diff --git a/changelog b/changelog index aa128b6..0bc440d 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080504 tpd src/input/schaum18.input post-mortem fixes 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 diff --git a/src/input/schaum18.input.pamphlet b/src/input/schaum18.input.pamphlet index 95fceae..f01b63d 100644 --- a/src/input/schaum18.input.pamphlet +++ b/src/input/schaum18.input.pamphlet @@ -257,7 +257,7 @@ cc1:=aa-bb1 --R Type: Expression Integer --E ---S 19 14:375 Axiom cannot simplify these expressions +--S 19 cc2:=aa-bb2 --R --R (5) @@ -273,6 +273,15 @@ cc2:=aa-bb2 --R Type: Expression Integer --E +--S 20 14:375 Schaums and Axiom differ by a constant +complexNormalize cc1 +--R +--R log(- 1) +--R (6) -------- +--R a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.376~~~~~$\displaystyle @@ -286,7 +295,7 @@ $$ <<*>>= )clear all ---S 20 14:376 Axiom cannot compute this integral +--S 21 14:376 Axiom cannot compute this integral aa:=integrate(x/cos(a*x),x) --R --R @@ -306,7 +315,7 @@ $$ <<*>>= )clear all ---S 21 +--S 22 aa:=integrate(cos(a*x)^2,x) --R --R @@ -316,7 +325,7 @@ aa:=integrate(cos(a*x)^2,x) --R Type: Union(Expression Integer,...) --E ---S 22 +--S 23 bb:=x/2+sin(2*a*x)/(4*a) --R --R sin(2a x) + 2a x @@ -325,7 +334,7 @@ bb:=x/2+sin(2*a*x)/(4*a) --R Type: Expression Integer --E ---S 23 +--S 24 cc:=aa-bb --R --R - sin(2a x) + 2cos(a x)sin(a x) @@ -334,16 +343,17 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 24 +--S 25 cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b))) +--R --R ---I %S sin(b + a) - %S sin(b - a) ---I (4) %S cos(b)sin(a) == ----------------------------- +--I %M sin(b + a) - %M sin(b - a) +--I (4) %M cos(b)sin(a) == ----------------------------- --R 2 --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 25 14:377 Schaums and Axiom agree +--S 26 14:377 Schaums and Axiom agree dd:=cossinrule cc --R --R (5) 0 @@ -359,7 +369,7 @@ $$ <<*>>= )clear all ---S 26 +--S 27 aa:=integrate(x*cos(a*x)^2,x) --R --R @@ -371,7 +381,7 @@ aa:=integrate(x*cos(a*x)^2,x) --R Type: Union(Expression Integer,...) --E ---S 27 +--S 28 bb:=x^2/4+(x*sin(2*a*x))/(4*a)+cos(2*a*x)/(8*a^2) --R --R 2 2 @@ -382,7 +392,7 @@ bb:=x^2/4+(x*sin(2*a*x))/(4*a)+cos(2*a*x)/(8*a^2) --R Type: Expression Integer --E ---S 28 +--S 29 cc:=aa-bb --R --R 2 @@ -393,16 +403,17 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 29 +--S 30 cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b))) +--R --R ---I %T sin(b + a) - %T sin(b - a) ---I (4) %T cos(b)sin(a) == ----------------------------- +--I %N sin(b + a) - %N sin(b - a) +--I (4) %N cos(b)sin(a) == ----------------------------- --R 2 --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 30 +--S 31 dd:=cossinrule cc --R --R 2 @@ -413,16 +424,17 @@ dd:=cossinrule cc --R Type: Expression Integer --E ---S 31 +--S 32 coscosrule:=rule(cos(a)*cos(b) == 1/2*(cos(a-b)+cos(a+b))) +--R --R ---I %U cos(b + a) + %U cos(b - a) ---I (6) %U cos(a)cos(b) == ----------------------------- +--R %O cos(b + a) + %O cos(b - a) +--R (6) %O cos(a)cos(b) == ----------------------------- --R 2 --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 32 +--S 33 ee:=coscosrule dd --R --R 2 @@ -433,7 +445,7 @@ ee:=coscosrule dd --R Type: Expression Integer --E ---S 33 +--S 34 cossqrrule1:=rule(cos(a)^2 == 1/2+1/2*cos(2*a)) --R --R 2 cos(2a) + 1 @@ -442,7 +454,7 @@ cossqrrule1:=rule(cos(a)^2 == 1/2+1/2*cos(2*a)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 34 14:378 Schaums and Axiom differ by a constant +--S 35 14:378 Schaums and Axiom differ by a constant ff:=cossqrrule1 ee --R --R 1 @@ -461,7 +473,7 @@ $$ <<*>>= )clear all ---S 35 +--S 36 aa:=integrate(cos(a*x)^3,x) --R --R @@ -472,7 +484,7 @@ aa:=integrate(cos(a*x)^3,x) --R Type: Union(Expression Integer,...) --E ---S 36 +--S 37 bb:=sin(a*x)/a-sin(a*x)^3/(3*a) --R --R 3 @@ -482,7 +494,7 @@ bb:=sin(a*x)/a-sin(a*x)^3/(3*a) --R Type: Expression Integer --E ---S 37 +--S 38 cc:=aa-bb --R --R 3 2 @@ -492,7 +504,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 38 +--S 39 cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2) --R --R 2 2 @@ -500,7 +512,7 @@ cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 39 14:379 Schaums and Axiom agree +--S 40 14:379 Schaums and Axiom agree dd:=cossqrrule cc --R --R (5) 0 @@ -516,7 +528,7 @@ $$ <<*>>= )clear all ---S 40 +--S 41 aa:=integrate(cos(a*x)^4,x) --R --R @@ -527,7 +539,7 @@ aa:=integrate(cos(a*x)^4,x) --R Type: Union(Expression Integer,...) --E ---S 41 +--S 42 bb:=(3*x)/8+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a) --R --R sin(4a x) + 8sin(2a x) + 12a x @@ -536,7 +548,7 @@ bb:=(3*x)/8+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a) --R Type: Expression Integer --E ---S 42 14:380 Axiom cannot simplify this expression +--S 43 cc:=aa-bb --R --R 3 @@ -545,6 +557,13 @@ cc:=aa-bb --R 32a --R Type: Expression Integer --E + +--S 44 14:380 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.381~~~~~$\displaystyle @@ -555,7 +574,7 @@ $$ <<*>>= )clear all ---S 43 +--S 45 aa:=integrate(1/cos(a*x)^2,x) --R --R @@ -565,7 +584,7 @@ aa:=integrate(1/cos(a*x)^2,x) --R Type: Union(Expression Integer,...) --E ---S 44 +--S 46 bb:=tan(a*x)/a --R --R tan(a x) @@ -574,7 +593,7 @@ bb:=tan(a*x)/a --R Type: Expression Integer --E ---S 45 +--S 47 cc:=aa-bb --R --R - cos(a x)tan(a x) + sin(a x) @@ -583,7 +602,7 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 46 +--S 48 tanrule:=rule(tan(a) == sin(a)/cos(a)) --R --R sin(a) @@ -592,7 +611,7 @@ tanrule:=rule(tan(a) == sin(a)/cos(a)) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 47 14:381 Schaums and Axiom agree +--S 49 14:381 Schaums and Axiom agree dd:=tanrule cc --R --R (5) 0 @@ -609,7 +628,7 @@ $$ <<*>>= )clear all ---S 48 +--S 50 aa:=integrate(1/cos(a*x)^3,x) --R --R @@ -627,7 +646,7 @@ aa:=integrate(1/cos(a*x)^3,x) --R Type: Union(Expression Integer,...) --E ---S 49 +--S 51 bb:=sin(a*x)/(2*a*cos(a*x)^2)+1/(2*a)*log(tan(%pi/4+(a*x)/2)) --R --R 2 2a x + %pi @@ -639,7 +658,7 @@ bb:=sin(a*x)/(2*a*cos(a*x)^2)+1/(2*a)*log(tan(%pi/4+(a*x)/2)) --R Type: Expression Integer --E ---S 50 14:382 Axiom cannot simplify this expression +--S 52 cc:=aa-bb --R --R (3) @@ -654,6 +673,15 @@ cc:=aa-bb --R 2a --R Type: Expression Integer --E + +--S 53 14:382 Schaums and Axiom differ by a constant +complexNormalize cc +--R +--R log(- 1) +--R (4) -------- +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.383~~~~~$\displaystyle @@ -664,42 +692,43 @@ $$ <<*>>= )clear all ---S 51 -aa:=integrate(cos(p*x)*cos(q*x),x) ---R +--S 54 +aa:=integrate(cos(a*x)*cos(p*x),x) --R ---R q cos(p x)sin(q x) - p cos(q x)sin(p x) +--R p cos(a x)sin(p x) - a cos(p x)sin(a x) --R (1) --------------------------------------- --R 2 2 ---R q - p +--R p - a --R Type: Union(Expression Integer,...) --E ---S 52 -bb:=(sin(a-p)*x)/(2*(a-p))+(sin(a+p)*x)/(2*(a+p)) +--S 55 +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 (p - a)sin((p + a)x) + (p + a)sin((p - a)x) +--R (2) ------------------------------------------- +--R 2 2 +--R 2p - 2a --R Type: Expression Integer --E ---S 53 14:383 Axiom cannot simplify this expression +--S 56 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 (- p + a)sin((p + a)x) + 2p cos(a x)sin(p x) + (- p - a)sin((p - a)x) --R + ---R 2 3 2 ---R ((- p - a)q + p + a p )x sin(p - a) +--R - 2a cos(p x)sin(a x) --R / ---R 2 2 2 4 2 2 ---R (2p - 2a )q - 2p + 2a p +--R 2 2 +--R 2p - 2a +--R Type: Expression Integer +--E + +--S 57 14:383 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 --R Type: Expression Integer --E @ @@ -712,7 +741,7 @@ $$ <<*>>= )clear all ---S 54 +--S 58 aa:=integrate(1/(1-cos(a*x)),x) --R --R @@ -722,21 +751,32 @@ aa:=integrate(1/(1-cos(a*x)),x) --R Type: Union(Expression Integer,...) --E ---S 55 -bb:=-1/a*cot(a*x)/2 +--S 59 +bb:=-1/a*cot((a*x)/2) --R ---R cot(a x) +--R a x +--R cot(---) +--R 2 --R (2) - -------- ---R 2a +--R a --R Type: Expression Integer --E ---S 56 14:384 Axiom cannot simplify this expression +--S 60 cc:=aa-bb --R ---R cot(a x)sin(a x) - 2cos(a x) - 2 ---R (3) -------------------------------- ---R 2a sin(a x) +--R a x +--R cot(---)sin(a x) - cos(a x) - 1 +--R 2 +--R (3) ------------------------------- +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 61 14:384 Schaums and Axiom agree +dd:=complexNormalize cc +--R +--R (4) 0 --R Type: Expression Integer --E @ @@ -750,38 +790,105 @@ $$ <<*>>= )clear all ---S 57 -aa:=integrate(x/(1-cos(ax)),x) ---R +--S 62 +aa:=integrate(x/(1-cos(a*x)),x) --R ---R 2 ---R x ---R (1) - ------------ ---R 2cos(ax) - 2 +--R (1) +--R sin(a x) 2 +--R 2sin(a x)log(------------) - sin(a x)log(------------) - a x cos(a x) - a x +--R cos(a x) + 1 cos(a x) + 1 +--R --------------------------------------------------------------------------- +--R 2 +--R a sin(a x) --R Type: Union(Expression Integer,...) --E ---S 58 -bb:=-x/a*cot(a*x)/2+2/a^2*log(sin((a*x)/2)) +--S 63 +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 a x a x +--R 2log(sin(---)) - a x cot(---) +--R 2 2 --R (2) ----------------------------- --R 2 ---R 2a +--R a --R Type: Expression Integer --E ---S 59 14:385 Axiom cannot simplify this expression +--S 64 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 (3) +--R sin(a x) a x +--R 2sin(a x)log(------------) - 2sin(a x)log(sin(---)) +--R cos(a x) + 1 2 +--R + +--R 2 a x +--R - sin(a x)log(------------) + a x cot(---)sin(a x) - a x cos(a x) - a x +--R cos(a x) + 1 2 +--R / +--R 2 +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 65 +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 66 +dd:=cotrule cc +--R +--R (5) +--R a x sin(a x) a x a x +--R 2sin(---)sin(a x)log(------------) - 2sin(---)sin(a x)log(sin(---)) +--R 2 cos(a x) + 1 2 2 +--R + +--R a x 2 a x +--R - sin(---)sin(a x)log(------------) + a x cos(---)sin(a x) +--R 2 cos(a x) + 1 2 +--R + +--R a x +--R (- a x cos(a x) - a x)sin(---) +--R 2 +--R / +--R 2 a x +--R a sin(---)sin(a x) +--R 2 +--R Type: Expression Integer +--E + +--S 67 +ee:=expandLog dd +--R +--R (6) +--R a x a x a x +--R 2sin(---)sin(a x)log(sin(a x)) - 2sin(---)sin(a x)log(sin(---)) +--R 2 2 2 +--R + +--R a x +--R - sin(---)sin(a x)log(cos(a x) + 1) +--R 2 +--R + +--R a x a x a x +--R (- log(2)sin(---) + a x cos(---))sin(a x) + (- a x cos(a x) - a x)sin(---) +--R 2 2 2 +--R / +--R 2 a x +--R a sin(---)sin(a x) +--R 2 +--R Type: Expression Integer +--E + +--S 68 14:385 Schaums and Axiom agree +complexNormalize ee +--R +--R (7) 0 --R Type: Expression Integer --E @ @@ -794,31 +901,41 @@ $$ <<*>>= )clear all ---S 60 -aa:=integrate(1/(1+cos(ax)),x) ---R +--S 69 +aa:=integrate(1/(1+cos(a*x)),x) --R ---R x ---R (1) ----------- ---R cos(ax) + 1 +--R sin(a x) +--R (1) -------------- +--R a cos(a x) + a --R Type: Union(Expression Integer,...) --E ---S 61 -bb:=1/a*tan(a*x)/2 +--S 70 +bb:=1/a*tan((a*x)/2) --R ---R tan(a x) +--R a x +--R tan(---) +--R 2 --R (2) -------- ---R 2a +--R a --R Type: Expression Integer --E ---S 62 14:386 Axiom cannot simplify