i q)\|- q + p +--R + +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (- %i p sin(a x) - %i q cos(a x) - %i q)\|- q + p +--R + +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R + +--R +---------+ +--R 1 | 2 2 2 2 +--R +---------+ (- %i p tan(- a x) - %i q)\|- q + p + q - p +--R | 2 2 2 +--R - %i\|- q + p log(------------------------------------------------) +--R 2 2 +--R q - p +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 93 +ff2:=expandLog ee2 +--R +--R (11) +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R - %i\|- q + p log((p tan(- a x) + q)\|- q + p + %i q - %i p ) +--R 2 +--R + +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R %i\|- q + p log((p tan(- a x) + q)\|- q + p - %i q + %i p ) +--R 2 +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (%i q - %i p )cos(a x) + %i q - %i p +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (- %i q + %i p )cos(a x) - %i q + %i p +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 94 +gg2:=numer(ff2)/denom(ff2) +--R +--R (12) +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R - %i\|- q + p log((p tan(- a x) + q)\|- q + p + %i q - %i p ) +--R 2 +--R + +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R %i\|- q + p log((p tan(- a x) + q)\|- q + p - %i q + %i p ) +--R 2 +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (%i q - %i p )cos(a x) + %i q - %i p +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (- %i q + %i p )cos(a x) - %i q + %i p +--R / +--R 2 2 +--R a q - a p +--RType: Fraction SparseMultivariatePolynomial(Complex Fraction Integer,Kernel Expression Complex Fraction Integer) +--E + +--S 95 +hh2:=gg2::Expression Complex Fraction Integer +--R +--R (13) +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R - %i\|- q + p log((p tan(- a x) + q)\|- q + p + %i q - %i p ) +--R 2 +--R + +--R +---------+ +---------+ +--R | 2 2 1 | 2 2 2 2 +--R %i\|- q + p log((p tan(- a x) + q)\|- q + p - %i q + %i p ) +--R 2 +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (%i q - %i p )cos(a x) + %i q - %i p +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 +--R (p sin(a x) + q cos(a x) + q)\|- q + p +--R + +--R 2 2 2 2 +--R (- %i q + %i p )cos(a x) - %i q + %i p +--R / +--R 2 2 +--R a q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 96 14:360 Schaums and Axiom agree +complexNormalize hh2 +--R +--R (14) 0 +--R Type: Expression Complex Fraction Integer +--E @ \section{\cite{1}:14.361~~~~~$\displaystyle @@ -1259,7 +1793,7 @@ $$ <<*>>= )clear all ---S 80 +--S 97 aa:=integrate(1/(p+q*sin(a*x))^2,x) --R --R @@ -1305,7 +1839,7 @@ aa:=integrate(1/(p+q*sin(a*x))^2,x) --R Type: Union(List Expression Integer,...) --E ---S 81 +--S 98 t1:=integrate(1/(p+q*sin(a*x)),x) --R --R (2) @@ -1337,7 +1871,7 @@ t1:=integrate(1/(p+q*sin(a*x)),x) --R Type: Union(List Expression Integer,...) --E ---S 82 +--S 99 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) @@ -1364,7 +1898,7 @@ bb1:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.1 --R Type: Expression Integer --E ---S 83 +--S 100 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) @@ -1385,7 +1919,7 @@ bb2:=(q*cos(a*x))/(a*(p^2-q^2)*(p+q*sin(a*x)))+p/(p^2-q^2)*t1.2 --R Type: Expression Integer --E ---S 84 +--S 101 cc1:=aa.1-bb1 --R --R (5) @@ -1425,7 +1959,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 85 +--S 102 cc2:=aa.2-bb1 --R --R (6) @@ -1460,7 +1994,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 86 +--S 103 cc3:=aa.1-bb2 --R --R (7) @@ -1495,7 +2029,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 87 14:361 Schaums and Axiom differ by a constant +--S 104 14:361 Schaums and Axiom differ by a constant cc4:=aa.2-bb2 --R --R q @@ -1514,41 +2048,31 @@ $$ <<*>>= )clear all ---S 88 -aa:=integrate(1/(p^2+a^2*sin(a*x)),x) ---R +--S 105 +aa:=integrate(1/(p^2+q^2*sin(a*x)^2),x) --R --R (1) ---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 | 4 4 ---R a\|- p + a ---R , ---R +-------+ ---R 2 2 2 | 4 4 ---R (p sin(a x) + a cos(a x) + a )\|p - a ---R 2atan(----------------------------------------) ---R 4 4 4 4 ---R (p - a )cos(a x) + p - a ---R -----------------------------------------------] ---R +-------+ ---R | 4 4 ---R a\|p - a ---R Type: Union(List Expression Integer,...) +--R +-------+ +--R | 2 2 +--R p sin(a x)\|q + p +--R atan(-------------------------------) +--R 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: Union(Expression Integer,...) --E ---S 89 +--S 106 bb:=1/(a*p*sqrt(p^2+q^2))*atan((sqrt(p^2+q^2)*tan(a*x))/p) --R --R +-------+ @@ -1563,60 +2087,368 @@ bb:=1/(a*p*sqrt(p^2+q^2))*atan((sqrt(p^2+q^2)*tan(a*x))/p) --R Type: Expression Integer --E ---S 90 -cc1:=aa.1-bb +--S 107 +cc:=aa-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 | 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 +-------+ ---R +---------+ | 2 2 ---R | 4 4 tan(a x)\|q + p ---R - \|- p + a atan(------------------) ---R p +--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 | 4 4 | 2 2 ---R a p\|- p + a \|q + p +--R +-------+ +--R | 2 2 +--R a p\|q + p --R Type: Expression Integer --E ---S 91 14:362 Axiom cannot simplify this expression -cc2:=aa.2-bb +--S 108 +dd:=ratDenom cc --R --R (4) --R +-------+ --R +-------+ | 2 2 ---R | 4 4 tan(a x)\|q + p ---R - \|p - a atan(------------------) +--R | 2 2 tan(a x)\|q + p +--R - \|q + p 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 +-------+ 2 2 2 2 | 2 2 +--R | 2 2 ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R \|q + p atan(--------------------------------------------------------) +--R 2 3 2 2 3 2 3 +--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p +--R + +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 p sin(a x)\|q + p +--R \|q + p atan(-------------------------------) +--R 2 2 2 2 +--R (2q + 2p )cos(a x) + 2q + 2p --R / ---R +-------+ +-------+ ---R | 4 4 | 2 2 ---R a p\|p - a \|q + p +--R 2 3 +--R a p q + a p --R Type: Expression Integer --E +--S 109 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (5) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 110 +ee:=atanrule dd +--R +--R (6) +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R | 2 2 2 2 2 +--R - p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R 2%i p +--R / +--R +-------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R 2%i p +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((- 2q - p )cos(a x) - 2q - 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q + %i p +--R / +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q + %i p +--R + +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 - tan(a x)\|q + p + %i p +--R %i\|q + p log(---------------------------) +--R +-------+ +--R | 2 2 +--R tan(a x)\|q + p + %i p +--R / +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E + +--S 111 +ff:=expandLog ee +--R +--R (7) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i\|q + p log(tan(a x)\|q + p + %i p) +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i\|q + p log(tan(a x)\|q + p - %i p) +--R + +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 2 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) + %i p q +--R + +--R 3 +--R %i p +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q - %i p )cos(a x) + (- 2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q - %i p +--R + +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R +-------+ +--R | 2 2 2 2 2 2 +--R log(p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q + 2%i p ) +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|q + p + (- 2%i q - 2%i p )cos(a x) - 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R +-------+ +--R | 2 2 +--R - %i log(- 1)\|q + p +--R / +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E + +--S 112 +tanrule2:RewriteRule(INT,COMPLEX(INT),EXPR(COMPLEX(INT))):=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (8) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 113 +gg:=tanrule2 ff +--R +--R (9) +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 2 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) + %i p q +--R + +--R 3 +--R %i p +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q - %i p )cos(a x) + (- 2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q - %i p +--R + +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R +-------+ +--R | 2 2 2 2 2 2 +--R log(p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q + 2%i p ) +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|q + p + (- 2%i q - 2%i p )cos(a x) - 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 sin(a x)\|q + p + %i p cos(a x) +--R - %i\|q + p log(----------------------------------) +--R cos(a x) +--R + +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 sin(a x)\|q + p - %i p cos(a x) | 2 2 +--R %i\|q + p log(----------------------------------) - %i