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## [Axiom-developer] 20080418.01.tpd.patch (CATS Schaums-Axiom equivalence

 From: daly Subject: [Axiom-developer] 20080418.01.tpd.patch (CATS Schaums-Axiom equivalence testing (1-11)) Date: Sat, 19 Apr 2008 01:28:16 -0500

This patch revises several old files where it was claimed that Axiom
could not factor the results. As Shakespeare once said (but never
wrote down), "The fault, Dear Brutus, is not in our software but
in ourselves". Now that I'm a tad wiser about simplification in Axiom
I was able to simplify most of the results.

Also of interest, I believe that there is a typo in the Schaums book
for problem 14:91. Axiom cannot simplify this expression. It appears
that the trailing sqrt factor should be
sqrt((a*x+b)^3)
not
sqrt((a+b*x)^3)
factor the numerator of the Axiom result. Note that I'm using the
Schaums book published in 1968.

Other than 14:91, where Axiom could compute the integral, the results
to date have agreed with Schaums to within a constant of integration.

schaum11.input.pamphlet is new work. The other changes are rework
of prior results.

The new cumulative results are:

schaum1.input.pamphet
14:59 Schaums and Axiom agree
14:60 Schaums and Axiom agree
14:61 Schaums and Axiom differ by a constant
14:62 Schaums and Axiom differ by a constant
14:63 Schaums and Axiom agree
14:64 Schaums and Axiom agree
14:65 Schaums and Axiom agree
14:66 Schaums and Axiom agree
14:67 Schaums and Axiom agree
14:68 Schaums and Axiom differ by a constant
14:69 Schaums and Axiom differ by a constant
14:70 Schaums and Axiom agree
14:71 Schaums and Axiom agree
14:72 Schaums and Axiom differ by a constant
14:73 Schaums and Axiom differ by a constant
14:74 Schaums and Axiom agree
14:75 Schaums and Axiom agree
14:76 Schaums and Axiom agree
14:77 Schaums and Axiom differ by a constant
14:78 Schaums and Axiom agree
14:79 Schaums and Axiom agree
14:80 Schaums and Axiom agree
14:82 Schaums and Axiom agree
14:83 Axiom cannot do this integration

schaum2.input.pamphet
14:84 Schaums and Axiom agree
14:85 Schaums and Axiom agree
14:86 Schaums and Axiom agree
14:87a Schaums and Axiom differ by a constant
14:87b Schaums and Axiom differ by a constant
14:88 Schaums and Axiom differ by a constant
14:89 Schaums and Axiom differ by a constant
14:90 Schaums and Axiom agree
14:91 Axiom cannot simplify this expression. Schaums typo?
14:92 Axiom cannot simplify this expression
14:93 Schaums and Axiom agree
14:94 Axiom cannot do this integral
14:95 Axiom cannot do this integral
14:96 Axiom cannot do this integral
14:97 Axiom cannot do this integral
14:98 Axiom cannot do this integral
14:99 Schaums and Axiom agree
14:100 Schaums and Axiom agree
14:101 Schaums and Axiom agree
14:102 Axiom cannot do this integral
14:103 Axiom cannot do this integral
14:104 Axiom cannot do this integral

schaum3.input.pamphet
14:105 Schaums and Axiom agree
14:106 Schaums and Axiom agree
14:107 Schaums and Axiom agree
14:108 Schaums and Axiom agree
14:109 Schaums and Axiom agree
14:110 Axiom cannot do this integral
14:111 Schaums and Axiom agree
14:112 Axiom cannot do this integral

schaum4.input.pamphet
14:113 Schaums and Axiom agree
14:114 Axiom cannot simplify these answers
14:115 Axiom cannot simplify these answers
14:116 Axiom cannot compute this integral
14:117 Axiom cannot compute this integral
14:118 Axiom cannot compute this integral
14:119 Axiom cannot compute this integral

schaum5.input.pamphet
14:120 Axiom cannot simplify these answers
14:121 Axiom cannot simplify this answer
14:122 Axiom cannot simplify this answer
14:123 Axiom cannot simplify these results
14:124 Axiom cannot simplify this result

schaum6.input.pamphet
14:125 Schaums and Axiom agree
14:126 Schaums and Axiom agree
14:127 Schaums and Axiom agree
14:128 Schaums and Axiom agree
14:129 Schaums and Axiom agree
14:130 Schaums and Axiom agree
14:131 Schaums and Axiom agree
14:132 Schaums and Axiom agree
14:133 Schaums and Axiom agree
14:134 Schaums and Axiom differ by a constant
14:135 Schaums and Axiom agree
14:136 Schaums and Axiom agree
14:137 Schaums and Axiom agree
14:138 Schaums and Axiom agree
14:139 Axiom cannot do this integral
14:140 Schaums and Axiom agree
14:141 Axiom cannot do this integral
14:142 Axiom cannot do this integral
14:143 Axiom cannot do this integral

schaum7.input.pamphet
14:144 Schaums and Axiom agree
14:145 Schaums and Axiom agree
14:146 Schaums and Axiom agree
14:147 Schaums and Axiom agree
14:148 Schaums and Axiom agree
14:149 Schaums and Axiom agree
14:150 Schaums and Axiom agree
14:151 Schaums and Axiom agree
14:152 Schaums and Axiom agree
14:153 Schaums and Axiom agree
14:154 Schaums and Axiom agree
14:155 Schaums and Axiom agree
14:156 Schaums and Axiom agree
14:157 Schaums and Axiom agree
14:158 Axiom cannot do this integral
14:159 Schaums and Axiom agree
14:160 Axiom cannot compute this integral
14:161 Axiom cannot compute this integral
14:162 Axiom cannot compute this integral

schaum8.input.pamphet
14:163 Schaums and Axiom differ by a constant
14:164 Schaums and Axiom differ by a constant
14:165 Schaums and Axiom differ by a constant
14:166 Schaums and Axiom differ by a constant
14:167 Schaums and Axiom differ by a constant
14:168 Schaums and Axiom differ by a constant
14:169 Schaums and Axiom differ by a constant
14:170 Schaums and Axiom differ by a constant
14:171 Schaums and Axiom agree
14:172 Schaums and Axiom differ by a constant
14:173 Schaums and Axiom differ by a constant
14:174 Schaums and Axiom differ by a constant
14:175 Schaums and Axiom differ by a constant
14:176 Schaums and Axiom differ by a constant
14:177 Axiom cannot do this integration
14:178 Schaums and Axiom agree
14:179 Axiom cannot integrate this expression
14:180 Axiom cannot integrate this expression
14:181 Axiom cannot integrate this expression

schaum9.input.pamphet
14:182 Schaums and Axiom differ by a constant
14:183 Schaums and Axiom agree
14:184 Schaums and Axiom differ by a constant
14:185 Schaums and Axiom agree
14:186 Schaums and Axiom differ by a constant
14:187 Schaums and Axiom differ by a constant
14:188 Schaums and Axiom differ by a constant
14:189 Schaums and Axiom differ by a constant
14:190 Schaums and Axiom agree
14:191 Schaums and Axiom differ by a constant
14:192 Schaums and Axiom agree
14:193 Schaums and Axiom differ by a constant
14:194 Schaums and Axiom differ by a constant
14:195 Schaums and Axiom differ by a constant
14:196 Schaums and Axiom differ by a constant
14:197 Schaums and Axiom agree
14:198 Schaums and Axiom differ by a constant
14:199 Schaums and Axiom agree
14:200 Schaums and Axiom differ by a constant
14:201 Schaums and Axiom differ by a constant
14:202 Schaums and Axiom differ by a constant
14:203 Schaums and Axiom differ by a constant
14:204 Schaums and Axiom agree
14:205 Schaums and Axiom differ by a constant
14:206 Schaums and Axiom agree
14:207 Schaums and Axiom differ by a constant
14:208 Schaums and Axiom differ by a constant
14:209 Schaums and Axiom differ by a constant

schaum10.input.pamphet
14:210 Schaums and Axiom differ by a constant
14:xxx Schaums and Axiom agree
14:211 Schaums and Axiom differ by a constant
14:212 Schaums and Axiom agree
14:213 Schaums and Axiom differ by a constant
14:214 Schaums and Axiom differ by a constant
14:215 Schaums and Axiom differ by a constant
14:216 Schaums and Axiom differ by a constant
14:217 Schaums and Axiom agree
14:218 Schaums and Axiom differ by a constant
14:219 Schaums and Axiom agree
14:220 Schaums and Axiom differ by a constant
14:221 Schaums and Axiom differ by a constant
14:222 Schaums and Axiom differ by a constant
14:223 Schaums and Axiom differ by a constant
14:224 Schaums and Axiom agree
14:225 Schaums and Axiom differ by a constant
14:226 Schaums and Axiom agree
14:227 Schaums and Axiom differ by a constant
14:228 Schaums and Axiom differ by a constant
14:229 Schaums and Axiom differ by a constant
14:230 Schaums and Axiom differ by a constant
14:231 Schaums and Axiom agree
14:232 Schaums and Axiom differ by a constant
14:233 Schaums and Axiom agree
14:234 Schaums and Axiom differ by a constant
14:235 Schaums and Axiom differ by a constant
14:236 Schaums and Axiom differ by a constant

schaum11.input.pamphet
14:238 Schaums and Axiom agree
14:238 Schaums and Axiom differ by a constant
14:239 Schaums and Axiom agree
14:240 Schaums and Axiom differ by a constant
14:241 Schaums and Axiom differ by a constant
14:242 Schaums and Axiom agree
14:243 Schaums and Axiom differ by a constant
14:244 Schaums and Axiom agree
14:245 Schaums and Axiom differ by a constant
14:246 Schaums and Axiom agree
14:247 Schaums and Axiom differ by a constant
14:248 Schaums and Axiom differ by a constant
14:249 Schaums and Axiom agree
14:250 Schaums and Axiom differ by a constant
14:251 Schaums and Axiom agree
14:252 Schaums and Axiom differ by a constant
14:253 Schaums and Axiom agree
14:254 Schaums and Axiom differ by a constant
14:255 Schaums and Axiom differ by a constant
14:256 Schaums and Axiom agree
14:257 Schaums and Axiom differ by a constant
14:258 Schaums and Axiom agree
14:259 Schaums and Axiom differ by a constant
14:260 Schaums and Axiom agree
14:261 Schaums and Axiom differ by a constant
14:262 Schaums and Axiom differ by a constant
14:263 Schaums and Axiom agree
14:264 Schaums and Axiom differ by a constant