this expression +--S 71 cc:=aa-bb --R ---R (- cos(ax) - 1)tan(a x) + 2a x ---R (3) ------------------------------ ---R 2a cos(ax) + 2a +--R a x +--R (- cos(a x) - 1)tan(---) + sin(a x) +--R 2 +--R (3) ----------------------------------- +--R a cos(a x) + a +--R Type: Expression Integer +--E + +--S 72 14:386 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 --R Type: Expression Integer --E @ @@ -832,7 +949,7 @@ $$ <<*>>= )clear all ---S 63 +--S 73 aa:=integrate(x/(1+cos(a*x)),x) --R --R @@ -845,7 +962,7 @@ aa:=integrate(x/(1+cos(a*x)),x) --R Type: Union(Expression Integer,...) --E ---S 64 +--S 74 bb:=x/a*tan((a*x)/2)+2/a^2*log(cos((a*x)/2)) --R --R a x a x @@ -857,7 +974,7 @@ bb:=x/a*tan((a*x)/2)+2/a^2*log(cos((a*x)/2)) --R Type: Expression Integer --E ---S 65 14:387 Axiom cannot simplify this expression +--S 75 cc:=aa-bb --R --R (3) @@ -873,6 +990,30 @@ cc:=aa-bb --R a cos(a x) + a --R Type: Expression Integer --E + +--S 76 +dd:=expandLog cc +--R +--R (4) +--R a x +--R (cos(a x) + 1)log(cos(a x) + 1) + (- 2cos(a x) - 2)log(cos(---)) +--R 2 +--R + +--R a x +--R (- a x cos(a x) - a x)tan(---) + a x sin(a x) - log(2)cos(a x) - log(2) +--R 2 +--R / +--R 2 2 +--R a cos(a x) + a +--R Type: Expression Integer +--E + +--S 77 14:387 Schaums and Axiom agree +complexNormalize dd +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.388~~~~~$\displaystyle @@ -884,7 +1025,7 @@ $$ <<*>>= )clear all ---S 66 +--S 78 aa:=integrate(1/(1-cos(a*x))^2,x) --R --R @@ -895,7 +1036,7 @@ aa:=integrate(1/(1-cos(a*x))^2,x) --R Type: Union(Expression Integer,...) --E ---S 67 +--S 79 bb:=-1/(2*a)*cot((a*x)/2)-1/(6*a)*cot((a*x)/2)^3 --R --R a x 3 a x @@ -906,7 +1047,7 @@ bb:=-1/(2*a)*cot((a*x)/2)-1/(6*a)*cot((a*x)/2)^3 --R Type: Expression Integer --E ---S 68 14:388 Axiom cannot simplify this expression +--S 80 cc:=aa-bb --R --R (3) @@ -919,6 +1060,13 @@ cc:=aa-bb --R (6a cos(a x) - 6a)sin(a x) --R Type: Expression Integer --E + +--S 81 14:388 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.389~~~~~$\displaystyle @@ -930,7 +1078,7 @@ $$ <<*>>= )clear all ---S 69 +--S 82 aa:=integrate(1/(1+cos(a*x))^2,x) --R --R @@ -941,10 +1089,10 @@ aa:=integrate(1/(1+cos(a*x))^2,x) --R Type: Union(Expression Integer,...) --E ---S 70 -bb:=1/(2*a)*tan((a*x)/2)+1/(6*a)*tan((a*x)/2)^2 +--S 83 +bb:=1/(2*a)*tan((a*x)/2)+1/(6*a)*tan((a*x)/2)^3 --R ---R a x 2 a x +--R a x 3 a x --R tan(---) + 3tan(---) --R 2 2 --R (2) --------------------- @@ -952,11 +1100,11 @@ bb:=1/(2*a)*tan((a*x)/2)+1/(6*a)*tan((a*x)/2)^2 --R Type: Expression Integer --E ---S 71 14:389 Axiom cannot simplify this expression +--S 84 cc:=aa-bb --R --R (3) ---R 2 a x 2 +--R 2 a x 3 --R (- cos(a x) - 2cos(a x) - 1)tan(---) --R 2 --R + @@ -968,6 +1116,13 @@ cc:=aa-bb --R 6a cos(a x) + 12a cos(a x) + 6a --R Type: Expression Integer --E + +--S 85 14:389 Schaums and Axiom agree +complexNormalize cc +--R +--R (4) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.390~~~~~$\displaystyle @@ -989,9 +1144,8 @@ $$ <<*>>= )clear all ---S 72 +--S 86 aa:=integrate(1/(p+q*cos(a*x)),x) ---R --R --R (1) --R +-------+ @@ -1015,13 +1169,14 @@ aa:=integrate(1/(p+q*cos(a*x)),x) --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) +--S 87 +bb1:=2/(a*sqrt(p^2-q^2))*atan(sqrt((p-q)/(p+q))*tan(1/2*a*x)) +--R --R --R +-------+ ---R a x |- q + p ---R 2tan(---)atan( |------- ) ---R 2 \| q + p +--R a x |- q + p +--R 2atan(tan(---) |------- ) +--R 2 \| q + p --R (2) ------------------------- --R +---------+ --R | 2 2 @@ -1029,8 +1184,8 @@ bb1:=2/(a*sqrt(p^2-q^2))*atan(sqrt((p-q)/(p+q)))*tan(1/2*a*x) --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)))) +--S 88 +bb2:=1/(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 @@ -1042,14 +1197,15 @@ 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 --R |----- - tan(---) --R \|q - p 2 --R (3) -------------------------- ---R +-------+ ---R | 2 2 ---R \|q - p +--R +-------+ +--R | 2 2 +--R a\|q - p --R Type: Expression Integer --E ---S 75 +--S 89 cc1:=aa.1-bb1 +--R --R --R (4) --R +-------+ @@ -1058,10 +1214,10 @@ cc1:=aa.1-bb1 --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 | 2 2 a x |- q + p +--R - 2\|q - p atan(tan(---) |------- ) +--R 2 \| q + p --R / --R +---------+ +-------+ --R | 2 2 | 2 2 @@ -1069,14 +1225,15 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 76 +--S 90 cc2:=aa.2-bb1 +--R --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 a x |- q + p sin(a x)\|- q + p +--R - 2atan(tan(---) |------- ) + 2atan(-----------------------) +--R 2 \| q + p (q + p)cos(a x) + q + p --R (5) ------------------------------------------------------------ --R +---------+ --R | 2 2 @@ -1084,19 +1241,19 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 77 +--S 91 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 |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 + --R +-------+ --R | 2 2 2 2 @@ -1110,19 +1267,19 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 78 14:390 Axiom cannot simplify these expressions +--S 92 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 +---------+ - |----- - tan(---) +--R | 2 2 \|q - p 2 +--R - \|- q + p log(---------------------) +--R +-----+ +--R |q + p a x +--R |----- - tan(---) +--R \|q - p 2 --R + --R +---------+ --R +-------+ | 2 2 @@ -1147,7 +1304,7 @@ $$ <<*>>= )clear all ---S 79 +--S 93 aa:=integrate(1/(p+q*cos(a*x))^2,x) --R --R @@ -1188,7 +1345,7 @@ aa:=integrate(1/(p+q*cos(a*x))^2,x) --R Type: Union(List Expression Integer,...) --E ---S 80 +--S 94 t1:=integrate(1/(p+q*cos(a*x)),x) --R --R (2) @@ -1213,7 +1370,7 @@ t1:=integrate(1/(p+q*cos(a*x)),x) --R Type: Union(List Expression Integer,...) --E ---S 81 +--S 95 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) @@ -1236,7 +1393,7 @@ bb1:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.1 --R Type: Expression Integer --E ---S 82 +--S 96 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) @@ -1252,7 +1409,7 @@ bb2:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.2 --R Type: Expression Integer --E ---S 83 +--S 97 cc1:=aa.1-bb1 --R --R (5) @@ -1274,7 +1431,7 @@ cc1:=aa.1-bb1 --R Type: Expression Integer --E ---S 84 +--S 98 cc2:=aa.2-bb1 --R --R (6) @@ -1296,7 +1453,7 @@ cc2:=aa.2-bb1 --R Type: Expression Integer --E ---S 85 +--S 99 cc3:=aa.1-bb2 --R --R (7) @@ -1318,7 +1475,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 86 14:391 Schaums and Axiom agree +--S 100 14:391 Schaums and Axiom agree cc4:=aa.2-bb2 --R --R (8) 0 @@ -1334,35 +1491,26 @@ $$ <<*>>= )clear all ---S 87 -aa:=integrate(1/(p^2+a^2*cos(a*x)),x) +--S 101 +aa:=integrate(1/(p^2+q^2*cos(a*x)^2),x) --R --R --R (1) ---R +---------+ ---R 2 2 | 4 4 4 4 ---R (- p cos(a x) - a )\|- p + a + (p - a )sin(a x) ---R log(---------------------------------------------------) ---R 2 2 ---R a cos(a x) + p ---R [--------------------------------------------------------, ---R +---------+ ---R | 4 4 ---R a\|- p + a ---R +-------+ ---R | 4 4 ---R sin(a x)\|p - a ---R 2atan(---------------------------) ---R 2 2 2 2 ---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 2 2 2 +--R sin(a x)\|q + p ((q - p )cos(a x) - 2p )sin(a x) +--R atan(------------------) - atan(-----------------------------------------) +--R 2p cos(a x) + 2p +-------+ +--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 88 +--S 102 