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[Axiom-developer] 20080502.01.tpd.patch (CATS Schaums-Axiom post mortem
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080502.01.tpd.patch (CATS Schaums-Axiom post mortem fixup) |
|
Date: |
Sat, 3 May 2008 22:59:45 -0500 |
More agreement between Schaums and Axiom.
=====================================================================
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
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- [Axiom-developer] 20080502.01.tpd.patch (CATS Schaums-Axiom post mortem fixup),
daly <=