=========================================================================
diff --git a/changelog b/changelog
index 502aa1e..c10550c 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,11 @@
+20080418 tpd src/input/schaum11.input show Schaums-Axiom equivalence
+20080418 tpd src/input/schaum10.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum9.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum8.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum7.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum6.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum2.input fix Schaums-Axiom equivalence
+20080418 tpd src/input/schaum1.input fix Schaums-Axiom equivalence
20080417 tpd src/input/schaum10.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum9.input show Schaums-Axiom equivalence
20080417 tpd src/input/schaum8.input show Schaums-Axiom equivalence
diff --git a/src/input/schaum1.input.pamphlet b/src/input/schaum1.input.pamphlet
index 7e7e8c4..615d907 100644
--- a/src/input/schaum1.input.pamphlet
+++ b/src/input/schaum1.input.pamphlet
@@ -1201,7 +1201,7 @@ explog:=rule(%e^(n*log(x)) == x^n)
--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
--E

---S 94     14:80 Schaums and Axiom agree
+--S 94
dd:=explog cc
--R
--R                   n + 1                     n
@@ -1210,8 +1210,14 @@ dd:=explog cc
--R                        a n + a
--R                                                     Type: Expression
Integer
--E
+
+--S 95     14:80 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R   (6)  0
+--R                                                     Type: Expression
Integer
+--E
@
-The numerator is clearly zero but I cannot get Axiom to simplify it.

\section{\cite{1}:14.81~~~~~$\displaystyle \int{x(ax+b)^n~dx}$}
@@ -1221,7 +1227,7 @@ $$\int{x(ax+b)^n}=$$
<<*>>=
)clear all
---S 95
+--S 96
aa:=integrate(x*(a*x+b)^n,x)
--R
--R           2     2  2              2   n log(a x + b)
@@ -1232,7 +1238,7 @@ aa:=integrate(x*(a*x+b)^n,x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 96
+--S 97
bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2)
--R
--R                        n + 2                        n + 1
@@ -1243,7 +1249,7 @@
bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2)
--R                                                     Type: Expression
Integer
--E

---S 97
+--S 98
cc:=aa-bb
--R
--R   (3)
@@ -1258,7 +1264,7 @@ cc:=aa-bb
--R                                                     Type: Expression
Integer
--E

---S 98
+--S 99
explog:=rule(%e^(n*log(x)) == x^n)
--R
--R          n log(x)     n
@@ -1266,7 +1272,7 @@ explog:=rule(%e^(n*log(x)) == x^n)
--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
--E

---S 99     14:81 Schaums and Axiom agreement cannot be determined
+--S 100
dd:=explog cc
--R
--R   (5)
@@ -1280,6 +1286,13 @@ dd:=explog cc
--R     a n  + 3a n + 2a
--R                                                     Type: Expression
Integer
--E
+
+--S 101
+ee:=complexNormalize dd
+--R
+--R   (6)  0
+--R                                                     Type: Expression
Integer
+--E
@
\section{\cite{1}:14.82~~~~~$\displaystyle \int{x^2(ax+b)^n~dx}$}
@@ -1292,7 +1305,7 @@ $$<<*>>= )clear all ---S 100 +--S 102 aa:=integrate(x^2*(a*x+b)^n,x) --R --R (1) @@ -1304,7 +1317,7 @@ aa:=integrate(x^2*(a*x+b)^n,x) --R Type: Union(Expression Integer,...) --E ---S 101 +--S 103 bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1))/((n+1)*a^3) --R --R (2) @@ -1319,7 +1332,7 @@ bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1) --R Type: Expression Integer --E ---S 102 14:82 Schaums and Axiom agreement cannot be determined +--S 104 cc:=aa-bb --R --R (3) @@ -1339,6 +1352,39 @@ cc:=aa-bb --R a n + 6a n + 11a n + 6a --R Type: Expression Integer --E + +--S 105 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 106 +dd:=explog cc +--R +--R (5) +--R 2 n + 3 2 n + 2 +--R (- n - 3n - 2)(a x + b) + (2b n + 8b n + 6b)(a x + b) +--R + +--R 2 2 2 2 n + 1 +--R (- b n - 5b n - 6b )(a x + b) +--R + +--R 3 2 3 3 3 2 2 2 2 2 3 n +--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Expression Integer +--E + +--S 107 14:82 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.83~~~~~\displaystyle \int{x^m(ax+b)^n}~dx} @@ -1361,7 +1407,7 @@$$\int{x^m(ax+b)^n}
$$<<*>>= ---S 103 14:83 Axiom cannot do this integration +--S 108 14:83 Axiom cannot do this integration aa:=integrate(x^m*(a*x+b)^n,x) --R --R x diff --git a/src/input/schaum10.input.pamphlet b/src/input/schaum10.input.pamphlet index d0ba7d9..6149cc0 100644 --- a/src/input/schaum10.input.pamphlet +++ b/src/input/schaum10.input.pamphlet @@ -133,7 +133,7 @@ bb:=(x*sqrt(x^2-a^2))/2+a^2/2*log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 11 14:211 Axiom cannot simplify this expression +--S 11 cc:=aa-bb --R --R +-------+ +-------+ @@ -144,6 +144,16 @@ cc:=aa-bb --R Type: Expression Integer --E +--S 12 14:211 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(- a ) +--R (4) - ----------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.212~~~~~\displaystyle @@ -154,7 +164,7 @@$$
<<*>>=
)clear all

---S 12
+--S 13
aa:=integrate(x^3/sqrt(x^2-a^2),x)
--R
--R
@@ -168,7 +178,7 @@ aa:=integrate(x^3/sqrt(x^2-a^2),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 13
+--S 14
bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2)
--R
--R                   +-------+
@@ -179,7 +189,7 @@ bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2)
--R                                                     Type: Expression
Integer
--E

---S 14     14:212 Schaums and Axiom agree
+--S 15     14:212 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
@@ -194,7 +204,7 @@ $$<<*>>= )clear all ---S 15 +--S 16 aa:=integrate(1/(x*sqrt(x^2-a^2)),x) --R --R @@ -208,29 +218,134 @@ aa:=integrate(1/(x*sqrt(x^2-a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 16 -bb:=1/a*asec(abs(x/a)) +--S 17 +bb:=1/a*asec(x/a) --R ---R x ---R asec(abs(-)) ---R a ---R (2) ------------ ---R a +--R x +--R asec(-) +--R a +--R (2) ------- +--R a --R Type: Expression Integer --E ---S 17 14:213 Axiom cannot simplify this expression +--S 18 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R 2atan(--------------) - asec(abs(-)) ---R a a ---R (3) ------------------------------------ ---R a +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R a --R Type: Expression Integer --E + +--S 19 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 20 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x \|x - a - x +--R - 2%i log(------------------) + 4atan(--------------) - %pi +--R x a +--R (5) ----------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 21 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 22 +ee:=atanrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 23 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 24 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 25 14:213 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 2a +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.214~~~~~\displaystyle @@ -241,7 +356,7 @@$$
<<*>>=
)clear all

---S 18
+--S 26
aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x)
--R
--R
@@ -253,7 +368,7 @@ aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 19
+--S 27
bb:=sqrt(x^2-a^2)/(a^2*x)
--R
--R         +-------+
@@ -265,7 +380,7 @@ bb:=sqrt(x^2-a^2)/(a^2*x)
--R                                                     Type: Expression
Integer
--E

---S 20     14:214 Schaums and Axiom differ by a constant
+--S 28     14:214 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R         1
@@ -284,7 +399,7 @@ $$<<*>>= )clear all ---S 21 +--S 29 aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x) --R --R @@ -305,33 +420,143 @@ aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 22 -bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(abs(x/a)) +--S 30 +bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(x/a) --R ---R +-------+ ---R 2 x | 2 2 ---R x asec(abs(-)) + a\|x - a ---R a ---R (2) ---------------------------- ---R 3 2 ---R 2a x +--R +-------+ +--R | 2 2 2 x +--R a\|x - a + x asec(-) +--R a +--R (2) ----------------------- +--R 3 2 +--R 2a x --R Type: Expression Integer --E ---S 23 14:215 Axiom cannot simplify this expression +--S 31 cc:=aa-bb +--R --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R 2atan(--------------) - asec(abs(-)) ---R a a ---R (3) ------------------------------------ ---R 3 ---R 2a +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R 3 +--R 2a --R Type: Expression Integer --E +--S 32 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 33 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R - %i log(-----------------------) - asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 34 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 35 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 36 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 37 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 38 14:215 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + @ \section{\cite{1}:14.216~~~~~\displaystyle\int{\sqrt{x^2-a^2}}~dx} @@ -341,7 +566,7 @@$$
<<*>>=
)clear all

---S 24
+--S 39
aa:=integrate(sqrt(x^2-a^2),x)
--R
--R
@@ -360,7 +585,7 @@ aa:=integrate(sqrt(x^2-a^2),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 25
+--S 40
bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2))
--R
--R                 +-------+          +-------+
@@ -371,7 +596,7 @@ bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2))
--R                                                     Type: Expression
Integer
--E

---S 26     14:216 Axiom cannot simplify this expression
+--S 41
cc:=aa-bb
--R
--R               +-------+               +-------+
@@ -382,6 +607,15 @@ cc:=aa-bb
--R                                                     Type: Expression
Integer
--E

+--S 42     14:216 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R         2       2
+--R        a log(- a )
+--R   (4)  -----------
+--R             2
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$}
@@ -391,7 +625,7 @@ $$<<*>>= )clear all ---S 27 +--S 43 aa:=integrate(x*sqrt(x^2-a^2),x) --R --R @@ -405,7 +639,7 @@ aa:=integrate(x*sqrt(x^2-a^2),x) --R Type: Union(Expression Integer,...) --E ---S 28 +--S 44 bb:=(x^2-a^2)^(3/2)/3 --R --R +-------+ @@ -416,7 +650,7 @@ bb:=(x^2-a^2)^(3/2)/3 --R Type: Expression Integer --E ---S 29 14:217 Schaums and Axiom agree +--S 45 14:217 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -433,7 +667,7 @@$$
<<*>>=
)clear all