log(- 1)\|q + p +--R cos(a x) +--R / +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E + +--S 114 +hh:=expandLog gg +--R +--R (10) +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 2 +--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x) + %i p q +--R + +--R 3 +--R %i p +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R 2 2 2 2 | 2 2 +--R ((2q + p )cos(a x) + 2q + 2p )sin(a x)\|q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q - %i p )cos(a x) + (- 2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q - %i p +--R + +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R +-------+ +--R | 2 2 2 2 2 2 +--R log(p sin(a x)\|q + p + (2%i q + 2%i p )cos(a x) + 2%i q + 2%i p ) +--R + +--R - +--R +-------+ +--R | 2 2 +--R %i\|q + p +--R * +--R log +--R +-------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|q + p + (- 2%i q - 2%i p )cos(a x) - 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - %i\|q + p log(sin(a x)\|q + p + %i p cos(a x)) +--R + +--R +-------+ +-------+ +-------+ +--R | 2 2 | 2 2 | 2 2 +--R %i\|q + p log(sin(a x)\|q + p - %i p cos(a x)) - %i log(- 1)\|q + p +--R / +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E + +--S 115 14:362 Schaums and Axiom differ by a constant +ii:=complexNormalize hh +--R +--R +-------+ +--R | 2 2 +--R (%i log(%i) - %i log(- %i) - %i log(- 1))\|q + p +--R (11) --------------------------------------------------- +--R 2 3 +--R 2a p q + 2a p +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.363~~~~~$\displaystyle @@ -1636,9 +2468,8 @@ $$ <<*>>= )clear all ---S 92 +--S 116 aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x) ---R --R --R (1) --R [ @@ -1679,7 +2510,7 @@ aa:=integrate(1/(p^2-q^2*sin(a*x)^2),x) --R Type: Union(List Expression Integer,...) --E ---S 93 +--S 117 bb1:=1/(a*p*sqrt(p^2-q^2))*atan((sqrt(p^2-q^2)*tan(a*x))/p) --R --R +---------+ @@ -1694,7 +2525,7 @@ bb1:=1/(a*p*sqrt(p^2-q^2))*atan((sqrt(p^2-q^2)*tan(a*x))/p) --R Type: Expression Integer --E ---S 94 +--S 118 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 +-------+ @@ -1711,7 +2542,7 @@ 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 --R Type: Expression Integer --E ---S 95 +--S 119 cc1:=aa.1-bb1 --R --R (4) @@ -1742,7 +2573,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 96 +--S 120 cc2:=aa.2-bb1 --R --R (5) @@ -1766,7 +2597,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 97 +--S 121 cc3:=aa.1-bb2 --R --R (6) @@ -1795,7 +2626,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 98 14:363 Axiom cannot simplify this expression +--S 122 cc4:=aa.2-bb2 --R --R (7) @@ -1826,6 +2657,305 @@ cc4:=aa.2-bb2 --R 2a p\|- q + p \|q - p --R Type: Expression Integer --E + +--S 123 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (8) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 124 +dd2:=tanrule cc2 +--R +--R (9) +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R sin(a x)\|- q + p p sin(a x)\|- q + p +--R - atan(--------------------) - atan(-------------------------------) +--R p cos(a x) 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 125 +ee2:=ratDenom dd2 +--R +--R (10) +--R - +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p +--R atan(--------------------------------------------------------) +--R 2 3 2 2 3 2 3 +--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 sin(a x)\|- q + p +--R \|- q + p atan(--------------------) +--R p cos(a x) +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 p sin(a x)\|- q + p +--R \|- q + p atan(-------------------------------) +--R 2 2 2 2 +--R (2q - 2p )cos(a x) + 2q - 2p +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Integer +--E + +--S 126 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (11) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 127 +ff2:=atanrule ee2 +--R +--R (12) +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 2 2 2 +--R - p sin(a x)\|- q + p + (2%i q - 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R - 2%i p +--R / +--R +---------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|- q + p + (2%i q - 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 - sin(a x)\|- q + p + %i p cos(a x) +--R - %i\|- q + p log(--------------------------------------) +--R +---------+ +--R | 2 2 +--R sin(a x)\|- q + p + %i p cos(a x) +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((- 2q + p )cos(a x) - 2q + 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 +--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q - %i p +--R / +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 +--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q - %i p +--R / +--R 2 3 +--R 2a p q - 2a p +--R Type: Expression Complex Integer +--E + +--S 128 +gg2:=expandLog ff2 +--R +--R (13) +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 +--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q - %i p +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R 2 2 2 2 | 2 2 +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q + %i p )cos(a x) + (- 2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q + %i p +--R + +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R +---------+ +--R | 2 2 2 2 2 2 +--R log(p sin(a x)\|- q + p + (2%i q - 2%i p )cos(a x) + 2%i q - 2%i p ) +--R + +--R - +--R +---------+ +--R | 2 2 +--R %i\|- q + p +--R * +--R log +--R +---------+ +--R | 2 2 2 2 2 +--R p sin(a x)\|- q + p + (- 2%i q + 2%i p )cos(a x) - 2%i q +--R + +--R 2 +--R 2%i p +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R %i\|- q + p log(sin(a x)\|- q + p + %i p cos(a x)) +--R + +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - %i\|- q + p log(sin(a x)\|- q + p - %i p cos(a x)) +--R + +--R +---------+ +--R | 2 2 +--R - %i log(- 1)\|- q + p +--R / +--R 2 3 +--R 2a p q - 2a p +--R Type: Expression Complex Integer +--E + +--S 129 +rootrule4a:RewriteRule(INT,COMPLEX(INT),EXPR(COMPLEX(INT))):=rule(sqrt(p^2-q^2)==sqrt(p-q)*sqrt(q+p)) +--R +--R +---------+ +--R | 2 2 +-------+ +-----+ +--R (14) \|- q + p == \|- q + p \|q + p +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 130 +hh2:=rootrule4a gg2 +--R +--R (15) +--R - +--R +-------+ +-----+ +--R %i\|- q + p \|q + p +--R * +--R log +--R 2 2 2 2 +-------+ +-----+ +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p \|q + p +--R + +--R 2 3 2 2 3 +--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) +--R + +--R 2 3 +--R %i p q - %i p +--R + +--R +-------+ +-----+ +--R %i\|- q + p \|q + p +--R * +--R log +--R 2 2 2 2 +-------+ +-----+ +--R ((2q - p )cos(a x) + 2q - 2p )sin(a x)\|- q + p \|q + p +--R + +--R 2 3 2 2 3 +--R (- %i p q + %i p )cos(a x) + (- 2%i p q + 2%i p )cos(a x) +--R + +--R 2 3 +--R - %i p q + %i p +--R + +--R +-------+ +-----+ +--R %i\|- q + p \|q + p +--R * +--R log +--R +-------+ +-----+ 2 2 2 +--R p sin(a x)\|- q + p \|q + p + (2%i q - 2%i p )cos(a x) + 2%i q +--R + +--R 2 +--R - 2%i p +--R + +--R - +--R +-------+ +-----+ +--R %i\|- q + p \|q + p +--R * +--R log +--R +-------+ +-----+ 2 2 +--R p sin(a x)\|- q + p \|q + p + (- 2%i q + 2%i p )cos(a x) +--R + +--R 2 2 +--R - 2%i q + 2%i p +--R + +--R +-------+ +-----+ +-------+ +-----+ +--R %i\|- q + p \|q + p log(sin(a x)\|- q + p \|q + p + %i p cos(a x)) +--R + +--R +-------+ +-----+ +-------+ +-----+ +--R - %i\|- q + p \|q + p log(sin(a x)\|- q + p \|q + p - %i p cos(a x)) +--R + +--R +-------+ +-----+ +--R - %i log(- 1)\|- q + p \|q + p +--R / +--R 2 3 +--R 2a p q - 2a p +--R Type: Expression Complex Integer +--E + +--S 131 14:363 Schaums and Axiom differ by a constant +ii2:=complexNormalize hh2 +--R +--R +-------+ +-----+ +--R (%i log(%i) - %i log(- %i) - %i log(- 1))\|- q + p \|q + p +--R (16) ----------------------------------------------------------- +--R 2 3 +--R 2a p q - 2a p +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.364~~~~~$\displaystyle @@ -1837,7 +2967,7 @@ $$ <<*>>= )clear all ---S 99 14:364 Axiom cannot compute this integral +--S 132 14:364 Axiom cannot compute this integral aa:=integrate(x^m*sin(a*x),x) --R --R @@ -1857,7 +2987,7 @@ $$ <<*>>= )clear all ---S 100 14:365 Axiom cannot compute this integral +--S 133 14:365 Axiom cannot compute this integral aa:=integrate(sin(a*x)/x^n,x) --R --R @@ -1878,7 +3008,7 @@ $$ <<*>>= )clear all ---S 101 14:366 Axiom cannot compute this integral +--S 134 14:366 Axiom cannot compute this integral aa:=integrate(sin(a*x)^n,x) --R --R @@ -1899,7 +3029,7 @@ $$ <<*>>= )clear all ---S 102 14:367 Axiom cannot compute this integral +--S 135 14:367 Axiom cannot compute this integral aa:=integrate(1/(sin(a*x))^n,x) --R --R @@ -1922,7 +3052,7 @@ $$ <<*>>= )clear all ---S 103 14:368 Axiom cannot compute this integral +--S 136 14:368 Axiom cannot compute this integral aa:=integrate(x/sin(a*x)^n,x) --R --R