[Top][All Lists]
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
[Axiom-developer] 20080504.01.tpd.patch (CATS Schaums-Axiom post-mortem
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080504.01.tpd.patch (CATS Schaums-Axiom post-mortem fixes) |
|
Date: |
Sun, 4 May 2008 19:58:23 -0500 |
More fixes
=========================================================================
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
| [Prev in Thread] |
Current Thread |
[Next in Thread] |
- [Axiom-developer] 20080504.01.tpd.patch (CATS Schaums-Axiom post-mortem fixes),
daly <=