---S 30
+--S 46
aa:=integrate(x^2*sqrt(x^2-a^2),x)
--R
--R
@@ -452,7 +686,7 @@ aa:=integrate(x^2*sqrt(x^2-a^2),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 31
+--S 47
bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2))
--R
--R                 +-------+                    +-------+
@@ -463,7 +697,7 @@
bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2))
--R                                                     Type: Expression
Integer
--E

---S 32     14:218 Axiom cannot simplify this expression
+--S 48
cc:=aa-bb
--R
--R               +-------+               +-------+
@@ -473,6 +707,16 @@ cc:=aa-bb
--R                              8
--R                                                     Type: Expression
Integer
--E
+
+--S 49     14:218 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R         4       2
+--R        a log(- a )
+--R   (4)  -----------
+--R             8
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.219~~~~~$\displaystyle @@ -483,7 +727,7 @@ $$<<*>>= )clear all ---S 33 +--S 50 aa:=integrate(x^3*sqrt(x^2-a^2),x) --R --R @@ -501,7 +745,7 @@ aa:=integrate(x^3*sqrt(x^2-a^2),x) --R Type: Union(Expression Integer,...) --E ---S 34 +--S 51 bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3 --R --R +-------+ @@ -512,7 +756,7 @@ bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3 --R Type: Expression Integer --E ---S 35 14:219 Schaums and Axiom agree +--S 52 14:219 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -528,7 +772,7 @@$$ <<*>>= )clear all ---S 36 +--S 53 aa:=integrate(sqrt(x^2-a^2)/x,x) --R --R @@ -544,26 +788,129 @@ aa:=integrate(sqrt(x^2-a^2)/x,x) --R Type: Union(Expression Integer,...) --E ---S 37 -bb:=sqrt(x^2-a^2)-a*asec(abs(x/a)) +--S 54 +bb:=sqrt(x^2-a^2)-a*asec(x/a) --R ---R +-------+ ---R x | 2 2 ---R (2) - a asec(abs(-)) + \|x - a ---R a +--R +-------+ +--R | 2 2 x +--R (2) \|x - a - a asec(-) +--R a --R Type: Expression Integer --E ---S 38 14:220 Axiom cannot simplify this expression +--S 55 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R (3) - 2a atan(--------------) + a asec(abs(-)) ---R a a +--R \|x - a - x x +--R (3) - 2a atan(--------------) + a asec(-) +--R a a --R Type: Expression Integer --E + +--S 56 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 57 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R (5) %i a log(-----------------------) + a asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R Type: Expression Complex Integer +--E + +--S 58 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 59 +ee:=asecrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 2%i a log(------------------) + 2%i a log(-----------------------) + a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R -------------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 60 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i a log(\|x - a - x + %i a) + 2%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 2%i a log(x |------- + %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi +--R | 2 +--R \| x +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 61 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i a log(\|x - a + %i a) - 2%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 2%i a log(\|x - a - x - %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 62 14:220 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R a %pi +--R (10) ----- +--R 2 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.221~~~~~$\displaystyle
@@ -574,7 +921,7 @@ $$<<*>>= )clear all ---S 39 +--S 63 aa:=integrate(sqrt(x^2-a^2)/x^2,x) --R --R @@ -588,7 +935,7 @@ aa:=integrate(sqrt(x^2-a^2)/x^2,x) --R Type: Union(Expression Integer,...) --E ---S 40 +--S 64 bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -599,7 +946,7 @@ bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 41 14:221 Axiom cannot simplify this expression +--S 65 cc:=aa-bb --R --R +-------+ +-------+ @@ -608,6 +955,14 @@ cc:=aa-bb --R Type: Expression Integer --E +--S 66 14:221 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(- a ) - 1 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.222~~~~~\displaystyle @@ -619,7 +974,7 @@$$
<<*>>=
)clear all

---S 42
+--S 67
aa:=integrate(sqrt(x^2-a^2)/x^3,x)
--R
--R
@@ -640,31 +995,136 @@ aa:=integrate(sqrt(x^2-a^2)/x^3,x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 43
-bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(abs(x/a))
+--S 68
+bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(x/a)
--R
---R                           +-------+
---R         2         x       | 2    2
---R        x asec(abs(-)) - a\|x  - a
---R                   a
---R   (2)  ----------------------------
---R                        2
---R                    2a x
+--R            +-------+
+--R            | 2    2     2     x
+--R        - a\|x  - a   + x asec(-)
+--R                               a
+--R   (2)  -------------------------
+--R                      2
+--R                  2a x
--R                                                     Type: Expression
Integer
--E

---S 44     14:222 Axiom cannot simplify this expression
+--S 69
cc:=aa-bb
--R
--R               +-------+
--R               | 2    2
---R              \|x  - a   - x             x
---R        2atan(--------------) - asec(abs(-))
---R                     a                   a
---R   (3)  ------------------------------------
---R                         2a
+--R              \|x  - a   - x         x
+--R        2atan(--------------) - asec(-)
+--R                     a               a
+--R   (3)  -------------------------------
+--R                       2a
--R                                                     Type: Expression
Integer
--E
+
+--S 70
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R                             +------+
+--R                             | 2
+--R                             |x  - 1
+--R                           x |------  + %i
+--R                             |   2
+--R                            \|  x
+--R                   2%i log(---------------) + %pi
+--R                                  x
+--R   (4)  asec(x) == ------------------------------
+--R                                  2
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 71
+dd:=asecrule cc
+--R
+--R                    +-------+
+--R                    | 2    2
+--R                    |x  - a
+--R                  x |-------  + %i a           +-------+
+--R                    |    2                     | 2    2
+--R                   \|   x                     \|x  - a   - x
+--R        - 2%i log(------------------) + 4atan(--------------) - %pi
+--R                           x                         a
+--R   (5)  -----------------------------------------------------------
+--R                                     4a
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 72
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R                            - x + %i
+--R                     %i log(--------)
+--R                             x + %i
+--R   (6)  atan(x) == - ----------------
+--R                             2
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 73
+ee:=atanrule dd
+--R
+--R                    +-------+
+--R                    | 2    2
+--R                    |x  - a
+--R                  x |-------  + %i a               +-------+
+--R                    |    2                         | 2    2
+--R                   \|   x                       - \|x  - a   + x + %i a
+--R        - 2%i log(------------------) - 2%i log(-----------------------) -
%pi
+--R                           x                      +-------+
+--R                                                  | 2    2
+--R                                                 \|x  - a   - x + %i a
+--R   (7)
----------------------------------------------------------------------
+--R                                          4a
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 74
+ff:=expandLog ee
+--R
+--R   (8)
+--R                +-------+                        +-------+
+--R                | 2    2                         | 2    2
+--R       2%i log(\|x  - a   - x + %i a) - 2%i log(\|x  - a   - x - %i a)
+--R     +
+--R                   +-------+
+--R                   | 2    2
+--R                   |x  - a
+--R       - 2%i log(x |-------  + %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R                   |    2
+--R                  \|   x
+--R  /
+--R     4a
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 75
+gg:=rootSimp ff
+--R
+--R   (9)
+--R                  +-------+                    +-------+
+--R                  | 2    2                     | 2    2
+--R       - 2%i log(\|x  - a   + %i a) + 2%i log(\|x  - a   - x + %i a)
+--R     +
+--R                  +-------+
+--R                  | 2    2
+--R       - 2%i log(\|x  - a   - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi
+--R  /
+--R     4a
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 76     14:222 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R           %pi
+--R   (10)  - ---
+--R            4a
+--R                                             Type: Expression Complex
Integer
+--E
@

\section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$}
@@ -674,7 +1134,7 @@ $$<<*>>= )clear all ---S 45 +--S 77 aa:=integrate(1/(x^2-a^2)^(3/2),x) --R --R @@ -686,7 +1146,7 @@ aa:=integrate(1/(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 46 +--S 78 bb:=-x/(a^2*sqrt(x^2-a^2)) --R --R x @@ -697,7 +1157,7 @@ bb:=-x/(a^2*sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 47 14:223 Schaums and Axiom differ by a constant +--S 79 14:223 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 1 @@ -716,7 +1176,7 @@$$
<<*>>=
)clear all

---S 48
+--S 80
aa:=integrate(x/(x^2-a^2)^(3/2),x)
--R
--R
@@ -730,7 +1190,7 @@ aa:=integrate(x/(x^2-a^2)^(3/2),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 49
+--S 81
bb:=-1/sqrt(x^2-a^2)
--R
--R               1
@@ -741,7 +1201,7 @@ bb:=-1/sqrt(x^2-a^2)
--R                                                     Type: Expression
Integer
--E

---S 50     14:224 Schaums and Axiom agree
+--S 82     14:224 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
@@ -757,7 +1217,7 @@ $$<<*>>= )clear all ---S 51 +--S 83 aa:=integrate(x^2/(x^2-a^2)^(3/2),x) --R --R @@ -771,7 +1231,7 @@ aa:=integrate(x^2/(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 52 +--S 84 bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -784,7 +1244,7 @@ bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 53 14:225 Axiom cannot simplify this expression +--S 85 cc:=aa-bb --R --R +-------+ +-------+ @@ -792,6 +1252,14 @@ cc:=aa-bb --R (3) - log(\|x - a + x) - log(\|x - a - x) - 1 --R Type: Expression Integer --E + +--S 86 14:225 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(- a ) - 1 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.226~~~~~\displaystyle @@ -802,7 +1270,7 @@$$
<<*>>=
)clear all

---S 54
+--S 87
aa:=integrate(x^3/(x^2-a^2)^(3/2),x)
--R
--R
@@ -816,7 +1284,7 @@ aa:=integrate(x^3/(x^2-a^2)^(3/2),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 55
+--S 88
bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2)
--R
--R          2     2
@@ -828,7 +1296,7 @@ bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2)
--R                                                     Type: Expression
Integer
--E