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) @@ -1377,47 +1525,232 @@ bb:=1/(a*p*sqrt(p^2+q^2))*atan((p*tan(a*x))/sqrt(p^2+q^2)) --R Type: Expression Integer --E ---S 89 -cc1:=aa.1-bb +--S 103 +cc:=aa-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 | 2 2 +--R sin(a x)\|q + p p tan(a x) +--R atan(------------------) - atan(----------) +--R 2p cos(a x) + 2p +-------+ +--R | 2 2 +--R \|q + p --R + ---R +---------+ ---R | 4 4 p tan(a x) ---R - \|- p + a atan(----------) ---R +-------+ ---R | 2 2 ---R \|q + p +--R 2 2 2 +--R ((q - p )cos(a x) - 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 90 14:392 Axiom cannot simplify these expressions -cc2:=aa.2-bb +--S 104 +dd:=ratDenom cc --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 +-------+ +--R +-------+ | 2 2 +--R | 2 2 p tan(a x)\|q + p +--R - \|q + p atan(--------------------) +--R 2 2 +--R q + p +--R + +--R - +--R +-------+ +--R | 2 2 +--R \|q + p +--R * +--R +-------+ +--R 2 2 2 | 2 2 +--R ((q - p )cos(a x) - 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 2p cos(a x) + 2p +--R / +--R 2 3 +--R a p q + a p --R Type: Expression Integer --E + +--S 105 +atanrule2:=rule(atan(x) == 1/2*%i*(log(1-%i*x)-log(1+%i*x))) +--R +--R 1 1 +--R (5) 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 106 +ee:=atanrule2 dd +--R +--R (6) +--R +-------+ +--R +-------+ | 2 2 2 2 +--R 1 | 2 2 %i p tan(a x)\|q + p + q + p +--R - %i\|q + p log(---------------------------------) +--R 2 2 2 +--R q + p +--R + +--R +-------+ +--R 1 | 2 2 +--R - %i\|q + p +--R 2 +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R ((%i q - %i p )cos(a x) - 2%i p )sin(a x)\|q + p +--R + +--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 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 1 | 2 2 +--R +-------+ - %i sin(a x)\|q + p + p cos(a x) + p +--R 1 | 2 2 2 +--R - - %i\|q + p log(----------------------------------------) +--R 2 p cos(a x) + p +--R + +--R +-------+ +--R 1 | 2 2 +--R +-------+ - - %i sin(a x)\|q + p + p cos(a x) + p +--R 1 | 2 2 2 +--R - %i\|q + p log(------------------------------------------) +--R 2 p cos(a x) + p +--R + +--R - +--R +-------+ +--R 1 | 2 2 +--R - %i\|q + p +--R 2 +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R ((- %i q + %i p )cos(a x) + 2%i p )sin(a x)\|q + p +--R + +--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 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 2 2 +--R 1 | 2 2 - %i p tan(a x)\|q + p + q + p +--R - - %i\|q + p log(-----------------------------------) +--R 2 2 2 +--R q + p +--R / +--R 2 3 +--R a p q + a p +--R Type: Expression Complex Fraction Integer +--E + +--S 107 +ff:=expandLog ee +--R +--R (7) +--R +-------+ +-------+ +--R 1 | 2 2 | 2 2 2 2 +--R - - %i\|q + p log(p tan(a x)\|q + p + %i q + %i p ) +--R 2 +--R + +--R +-------+ +-------+ +--R 1 | 2 2 | 2 2 2 2 +--R - %i\|q + p log(p tan(a x)\|q + p - %i q - %i p ) +--R 2 +--R + +--R - +--R +-------+ +--R 1 | 2 2 +--R - %i\|q + p +--R 2 +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R ((q - p )cos(a x) - 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 1 | 2 2 +--R - %i\|q + p +--R 2 +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R ((q - p )cos(a x) - 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 1 | 2 2 | 2 2 +--R - %i\|q + p log(sin(a x)\|q + p + 2%i p cos(a x) + 2%i p) +--R 2 +--R + +--R +-------+ +-------+ +--R 1 | 2 2 | 2 2 +--R - - %i\|q + p log(sin(a x)\|q + p - 2%i p cos(a x) - 2%i p) +--R 2 +--R + +--R +-------+ +--R 1 1 1 1 | 2 2 +--R (%i log(%i) - - %i log(- %i) + - %i log(- - %i) - %i log(- %i))\|q + p +--R 2 2 2 2 +--R / +--R 2 3 +--R a p q + a p +--R Type: Expression Complex Fraction Integer +--E + +--S 108 14:392 Schaums and Axiom differ by a constant +complexNormalize ff +--R +--R (8) +--R 1 1 1 1 +--R %i log(%i) - - %i log(- %i) + - %i log(- - %i) - %i log(- %i) +--R 2 2 2 2 +--R + +--R 1 +--R - - %i log(- 1) +--R 2 +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R / +--R 2 3 +--R a p q + a p +--R Type: Expression Complex Fraction Integer +--E + @ \section{\cite{1}:14.393~~~~~$\displaystyle @@ -1437,7 +1770,7 @@ $$ <<*>>= )clear all ---S 91 +--S 109 aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x) --R --R @@ -1473,8 +1806,8 @@ aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x) --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)) +--S 110 +bb1:=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(------------) @@ -1482,13 +1815,13 @@ bb1:=1/(a*p*sqrt(p^2-a^2))*atan((p*tan(a*x))/sqrt(p^2-q^2)) --R | 2 2 --R \|- q + p --R (2) ------------------ ---R +-------+ ---R | 2 2 ---R a p\|p - a +--R +---------+ +--R | 2 2 +--R a p\|- q + p --R Type: Expression Integer --E ---S 93 +--S 111 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 +-------+ @@ -1505,13 +1838,13 @@ 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 --R Type: Expression Integer --E ---S 94 +--S 112 cc1:=aa.1-bb1 --R --R (4) ---R +-------+ ---R | 2 2 ---R \|p - a +--R +---------+ +--R | 2 2 +--R \|- q + p --R * --R log --R +-------+ @@ -1531,38 +1864,38 @@ cc1:=aa.1-bb1 --R | 2 2 --R \|- q + p --R / ---R +-------+ +-------+ ---R | 2 2 | 2 2 ---R 2a p\|p - a \|q - p +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R 2a p\|- q + p \|q - p --R Type: Expression Integer --E ---S 95 +--S 113 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 +--R sin(a x)\|- q + p p tan(a x) +--R atan(--------------------) - 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 2 2 2 +--R ((q + p )cos(a x) + 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 | 2 2 ---R a p\|- q + p \|p - a +--R +---------+ +--R | 2 2 +--R a p\|- q + p --R Type: Expression Integer --E ---S 96 +--S 114 cc3:=aa.1-bb2 --R --R (6) @@ -1588,7 +1921,7 @@ cc3:=aa.1-bb2 --R Type: Expression Integer --E ---S 97 14:393 Axiom cannot simplify these expressions +--S 115 cc4:=aa.2-bb2 --R --R (7) @@ -1618,6 +1951,244 @@ cc4:=aa.2-bb2 --R 2a p\|- q + p \|q - p --R Type: Expression Integer --E + +--S 116 +dd2:=ratDenom cc2 +--R +--R (8) +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 p tan(a x)\|- q + p +--R - \|- q + p atan(----------------------) +--R 2 2 +--R q - p +--R + +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R +---------+ +--R 2 2 2 | 2 2 +--R ((q + p )cos(a x) + 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 2p cos(a x) + 2p +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Integer +--E + +--S 117 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (9) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 118 +ee2:=tanrule dd2 +--R +--R (10) +--R +---------+ +--R | 2 2 +--R \|- q + p +--R * +--R +---------+ +--R 2 2 2 | 2 2 +--R ((q + p )cos(a x) + 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 2p cos(a x) + 2p +--R + +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 p sin(a x)\|- q + p +--R - \|- q + p atan(----------------------) +--R 2 2 +--R (q - p )cos(a x) +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Integer +--E + +--S 119 +atanrule2:=rule(atan(x) == 1/2*%i*(log(1-%i*x)-log(1+%i*x))) +--R +--R 1 1 +--R (11) 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 120 +ff2:=atanrule2 ee2 +--R +--R (12) +--R - +--R +---------+ +--R 1 | 2 2 +--R - %i\|- q + p +--R 2 +--R * +--R log +--R +---------+ +--R 2 2 2 | 2 2 +--R ((%i q + %i p )cos(a x) + 2%i p )sin(a x)\|- q + p +--R + +--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 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 1 | 2 2 +--R +---------+ - %i sin(a x)\|- q + p + p cos(a x) + p +--R 1 | 2 2 2 +--R - %i\|- q + p log(------------------------------------------) +--R 2 p cos(a x) + p +--R + +--R +---------+ +--R +---------+ | 2 2 2 2 +--R 1 | 2 2 %i p sin(a x)\|- q + p + (q - p )cos(a x) +--R - %i\|- q + p log(---------------------------------------------) +--R 2 2 2 +--R (q - p )cos(a x) +--R + +--R +---------+ +--R +---------+ | 2 2 2 2 +--R 1 | 2 2 - %i p sin(a x)\|- q + p + (q - p )cos(a x) +--R - - %i\|- q + p log(-----------------------------------------------) +--R 2 2 2 +--R (q - p )cos(a x) +--R + +--R +---------+ +--R 1 | 2 2 +--R +---------+ - - %i sin(a x)\|- q + p + p cos(a x) + p +--R 1 | 2 2 2 +--R - - %i\|- q + p log(--------------------------------------------) +--R 2 p cos(a x) + p +--R + +--R +---------+ +--R 1 | 2 2 +--R - %i\|- q + p +--R 2 +--R * +--R log +--R +---------+ +--R 2 2 2 | 2 2 +--R ((- %i q - %i p )cos(a x) - 2%i p )sin(a x)\|- q + p +--R + +--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 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 2 3 +--R a p q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 121 +gg2:=expandLog ff2 +--R +--R (13) +--R +---------+ +--R 1 | 2 2 +--R - %i\|- q + p +--R 2 +--R * +--R log +--R +---------+ +--R 2 2 2 | 2 2 +--R ((q + p )cos(a x) + 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 1 | 2 2 +--R - %i\|- q + p +--R 2 +--R * +--R log +--R +---------+ +--R 2 2 2 | 2 2 +--R ((q + p )cos(a x) + 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 1 | 2 2 | 2 2 2 2 +--R - - %i\|- q + p log(p sin(a x)\|- q + p + (%i q - %i p )cos(a x)) +--R 2 +--R + +--R +---------+ +---------+ +--R 1 | 2 2 | 2 2 2 2 +--R - %i\|- q + p log(p sin(a x)\|- q + p + (- %i q + %i p )cos(a x)) +--R 2 +--R + +--R +---------+ +---------+ +--R 1 | 2 2 | 2 2 +--R - - %i\|- q + p log(sin(a x)\|- q + p + 2%i p cos(a x) + 2%i p) +--R 2 +--R + +--R +---------+ +---------+ +--R 1 | 2 2 | 2 2 +--R - %i\|- q + p log(sin(a x)\|- q + p - 2%i p cos(a x) - 2%i p) +--R 2 +--R + +--R +---------+ +--R 1 1 1 1 | 2 2 +--R (- %i log(- %i) - - %i log(- - %i))\|- q + p +--R 2 2 2 2 +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Complex Fraction Integer +--E + +--S 122 14:393 Schaums and Axiom differ by a constant +hh2:=complexNormalize gg2 +--R +--R (14) +--R 1 1 1 1 1 1 +--R (- - %i log(%i) + - %i log(- %i) - - %i log(- - %i) + - %i log(- %i)) +--R 2 2 2 2 2 2 +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R / +--R 2 3 +--R a p q - a p +--R Type: Expression Complex Fraction Integer +--E @ \section{\cite{1}:14.394~~~~~$\displaystyle @@ -1629,7 +2200,7 @@ $$ <<*>>= )clear all ---S 98 14:394 Axiom cannot compute this integral +--S 123 14:394 Axiom cannot compute this integral aa:=integrate(x^m*cos(a*x),x) --R --R @@ -1649,7 +2220,7 @@ $$ <<*>>= )clear all ---S 99 14:395 Axiom cannot compute this integral +--S 124 14:395 Axiom cannot compute this integral aa:=integrate(cos(a*x)/x^n,x) --R --R @@ -1670,7 +2241,7 @@ $$ <<*>>= )clear all ---S 100 14:396 Axiom cannot compute this integral +--S 125 14:396 Axiom cannot compute this integral aa:=integrate(cos(a*x)^n,x) --R --R @@ -1691,7 +2262,7 @@ $$ <<*>>= )clear all ---S 101 14:397 Axiom cannot compute this integral +--S 126 14:397 Axiom cannot compute this integral aa:=integrate(1/(cos(a*x))^n,x) --R --R @@ -1714,7 +2285,7 @@ $$ <<*>>= )clear all ---S 102 14:398 Axiom cannot compute this integral +--S 127 14:398 Axiom cannot compute this integral aa:=integrate(x/cos(a*x)^n,x) --R --R