---S 56     14:226 Schaums and Axiom agree
+--S 89     14:226 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
@@ -845,7 +1313,7 @@ $$<<*>>= )clear all ---S 57 +--S 90 aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x) --R --R @@ -861,33 +1329,143 @@ aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 58 -bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(abs(x/a)) +--S 91 +bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(x/a) --R ---R +-------+ ---R | 2 2 x ---R - \|x - a asec(abs(-)) - a ---R a ---R (2) ---------------------------- ---R +-------+ ---R 3 | 2 2 ---R a \|x - a +--R +-------+ +--R x | 2 2 +--R - asec(-)\|x - a - a +--R a +--R (2) ----------------------- +--R +-------+ +--R 3 | 2 2 +--R a \|x - a --R Type: Expression Integer --E ---S 59 14:227 Axiom cannot simplify this expression +--S 92 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R - 2atan(--------------) + asec(abs(-)) ---R a a ---R (3) -------------------------------------- ---R 3 ---R a +--R \|x - a - x x +--R - 2atan(--------------) + asec(-) +--R a a +--R (3) --------------------------------- +--R 3 +--R a --R Type: Expression Integer --E + +--S 93 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 94 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R %i log(-----------------------) + asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ----------------------------------------- +--R 3 +--R a +--R Type: Expression Complex Integer +--E + +--S 95 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 96 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 2%i log(------------------) + 2%i log(-----------------------) + %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) -------------------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 97 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a - x + %i a) + 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 2%i log(x |------- + %i a) - 2%i log(x) + 2%i log(- 1) + %pi +--R | 2 +--R \| x +--R / +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 98 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a + %i a) - 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 2%i log(\|x - a - x - %i a) - 2%i log(x) + 2%i log(- 1) + %pi +--R / +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 99 14:227 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) --- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + @ \section{\cite{1}:14.228~~~~~\displaystyle @@ -898,7 +1476,7 @@$$
<<*>>=
)clear all

---S 60
+--S 100
aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x)
--R
--R
@@ -910,7 +1488,7 @@ aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 61
+--S 101
bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2))
--R
--R              2    2
@@ -922,7 +1500,7 @@ bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2))
--R                                                     Type: Expression
Integer
--E

---S 62     14:228 Schaums and Axiom differ by a constant
+--S 102    14:228 Schaums and Axiom differ by a constant
cc:=aa-bb
--R
--R           2
@@ -944,7 +1522,7 @@ $$<<*>>= )clear all ---S 63 +--S 103 aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x) --R --R @@ -965,35 +1543,143 @@ aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 64 -bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(abs(x/a)) +--S 104 +bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(x/a) --R ---R +-------+ ---R 2 | 2 2 x 2 3 ---R - 3x \|x - a asec(abs(-)) - 3a x + a ---R a ---R (2) ---------------------------------------- ---R +-------+ ---R 5 2 | 2 2 ---R 2a x \|x - a +--R +-------+ +--R 2 x | 2 2 2 3 +--R - 3x asec(-)\|x - a - 3a x + a +--R a +--R (2) ----------------------------------- +--R +-------+ +--R 5 2 | 2 2 +--R 2a x \|x - a --R Type: Expression Integer --E ---S 65 14:229 Axiom cannot simplify this expression +--S 105 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R - 6atan(--------------) + 3asec(abs(-)) ---R a a ---R (3) --------------------------------------- ---R 5 ---R 2a +--R \|x - a - x x +--R - 6atan(--------------) + 3asec(-) +--R a a +--R (3) ---------------------------------- +--R 5 +--R 2a --R Type: Expression Integer --E -@ +--S 106 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 107 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R 3%i log(-----------------------) + 3asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 108 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 109 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 6%i log(------------------) + 6%i log(-----------------------) + 3%pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) --------------------------------------------------------------------- +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 110 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 6%i log(\|x - a - x + %i a) + 6%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 6%i log(x |------- + %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi +--R | 2 +--R \| x +--R / +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 111 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 6%i log(\|x - a + %i a) - 6%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 6%i log(\|x - a - x - %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi +--R / +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 112 14:229 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3%pi +--R (10) ---- +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E +@ \section{\cite{1}:14.230~~~~~\displaystyle\int{(x^2-a^2)^{3/2}}~dx}$$\int{(x^2-a^2)^{3/2}}=
\frac{x(x^2-a^2)^{3/2}}{4}-\frac{3a^2x\sqrt{x^2-a^2}}{8}+
@@ -1002,7 +1688,7 @@ $$<<*>>= )clear all ---S 66 +--S 113 aa:=integrate((x^2-a^2)^(3/2),x) --R --R @@ -1024,7 +1710,7 @@ aa:=integrate((x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 67 +--S 114 bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -1035,7 +1721,7 @@ bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 68 14:230 Axiom cannot simplify this expression +--S 115 cc:=aa-bb --R --R +-------+ +-------+ @@ -1046,6 +1732,16 @@ cc:=aa-bb --R Type: Expression Integer --E +--S 116 14:230 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R 3a log(- a ) +--R (4) - ------------ +--R 8 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.231~~~~~\displaystyle\int{x(x^2-a^2)^{3/2}}~dx} @@ -1053,7 +1749,7 @@$$\int{x(x^2-a^2)^{3/2}}=\frac{(x^2-a^2)^{5/2}}{5}$$<<*>>= )clear all ---S 69 +--S 117 aa:=integrate(x*(x^2-a^2)^(3/2),x) --R --R @@ -1071,7 +1767,7 @@ aa:=integrate(x*(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 70 +--S 118 bb:=(x^2-a^2)^(5/2)/5 --R --R +-------+ @@ -1082,7 +1778,7 @@ bb:=(x^2-a^2)^(5/2)/5 --R Type: Expression Integer --E ---S 71 14:231 Schaums and Axiom agree +--S 119 14:231 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -1099,7 +1795,7 @@$$
<<*>>=
)clear all

---S 72
+--S 120
aa:=integrate(x^2*(x^2-a^2)^(3/2),x)
--R
--R
@@ -1128,7 +1824,7 @@ aa:=integrate(x^2*(x^2-a^2)^(3/2),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 73
+--S 121

bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^6/16*log(x+sqrt(x^2-a^2))
--R
--R                +-------+                              +-------+
@@ -1139,7 +1835,7 @@
bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^
--R                                                     Type: Expression
Integer
--E

---S 74     14:232 Axiom cannot simplify this expression
+--S 122
cc:=aa-bb
--R
--R                 +-------+               +-------+
@@ -1149,6 +1845,16 @@ cc:=aa-bb
--R                               16
--R                                                     Type: Expression
Integer
--E
+
+--S 123    14:232 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R           6       2
+--R          a log(- a )
+--R   (4)  - -----------
+--R               16
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$}
@@ -1158,7 +1864,7 @@ $$<<*>>= )clear all ---S 75 +--S 124 aa:=integrate(x^3*(x^2-a^2)^(3/2),x) --R --R @@ -1188,7 +1894,7 @@ aa:=integrate(x^3*(x^2-a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 76 +--S 125 bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5 --R --R +-------+ @@ -1199,7 +1905,7 @@ bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5 --R Type: Expression Integer --E ---S 77 14:233 Schaums and Axiom agree +--S 126 14:233 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -1216,7 +1922,7 @@$$
<<*>>=
)clear all

---S 78
+--S 127
aa:=integrate((x^2-a^2)^(3/2)/x,x)
--R
--R
@@ -1237,28 +1943,135 @@ aa:=integrate((x^2-a^2)^(3/2)/x,x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 79
-bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(abs(x/a))
+--S 128
+bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(x/a)
--R
---R                                     +-------+
---R          3         x       2     2  | 2    2
---R        3a asec(abs(-)) + (x  - 4a )\|x  - a
---R                    a
---R   (2)  --------------------------------------
---R                           3
+--R                   +-------+
+--R          2     2  | 2    2      3     x
+--R        (x  - 4a )\|x  - a   + 3a asec(-)
+--R                                       a
+--R   (2)  ---------------------------------
+--R                        3
--R                                                     Type: Expression
Integer
--E

---S 80     14:234 Axiom cannot simplify this expression
+--S 129
cc:=aa-bb
--R
--R                 +-------+
--R                 | 2    2
---R          3     \|x  - a   - x     3         x
---R   (3)  2a atan(--------------) - a asec(abs(-))
---R                       a                     a
+--R          3     \|x  - a   - x     3     x
+--R   (3)  2a atan(--------------) - a asec(-)
+--R                       a                 a
--R                                                     Type: Expression
Integer
--E
+
+--S 130
+asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x))
+--R
+--R                             +------+
+--R                             | 2
+--R                             |x  - 1
+--R                           x |------  + %i
+--R                             |   2
+--R                            \|  x
+--R                   2%i log(---------------) + %pi
+--R                                  x
+--R   (4)  asec(x) == ------------------------------
+--R                                  2
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 131
+dd:=asecrule cc
+--R
+--R                      +-------+
+--R                      | 2    2
+--R                      |x  - a
+--R                    x |-------  + %i a             +-------+
+--R                      |    2                       | 2    2
+--R               3     \|   x                 3     \|x  - a   - x     3
+--R        - 2%i a log(------------------) + 4a atan(--------------) - a %pi
+--R                             x                           a
+--R   (5)  -----------------------------------------------------------------
+--R                                        2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 132
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R                            - x + %i
+--R                     %i log(--------)
+--R                             x + %i
+--R   (6)  atan(x) == - ----------------
+--R                             2
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 133
+ee:=atanrule dd
+--R
+--R   (7)
+--R                 +-------+
+--R                 | 2    2
+--R                 |x  - a
+--R               x |-------  + %i a                 +-------+
+--R                 |    2                           | 2    2
+--R          3     \|   x                    3    - \|x  - a   + x + %i a     3
+--R   - 2%i a log(------------------) - 2%i a log(-----------------------) - a
%pi
+--R                        x                        +-------+
+--R                                                 | 2    2
+--R                                                \|x  - a   - x + %i a
+--R
----------------------------------------------------------------------------
+--R                                         2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 134
+ff:=expandLog ee
+--R
+--R   (8)
+--R                  +-------+                          +-------+
+--R            3     | 2    2                     3     | 2    2
+--R       2%i a log(\|x  - a   - x + %i a) - 2%i a log(\|x  - a   - x - %i a)
+--R     +
+--R                     +-------+
+--R                     | 2    2
+--R              3      |x  - a                  3              3            3
+--R       - 2%i a log(x |-------  + %i a) + 2%i a log(x) - 2%i a log(- 1) - a
%pi
+--R                     |    2
+--R                    \|   x
+--R  /
+--R     2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 135
+gg:=rootSimp ff
+--R
+--R   (9)
+--R                    +-------+                      +-------+
+--R              3     | 2    2                 3     | 2    2
+--R       - 2%i a log(\|x  - a   + %i a) + 2%i a log(\|x  - a   - x + %i a)
+--R     +
+--R                  +-------+
+--R            3     | 2    2                     3              3            3
+--R     - 2%i a log(\|x  - a   - x - %i a) + 2%i a log(x) - 2%i a log(- 1) - a
%pi
+--R  /
+--R     2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 136    14:234 Schaums and Axiom differ by a constant
+hh:=complexNormalize gg
+--R
+--R            3
+--R           a %pi
+--R   (10)  - -----
+--R             2
+--R                                             Type: Expression Complex
Integer
+--E
@

\section{\cite{1}:14.235~~~~~$\displaystyle @@ -1270,7 +2083,7 @@ $$<<*>>= )clear all ---S 81 +--S 137 aa:=integrate((x^2-a^2)^{3/2}/x^2,x) --R --R @@ -1289,7 +2102,7 @@ aa:=integrate((x^2-a^2)^{3/2}/x^2,x) --R Type: Union(Expression Integer,...) --E ---S 82 +--S 138 bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2)) --R --R +-------+ +-------+ @@ -1300,7 +2113,7 @@ bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2)) --R Type: Expression Integer --E ---S 83 14:235 Axiom cannot simplify this expression +--S 139 cc:=aa-bb --R --R +-------+ +-------+ @@ -1310,6 +2123,17 @@ cc:=aa-bb --R 2 --R Type: Expression Integer --E + +--S 140 14:235 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 2 +--R 3a log(- a ) + 2a +--R (4) ------------------ +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.236~~~~~\displaystyle @@ -1321,7 +2145,7 @@$$ <<*>>= )clear all ---S 84 +--S 141 aa:=integrate((x^2-a^2)^(3/2)/x^3,x) --R --R @@ -1342,31 +2166,137 @@ aa:=integrate((x^2-a^2)^(3/2)/x^3,x) --R Type: Union(Expression Integer,...) --E ---S 85 -bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(abs(x/a)) +--S 142 +bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(x/a) --R ---R +-------+ ---R 2 x 2 2 | 2 2 ---R - 3a x asec(abs(-)) + (2x + a )\|x - a ---R a ---R (2) ------------------------------------------ ---R 2 ---R 2x +--R +-------+ +--R 2 2 | 2 2 2 x +--R (2x + a )\|x - a - 3a x asec(-) +--R a +--R (2) ----------------------------------- +--R 2 +--R 2x --R Type: Expression Integer --E ---S 86 14:236 Axiom cannot simplify this expression +--S 143 cc:=aa-bb --R --R +-------+ --R | 2 2 ---R \|x - a - x x ---R - 6a atan(--------------) + 3a asec(abs(-)) ---R a a ---R (3) ------------------------------------------- ---R 2 +--R \|x - a - x x +--R - 6a atan(--------------) + 3a asec(-) +--R a a +--R (3) -------------------------------------- +--R 2 --R Type: Expression Integer --E + +--S 144 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 145 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R 3%i a log(-----------------------) + 3a asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ----------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 146 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 147 +ee:=asecrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 6%i a log(------------------) + 6%i a log(-----------------------) + 3a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R --------------------------------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 148 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 6%i a log(\|x - a - x + %i a) + 6%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 6%i a log(x |------- + %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a %pi +--R | 2 +--R \| x +--R / +--R 4 +--R Type: Expression Complex Integer +--E + +--S 149 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 6%i a log(\|x - a + %i a) - 6%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 6%i a log(\|x - a - x - %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a %pi +--R / +--R 4 +--R Type: Expression Complex Integer +--E + +--S 150 14:236 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3a %pi +--R (10) ------ +--R 4 +--R Type: Expression Complex Integer +--E + )spool )lisp (bye) @ diff --git a/src/input/schaum11.input.pamphlet b/src/input/schaum11.input.pamphlet index 6cd59a6..fc3117e 100644 --- a/src/input/schaum11.input.pamphlet +++ b/src/input/schaum11.input.pamphlet @@ -15,7 +15,7 @@ $$\int{\frac{1}{\sqrt{a^2-x^2}}}=\ln\left(x+\sqrt{a^2-x^2}\right)$$ )set message auto off )clear all ---S 1 of 28 +--S 1 aa:=integrate(1/(sqrt(a^2-x^2)),x) --R --R @@ -26,6 +26,96 @@ aa:=integrate(1/(sqrt(a^2-x^2)),x) --R x --R Type: Union(Expression Integer,...) --E + +--S 2 +bb:=asin(x/a) +--R +--R x +--R (2) asin(-) +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) - 2atan(----------------) - asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 4 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 5 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R - \|- x + a + %i x + a x +--R (5) %i log(-------------------------) - asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 6 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 7 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) - %i log(--------------------) + %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 8 +ff:=rootSimp ee +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i\|x - a - %i x - \|x - a + x - %i a +--R (8) - %i log(-------------------) + %i log(-----------------------) +--R a +-------+ +--R | 2 2 +--R \|x - a + x + %i a +--R Type: Expression Complex Integer +--E + +--S 9 14:238 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (9) 0 +--R Type: Expression Complex Integer +--E + @ \section{\cite{1}:14.238~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{a^2-x^2}}}$} @@ -33,7 +123,7 @@ $$\int{\frac{x}{\sqrt{a^2-x^2}}}=\sqrt{a^2-x^2}$$ <<*>>= )clear all ---S 2 of 28 +--S 10 aa:=integrate(x/(sqrt(a^2-x^2)),x) --R --R @@ -45,6 +135,22 @@ aa:=integrate(x/(sqrt(a^2-x^2)),x) --R \|- x + a - a --R Type: Union(Expression Integer,...) --E + +--S 11 +bb:=-sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 +--R (2) - \|- x + a +--R Type: Expression Integer +--E + +--S 12 14:238 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R (3) - a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.239~~~~~$\displaystyle
@@ -55,7 +161,7 @@ $$<<*>>= )clear all ---S 3 of 28 +--S 13 aa:=integrate(x^2/sqrt(a^2-x^2),x) --R --R @@ -75,6 +181,127 @@ aa:=integrate(x^2/sqrt(a^2-x^2),x) --R 4a\|- x + a + 2x - 4a --R Type: Union(Expression Integer,...) --E + +--S 14 +bb:=-(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a) +--R +--R +---------+ +--R | 2 2 2 x +--R - x\|- x + a + a asin(-) +--R a +--R (2) --------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 16 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 17 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 2 - \|- x + a + %i x + a 2 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 18 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 19 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 20 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 21 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 22 14:239 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E + @ \section{\cite{1}:14.240~~~~~\displaystyle @@ -85,7 +312,7 @@$$
<<*>>=
)clear all

---S 4 of 28
+--S 23
aa:=integrate(x^3/sqrt(a^2-x^2),x)
--R
--R
@@ -98,6 +325,27 @@ aa:=integrate(x^3/sqrt(a^2-x^2),x)
--R        (3x  - 12a )\|- x  + a   - 9a x  + 12a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 24
+bb:=(a^2-x^2)^(3/2)/3-a^2*sqrt(a^2-x^2)
+--R
+--R                     +---------+
+--R            2     2  |   2    2
+--R        (- x  - 2a )\|- x  + a
+--R   (2)  ------------------------
+--R                    3
+--R                                                     Type: Expression
Integer
+--E
+
+--S 25     14:240 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R            3
+--R          2a
+--R   (3)  - ---
+--R           3
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.241~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{a^2-x^2}}}$}
@@ -107,7 +355,7 @@ $$<<*>>= )clear all ---S 5 of 28 +--S 26 aa:=integrate(1/(x*sqrt(a^2-x^2)),x) --R --R @@ -120,6 +368,66 @@ aa:=integrate(1/(x*sqrt(a^2-x^2)),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 27 +bb:=-1/a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a + a +--R log(----------------) +--R x +--R (2) - --------------------- +--R a +--R Type: Expression Integer +--E + +--S 28 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) +--R x x +--R (3) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 29 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) +--R (4) ------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 30 +ee:=complexNormalize dd +--R +--R x +--R 2log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 31 14:241 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 2log(\|- 1 ) +--R (6) ------------ +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.242~~~~~\displaystyle @@ -130,7 +438,7 @@$$
<<*>>=
)clear all

---S 6 of 28
+--S 32
aa:=integrate(1/(x^2*sqrt(a^2-x^2)),x)
--R
--R
@@ -143,6 +451,25 @@ aa:=integrate(1/(x^2*sqrt(a^2-x^2)),x)
--R         a x\|- x  + a   - a x
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 33
+bb:=-sqrt(a^2-x^2)/(a^2*x)
+--R
+--R           +---------+
+--R           |   2    2
+--R          \|- x  + a
+--R   (2)  - ------------
+--R                2
+--R               a x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 34     14:242 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R   (3)  0
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.243~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{a^2-x^2}}}$}
@@ -153,7 +480,7 @@ $$<<*>>= )clear all ---S 7 of 28 +--S 35 aa:=integrate(1/(x^3*sqrt(a^2-x^2)),x) --R --R @@ -173,6 +500,71 @@ aa:=integrate(1/(x^3*sqrt(a^2-x^2)),x) --R 4a x \|- x + a + 2a x - 4a x --R Type: Union(Expression Integer,...) --E + +--S 36 +bb:=-sqrt(a^2-x^2)/(2*a^2*x^2)-1/(2*a^3)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a | 2 2 +--R - x log(----------------) - a\|- x + a +--R x +--R (2) ----------------------------------------- +--R 3 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 37 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) +--R x x +--R (3) --------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 38 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) +--R (4) ------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 39 +ee:=complexNormalize dd +--R +--R x +--R log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) - ------------ +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 40 14:243 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R log(\|- 1 ) +--R (6) ----------- +--R 3 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.244~~~~~\displaystyle\int{\sqrt{a^2-x^2}}~dx} @@ -182,7 +574,7 @@$$
<<*>>=
)clear all

---S 8 of 28
+--S 41
aa:=integrate(sqrt(a^2-x^2),x)
--R
--R
@@ -202,6 +594,127 @@ aa:=integrate(sqrt(a^2-x^2),x)
--R     4a\|- x  + a   + 2x  - 4a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 42
+bb:=(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a)
+--R
+--R          +---------+
+--R          |   2    2     2     x
+--R        x\|- x  + a   + a asin(-)
+--R                               a
+--R   (2)  -------------------------
+--R                    2
+--R                                                     Type: Expression
Integer
+--E
+
+--S 43
+cc:=aa-bb
+--R
+--R                   +---------+
+--R                   |   2    2
+--R            2     \|- x  + a   - a     2     x
+--R        - 2a atan(----------------) - a asin(-)
+--R                          x                  a
+--R   (3)  ---------------------------------------
+--R                           2
+--R                                                     Type: Expression
Integer
+--E
+
+--S 44
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R                           +--------+
+--R                           |   2
+--R   (4)  asin(x) == %i log(\|- x  + 1  - %i x)
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 45
+dd:=asinrule cc
+--R
+--R                     +---------+
+--R                     |   2    2
+--R                     |- x  + a
+--R                   a |---------  - %i x             +---------+
+--R                     |     2                        |   2    2
+--R              2     \|    a                  2     \|- x  + a   - a
+--R        - %i a log(--------------------) - 2a atan(----------------)
+--R                             a                             x
+--R   (5)  ------------------------------------------------------------
+--R                                      2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 46
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R                            - x + %i
+--R                     %i log(--------)
+--R                             x + %i
+--R   (6)  atan(x) == - ----------------
+--R                             2
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 47
+ee:=atanrule dd
+--R
+--R                     +---------+
+--R                     |   2    2
+--R                     |- x  + a
+--R                   a |---------  - %i x                +---------+
+--R                     |     2                           |   2    2
+--R              2     \|    a                    2    - \|- x  + a   + %i x +
a
+--R        - %i a log(--------------------) + %i a
log(-------------------------)
+--R                             a                        +---------+
+--R                                                      |   2    2
+--R                                                     \|- x  + a   + %i x - a
+--R   (7)
----------------------------------------------------------------------
+--R                                           2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 48
+ff:=expandLog ee
+--R
+--R   (8)
+--R                    +---------+
+--R                    |   2    2                      +---------+
+--R             2      |- x  + a                 2     |   2    2
+--R       - %i a log(a |---------  - %i x) - %i a log(\|- x  + a   + %i x - a)
+--R                    |     2
+--R                   \|    a
+--R     +
+--R                 +---------+
+--R           2     |   2    2                    2             2
+--R       %i a log(\|- x  + a   - %i x - a) + %i a log(a) + %i a log(- 1)
+--R  /
+--R     2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 49
+gg:=rootSimp ff
+--R
+--R   (9)
+--R                     +-------+                           +-------+
+--R             2       | 2    2                    2       | 2    2
+--R       - %i a log(%i\|x  - a   + %i x - a) - %i a log(%i\|x  - a   - %i x)
+--R     +
+--R                   +-------+
+--R           2       | 2    2                    2             2
+--R       %i a log(%i\|x  - a   - %i x - a) + %i a log(a) + %i a log(- 1)
+--R  /
+--R     2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 50     14:244 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R   (10)  0
+--R                                             Type: Expression Complex
Integer
+--E
@

\section{\cite{1}:14.245~~~~~$\displaystyle\int{x\sqrt{a^2-x^2}}~dx$}
@@ -211,7 +724,7 @@ $$<<*>>= )clear all ---S 9 of 28 +--S 51 aa:=integrate(x*sqrt(a^2-x^2),x) --R --R @@ -224,6 +737,27 @@ aa:=integrate(x*sqrt(a^2-x^2),x) --R (3x - 12a )\|- x + a - 9a x + 12a --R Type: Union(Expression Integer,...) --E + +--S 52 +bb:=-(a^2-x^2)^(3/2)/3 +--R +--R +---------+ +--R 2 2 | 2 2 +--R (x - a )\|- x + a +--R (2) --------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 53 14:245 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 3 +--R a +--R (3) - -- +--R 3 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.246~~~~~\displaystyle @@ -235,7 +769,7 @@$$
<<*>>=
)clear all

---S 10 of 28
+--S 54
aa:=integrate(x^2*sqrt(a^2-x^2),x)
--R
--R
@@ -259,6 +793,126 @@ aa:=integrate(x^2*sqrt(a^2-x^2),x)
--R     (32a x  - 64a )\|- x  + a   + 8x  - 64a x  + 64a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 55
+bb:=-(x*(a^2-x^2)^(3/2))/4+(a^2*x*sqrt(a^2-x^2))/8+a^4/8*asin(x/a)
+--R
+--R                    +---------+
+--R           3    2   |   2    2     4     x
+--R        (2x  - a x)\|- x  + a   + a asin(-)
+--R                                         a
+--R   (2)  -----------------------------------
+--R                         8
+--R                                                     Type: Expression
Integer
+--E
+
+--S 56
+cc:=aa-bb
+--R
+--R                   +---------+
+--R                   |   2    2
+--R            4     \|- x  + a   - a     4     x
+--R        - 2a atan(----------------) - a asin(-)
+--R                          x                  a
+--R   (3)  ---------------------------------------
+--R                           8
+--R                                                     Type: Expression
Integer
+--E
+
+--S 57
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R                            - x + %i
+--R                     %i log(--------)
+--R                             x + %i
+--R   (4)  atan(x) == - ----------------
+--R                             2
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 58
+dd:=atanrule cc
+--R
+--R                    +---------+
+--R                    |   2    2
+--R            4    - \|- x  + a   + %i x + a     4     x
+--R        %i a log(-------------------------) - a asin(-)
+--R                   +---------+                       a
+--R                   |   2    2
+--R                  \|- x  + a   + %i x - a
+--R   (5)  -----------------------------------------------
+--R                               8
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 59
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R                           +--------+
+--R                           |   2
+--R   (6)  asin(x) == %i log(\|- x  + 1  - %i x)
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 60
+ee:=asinrule dd
+--R
+--R                     +---------+
+--R                     |   2    2
+--R                     |- x  + a
+--R                   a |---------  - %i x                +---------+
+--R                     |     2                           |   2    2
+--R              4     \|    a                    4    - \|- x  + a   + %i x +
a
+--R        - %i a log(--------------------) + %i a
log(-------------------------)
+--R                             a                        +---------+
+--R                                                      |   2    2
+--R                                                     \|- x  + a   + %i x - a
+--R   (7)
----------------------------------------------------------------------
+--R                                           8
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 61
+ff:=expandLog ee
+--R
+--R   (8)
+--R                    +---------+
+--R                    |   2    2                      +---------+
+--R             4      |- x  + a                 4     |   2    2
+--R       - %i a log(a |---------  - %i x) - %i a log(\|- x  + a   + %i x - a)
+--R                    |     2
+--R                   \|    a
+--R     +
+--R                 +---------+
+--R           4     |   2    2                    4             4
+--R       %i a log(\|- x  + a   - %i x - a) + %i a log(a) + %i a log(- 1)
+--R  /
+--R     8
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 62
+gg:=rootSimp ff
+--R
+--R   (9)
+--R                     +-------+                           +-------+
+--R             4       | 2    2                    4       | 2    2
+--R       - %i a log(%i\|x  - a   + %i x - a) - %i a log(%i\|x  - a   - %i x)
+--R     +
+--R                   +-------+
+--R           4       | 2    2                    4             4
+--R       %i a log(%i\|x  - a   - %i x - a) + %i a log(a) + %i a log(- 1)
+--R  /
+--R     8
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 63     14:246 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R   (10)  0
+--R                                             Type: Expression Complex
Integer
+--E
@

\section{\cite{1}:14.247~~~~~$\displaystyle @@ -269,7 +923,7 @@ $$<<*>>= )clear all ---S 11 of 28 +--S 64 aa:=integrate(x^3*sqrt(a^2-x^2),x) --R --R @@ -283,6 +937,27 @@ aa:=integrate(x^3*sqrt(a^2-x^2),x) --R (15x - 180a x + 240a )\|- x + a - 75a x + 300a x - 240a --R Type: Union(Expression Integer,...) --E + +--S 65 +bb:=(a^2-x^2)^(5/2)/5-(a^2*(a^2-x^2)^(3/2))/3 +--R +--R +---------+ +--R 4 2 2 4 | 2 2 +--R (3x - a x - 2a )\|- x + a +--R (2) ------------------------------ +--R 15 +--R Type: Expression Integer +--E + +--S 66 14:247 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 5 +--R 2a +--R (3) - --- +--R 15 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.248~~~~~\displaystyle @@ -293,7 +968,7 @@$$ <<*>>= )clear all ---S 12 of 28 +--S 67 aa:=integrate(sqrt(a^2-x^2)/x,x) --R --R @@ -308,6 +983,56 @@ aa:=integrate(sqrt(a^2-x^2)/x,x) --R \|- x + a - a --R Type: Union(Expression Integer,...) --E + +--S 68 +bb:=sqrt(a^2-x^2)-a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R \|- x + a + a | 2 2 +--R (2) - a log(----------------) + \|- x + a +--R x +--R Type: Expression Integer +--E + +--S 69 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R (3) a log(----------------) + a log(----------------) + a +--R x x +--R Type: Expression Integer +--E + +--S 70 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R (4) a log(\|- x + a + a) + a log(\|- x + a - a) - 2a log(x) + a +--R Type: Expression Integer +--E + +--S 71 +ee:=complexNormalize dd +--R +--R x +--R (5) - 2a log(-------) + a +--R +----+ +--R | 2 +--R \|- x +--R Type: Expression Integer +--E + +--S 72 14:248 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R (6) 2a log(\|- 1 ) + a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.249~~~~~$\displaystyle
@@ -318,7 +1043,7 @@ $$<<*>>= )clear all ---S 13 of 28 +--S 73 aa:=integrate(sqrt(a^2-x^2)/x^2,x) --R --R @@ -334,6 +1059,117 @@ aa:=integrate(sqrt(a^2-x^2)/x^2,x) --R x\|- x + a - a x --R Type: Union(Expression Integer,...) --E + +--S 74 +bb:=-sqrt(a^2-x^2)/x-asin(x/a) +--R +--R +---------+ +--R | 2 2 x +--R - \|- x + a - x asin(-) +--R a +--R (2) -------------------------- +--R x +--R Type: Expression Integer +--E + +--S 75 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) 2atan(----------------) + asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 76 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 77 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a \|- x + a - a +--R (5) %i log(--------------------) + 2atan(----------------) +--R a x +--R Type: Expression Complex Integer +--E + +--S 78 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 79 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) %i log(--------------------) - %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 80 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R |- x + a | 2 2 +--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R | 2 2 +--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 81 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 82 14:249 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.250~~~~~\displaystyle @@ -345,7 +1181,7 @@$$
<<*>>=
)clear all

---S 14 of 28
+--S 83
aa:=integrate(sqrt(a^2-x^2)/x^3,x)
--R
--R
@@ -365,6 +1201,67 @@ aa:=integrate(sqrt(a^2-x^2)/x^3,x)
--R     4a x \|- x  + a   + 2a x  - 4a x
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 84
+bb:=-sqrt(a^2-x^2)/(2*x^2)+1/(2*a)*log((a+sqrt(a^2-x^2))/x)
+--R
+--R               +---------+
+--R               |   2    2           +---------+
+--R         2    \|- x  + a   + a      |   2    2
+--R        x log(----------------) - a\|- x  + a
+--R                      x
+--R   (2)  ---------------------------------------
+--R                             2
+--R                         2a x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 85
+cc:=aa-bb
+--R
+--R               +---------+             +---------+
+--R               |   2    2              |   2    2
+--R              \|- x  + a   + a        \|- x  + a   - a
+--R        - log(----------------) - log(----------------)
+--R                      x                       x
+--R   (3)  -----------------------------------------------
+--R                               2a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 86
+dd:=expandLog cc
+--R
+--R               +---------+             +---------+
+--R               |   2    2              |   2    2
+--R        - log(\|- x  + a   + a) - log(\|- x  + a   - a) + 2log(x)
+--R   (4)  ---------------------------------------------------------
+--R                                    2a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 87
+ee:=complexNormalize dd
+--R
+--R               x
+--R        log(-------)
+--R             +----+
+--R             |   2
+--R            \|- x
+--R   (5)  ------------
+--R              a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 88     14:250 Schaums and Axiom differ by a constant
+ff:=rootSimp ee
+--R
+--R               +---+
+--R          log(\|- 1 )
+--R   (6)  - -----------
+--R               a
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.251~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^{3/2}}}$}
@@ -374,7 +1271,7 @@ $$<<*>>= )clear all ---S 15 of 28 +--S 89 aa:=integrate(1/(a^2-x^2)^(3/2),x) --R --R @@ -387,6 +1284,24 @@ aa:=integrate(1/(a^2-x^2)^(3/2),x) --R a \|- x + a + a x - a --R Type: Union(Expression Integer,...) --E + +--S 90 +bb:=x/(a^2*sqrt(a^2-x^2)) +--R +--R x +--R (2) -------------- +--R +---------+ +--R 2 | 2 2 +--R a \|- x + a +--R Type: Expression Integer +--E + +--S 91 14:251 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.252~~~~~\displaystyle @@ -397,7 +1312,7 @@$$
<<*>>=
)clear all

---S 16 of 28
+--S 92
aa:=integrate(x/(a^2-x^2)^(3/2),x)
--R
--R
@@ -409,6 +1324,26 @@ aa:=integrate(x/(a^2-x^2)^(3/2),x)
--R        a \|- x  + a   + a x  - a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 93
+bb:=1/sqrt(a^2-x^2)
+--R
+--R              1
+--R   (2)  ------------
+--R         +---------+
+--R         |   2    2
+--R        \|- x  + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 94     14:252 Schaums and Axiom differ by a constant
+cc:=aa-bb
+--R
+--R        1
+--R   (3)  -
+--R        a
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.253~~~~~$\displaystyle @@ -419,7 +1354,7 @@ $$<<*>>= )clear all ---S 17 of 28 +--S 95 aa:=integrate(x^2/(a^2-x^2)^(3/2),x) --R --R @@ -435,6 +1370,118 @@ aa:=integrate(x^2/(a^2-x^2)^(3/2),x) --R a\|- x + a + x - a --R Type: Union(Expression Integer,...) --E + +--S 96 +bb:=x/sqrt(a^2-x^2)-asin(x/a) +--R +--R +---------+ +--R x | 2 2 +--R - asin(-)\|- x + a + x +--R a +--R (2) ------------------------- +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 97 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) 2atan(----------------) + asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 98 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 99 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R - \|- x + a + %i x + a x +--R (5) - %i log(-------------------------) + asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 100 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 101 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) %i log(--------------------) - %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 102 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R |- x + a | 2 2 +--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R | 2 2 +--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 103 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 104 14:253 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.254~~~~~\displaystyle @@ -445,7 +1492,7 @@$$ <<*>>= )clear all ---S 18 of 28 +--S 105 aa:=integrate(x^3/(a^2-x^2)^(3/2),x) --R --R @@ -457,6 +1504,25 @@ aa:=integrate(x^3/(a^2-x^2)^(3/2),x) --R (x - 2a )\|- x + a - 2a x + 2a --R Type: Union(Expression Integer,...) --E + +--S 106 +bb:=sqrt(a^2-x^2)+a^2/sqrt(a^2-x^2) +--R +--R 2 2 +--R - x + 2a +--R (2) ------------ +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 107 14:254 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R (3) 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.255~~~~~$\displaystyle
@@ -468,7 +1534,7 @@ $$<<*>>= )clear all ---S 19 of 28 +--S 108 aa:=integrate(1/(x*(a^2-x^2)^(3/2)),x) --R --R @@ -483,6 +1549,72 @@ aa:=integrate(1/(x*(a^2-x^2)^(3/2)),x) --R a \|- x + a + a x - a --R Type: Union(Expression Integer,...) --E + +--S 109 +bb:=1/(a^2*sqrt(a^2-x^2))-1/a^3*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 \|- x + a + a +--R - \|- x + a log(----------------) + a +--R x +--R (2) --------------------------------------- +--R +---------+ +--R 3 | 2 2 +--R a \|- x + a +--R Type: Expression Integer +--E + +--S 110 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) + 1 +--R x x +--R (3) ------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 111 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) + 1 +--R (4) ----------------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 112 +ee:=complexNormalize dd +--R +--R x +--R - 2log(-------) + 1 +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 113 14:255 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 2log(\|- 1 ) + 1 +--R (6) ---------------- +--R 3 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.256~~~~~\displaystyle @@ -493,7 +1625,7 @@$$
<<*>>=
)clear all

---S 20 of 28
+--S 114
aa:=integrate(1/(x^2*(a^2-x^2)^(3/2)),x)
--R
--R
@@ -506,6 +1638,25 @@ aa:=integrate(1/(x^2*(a^2-x^2)^(3/2)),x)
--R           (a x  - 2a x)\|- x  + a   - 2a x  + 2a x
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 115
+bb:=-sqrt(a^2-x^2)/(a^4*x)+x/(a^4*sqrt(a^2-x^2))
+--R
+--R              2    2
+--R            2x  - a
+--R   (2)  ---------------
+--R            +---------+
+--R         4  |   2    2
+--R        a x\|- x  + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 116    14:256 Schaums and Axiom agree
+cc:=aa-bb
+--R
+--R   (3)  0
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.257~~~~~$\displaystyle @@ -518,7 +1669,7 @@ $$<<*>>= )clear all ---S 21 of 28 +--S 117 aa:=integrate(1/(x^3*(a^2-x^2)^(3/2)),x) --R --R @@ -542,6 +1693,72 @@ aa:=integrate(1/(x^3*(a^2-x^2)^(3/2)),x) --R (6a x - 8a x )\|- x + a + 2a x - 10a x + 8a x --R Type: Union(Expression Integer,...) --E + +--S 118 +bb:=-1/(2*a^2*x^2*sqrt(a^2-x^2))+3/(2*a^4*sqrt(a^2-x^2))-3/(2*a^5)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 \|- x + a + a 2 3 +--R - 3x \|- x + a log(----------------) + 3a x - a +--R x +--R (2) --------------------------------------------------- +--R +---------+ +--R 5 2 | 2 2 +--R 2a x \|- x + a +--R Type: Expression Integer +--E + +--S 119 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R 3log(----------------) + 3log(----------------) + 2 +--R x x +--R (3) --------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 120 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R 3log(\|- x + a + a) + 3log(\|- x + a - a) - 6log(x) + 2 +--R (4) ------------------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 121 +ee:=complexNormalize dd +--R +--R x +--R - 3log(-------) + 1 +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------------- +--R 5 +--R a +--R Type: Expression Integer +--E + +--S 122 14:257 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 3log(\|- 1 ) + 1 +--R (6) ---------------- +--R 5 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.258~~~~~\displaystyle\int{(a^2-x^2)^{3/2}}~dx} @@ -552,7 +1769,7 @@$$ <<*>>= )clear all ---S 22 of 28 +--S 123 aa:=integrate((a^2-x^2)^(3/2),x) --R --R @@ -579,6 +1796,128 @@ aa:=integrate((a^2-x^2)^(3/2),x) --R (32a x - 64a )\|- x + a + 8x - 64a x + 64a --R Type: Union(Expression Integer,...) --E + +--S 124 +bb:=(x*(a^2-x^2)^(3/2))/4+(3*a^2*x*sqrt(a^2-x^2))/8+3/8*a^4*asin(x/a) +--R +--R +---------+ +--R 3 2 | 2 2 4 x +--R (- 2x + 5a x)\|- x + a + 3a asin(-) +--R a +--R (2) --------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 125 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 4 \|- x + a - a 4 x +--R - 6a atan(----------------) - 3a asin(-) +--R x a +--R (3) ---------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 126 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 127 +ee:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 \|- x + a - a +--R - 3%i a log(--------------------) - 6a atan(----------------) +--R a x +--R (5) ------------------------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 128 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 129 +ff:=atanrule ee +--R +--R (7) +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 - \|- x + a + %i x + a +--R - 3%i a log(--------------------) + 3%i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R ------------------------------------------------------------------------ +--R 8 +--R Type: Expression Complex Integer +--E + +--S 130 +gg:=expandLog ff +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 4 |- x + a 4 | 2 2 +--R - 3%i a log(a |--------- - %i x) - 3%i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 4 | 2 2 4 4 +--R 3%i a log(\|- x + a - %i x - a) + 3%i a log(a) + 3%i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 131 +hh:=rootSimp gg +--R +--R (9) +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - 3%i a log(%i\|x - a + %i x - a) - 3%i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 4 | 2 2 4 4 +--R 3%i a log(%i\|x - a - %i x - a) + 3%i a log(a) + 3%i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 132 14:258 Schaums and Axiom agree +ii:=complexNormalize hh +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.259~~~~~$\displaystyle\int{x(a^2-x^2)^{3/2}}~dx$} @@ -586,7 +1925,7 @@ $$\int{x(a^2-x^2)^{3/2}}=\frac{(a^2-x^2)^{5/2}}{5}$$ <<*>>= )clear all ---S 23 of 28 +--S 133 aa:=integrate(x*(a^2-x^2)^(3/2),x) --R --R @@ -603,6 +1942,27 @@ aa:=integrate(x*(a^2-x^2)^(3/2),x) --R (5x - 60a x + 80a )\|- x + a - 25a x + 100a x - 80a --R Type: Union(Expression Integer,...) --E + +--S 134 +bb:=-(a^2-x^2)^(5/2)/5 +--R +--R +---------+ +--R 4 2 2 4 | 2 2 +--R (- x + 2a x - a )\|- x + a +--R (2) ------------------------------- +--R 5 +--R Type: Expression Integer +--E + +--S 135 14:259 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 5 +--R a +--R (3) - -- +--R 5 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.260~~~~~$\displaystyle\int{x^2(a^2-x^2)^{3/2}}~dx$} @@ -614,7 +1974,7 @@ $$<<*>>= )clear all ---S 24 of 28 +--S 136 aa:=integrate(x^2*(a^2-x^2)^(3/2),x) --R --R @@ -647,6 +2007,126 @@ aa:=integrate(x^2*(a^2-x^2)^(3/2),x) --R - 1536a --R Type: Union(Expression Integer,...) --E + +--S 137 +bb:=-(x*(a^2-x^2)^(5/2))/6+(a^2*x*(a^2-x^2)^(3/2))/24+(a^4*x*sqrt(a^2-x^2))/16+a^6/16*asin(x/a) +--R +--R +---------+ +--R 5 2 3 4 | 2 2 6 x +--R (- 8x + 14a x - 3a x)\|- x + a + 3a asin(-) +--R a +--R (2) ------------------------------------------------ +--R 48 +--R Type: Expression Integer +--E + +--S 138 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 6 \|- x + a - a 6 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 16 +--R Type: Expression Integer +--E + +--S 139 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 140 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 6 - \|- x + a + %i x + a 6 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 16 +--R Type: Expression Complex Integer +--E + +--S 141 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 142 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 6 \| a 6 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 16 +--R Type: Expression Complex Integer +--E + +--S 143 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 6 |- x + a 6 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 6 | 2 2 6 6 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 16 +--R Type: Expression Complex Integer +--E + +--S 144 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 6 | 2 2 6 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 6 | 2 2 6 6 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 16 +--R Type: Expression Complex Integer +--E + +--S 145 14:260 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E @ \section{\cite{1}:14.261~~~~~\displaystyle\int{x^3(a^2-x^2)^{3/2}}~dx} @@ -656,7 +2136,7 @@$$ <<*>>= )clear all ---S 25 of 28 +--S 146 aa:=integrate(x^3*(a^2-x^2)^(3/2),x) --R --R @@ -676,6 +2156,27 @@ aa:=integrate(x^3*(a^2-x^2)^(3/2),x) --R - 3920a x + 2240a --R Type: Union(Expression Integer,...) --E + +--S 147 +bb:=(a^2-x^2)^(7/2)/7-(a^2*(a^2-x^2)^(5/2))/5 +--R +--R +---------+ +--R 6 2 4 4 2 6 | 2 2 +--R (- 5x + 8a x - a x - 2a )\|- x + a +--R (2) ---------------------------------------- +--R 35 +--R Type: Expression Integer +--E + +--S 148 14:261 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 7 +--R 2a +--R (3) - --- +--R 35 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.262~~~~~$\displaystyle
@@ -687,7 +2188,7 @@ $$<<*>>= )clear all ---S 26 of 28 +--S 149 aa:=integrate((a^2-x^2)^(3/2)/x,x) --R --R @@ -707,6 +2208,66 @@ aa:=integrate((a^2-x^2)^(3/2)/x,x) --R (3x - 12a )\|- x + a - 9a x + 12a --R Type: Union(Expression Integer,...) --E + +--S 150 +bb:=(a^2-x^2)^(3/2)/3+a^2*sqrt(a^2-x^2)-a^3*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 3 \|- x + a + a 2 2 | 2 2 +--R - 3a log(----------------) + (- x + 4a )\|- x + a +--R x +--R (2) ----------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 151 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R 3 \|- x + a + a 3 \|- x + a - a 3 +--R 3a log(----------------) + 3a log(----------------) + 4a +--R x x +--R (3) --------------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 152 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R 3 | 2 2 3 | 2 2 3 3 +--R 3a log(\|- x + a + a) + 3a log(\|- x + a - a) - 6a log(x) + 4a +--R (4) --------------------------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 153 +ee:=complexNormalize dd +--R +--R 3 x 3 +--R - 6a log(-------) + 4a +--R +----+ +--R | 2 +--R \|- x +--R (5) ----------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 154 14:262 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R 3 +---+ 3 +--R 6a log(\|- 1 ) + 4a +--R (6) -------------------- +--R 3 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.263~~~~~\displaystyle @@ -718,7 +2279,7 @@$$
<<*>>=
)clear all

---S 27 of 28
+--S 155
aa:=integrate((a^2-x^2)^{3/2}/x^2,x)
--R
--R
@@ -738,6 +2299,127 @@ aa:=integrate((a^2-x^2)^{3/2}/x^2,x)
--R     (2x  - 8a x)\|- x  + a   - 6a x  + 8a x
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 156
+bb:=-(a^2-x^2)^(3/2)/x-(3*x*sqrt(a^2-x^2))/2-3/2*a^2*asin(x/a)
+--R
+--R                     +---------+
+--R            2     2  |   2    2      2       x
+--R        (- x  - 2a )\|- x  + a   - 3a x asin(-)
+--R                                             a
+--R   (2)  ---------------------------------------
+--R                           2x
+--R                                                     Type: Expression
Integer
+--E
+
+--S 157
+cc:=aa-bb
+--R
+--R                 +---------+
+--R                 |   2    2
+--R          2     \|- x  + a   - a      2     x
+--R        6a atan(----------------) + 3a asin(-)
+--R                        x                   a
+--R   (3)  --------------------------------------
+--R                           2
+--R                                                     Type: Expression
Integer
+--E
+
+--S 158
+asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2)))
+--R
+--R                           +--------+
+--R                           |   2
+--R   (4)  asin(x) == %i log(\|- x  + 1  - %i x)
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 159
+dd:=asinrule cc
+--R
+--R                    +---------+
+--R                    |   2    2
+--R                    |- x  + a
+--R                  a |---------  - %i x             +---------+
+--R                    |     2                        |   2    2
+--R             2     \|    a                  2     \|- x  + a   - a
+--R        3%i a log(--------------------) + 6a atan(----------------)
+--R                            a                             x
+--R   (5)  -----------------------------------------------------------
+--R                                     2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 160
+atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x)))
+--R
+--R                            - x + %i
+--R                     %i log(--------)
+--R                             x + %i
+--R   (6)  atan(x) == - ----------------
+--R                             2
+--R        Type: RewriteRule(Integer,Complex Integer,Expression Complex
Integer)
+--E
+
+--S 161
+ee:=atanrule dd
+--R
+--R                    +---------+
+--R                    |   2    2
+--R                    |- x  + a
+--R                  a |---------  - %i x                 +---------+
+--R                    |     2                            |   2    2
+--R             2     \|    a                     2    - \|- x  + a   + %i x +
a
+--R        3%i a log(--------------------) - 3%i a
log(-------------------------)
+--R                            a                         +---------+
+--R                                                      |   2    2
+--R                                                     \|- x  + a   + %i x - a
+--R   (7)
----------------------------------------------------------------------
+--R                                           2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 162
+ff:=expandLog ee
+--R
+--R   (8)
+--R                   +---------+
+--R                   |   2    2                       +---------+
+--R            2      |- x  + a                  2     |   2    2
+--R       3%i a log(a |---------  - %i x) + 3%i a log(\|- x  + a   + %i x - a)
+--R                   |     2
+--R                  \|    a
+--R     +
+--R                    +---------+
+--R              2     |   2    2                     2              2
+--R       - 3%i a log(\|- x  + a   - %i x - a) - 3%i a log(a) - 3%i a log(- 1)
+--R  /
+--R     2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 163
+gg:=rootSimp ff
+--R
+--R   (9)
+--R                    +-------+                            +-------+
+--R            2       | 2    2                     2       | 2    2
+--R       3%i a log(%i\|x  - a   + %i x - a) + 3%i a log(%i\|x  - a   - %i x)
+--R     +
+--R                      +-------+
+--R              2       | 2    2                     2              2
+--R       - 3%i a log(%i\|x  - a   - %i x - a) - 3%i a log(a) - 3%i a log(- 1)
+--R  /
+--R     2
+--R                                             Type: Expression Complex
Integer
+--E
+
+--S 164    14:263 Schaums and Axiom agree
+hh:=complexNormalize gg
+--R
+--R   (10)  0
+--R                                             Type: Expression Complex
Integer
+--E
@

\section{\cite{1}:14.264~~~~~$\displaystyle @@ -749,7 +2431,7 @@ $$<<*>>= )clear all ---S 28 of 28 +--S 165 aa:=integrate((a^2-x^2)^(3/2)/x^3,x) --R --R @@ -770,6 +2452,63 @@ aa:=integrate((a^2-x^2)^(3/2)/x^3,x) --R Type: Union(Expression Integer,...) --E +--S 166 +bb:=-(a^2-x^2)^(3/2)/(2*x^2)-(3*sqrt(a^2-x^2))/2+3/2*a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a 2 2 | 2 2 +--R 3a x log(----------------) + (- 2x - a )\|- x + a +--R x +--R (2) ----------------------------------------------------- +--R 2 +--R 2x +--R Type: Expression Integer +--E + +--S 167 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R - 3a log(----------------) - 3a log(----------------) - 2a +--R x x +--R (3) ---------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 168 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - 3a log(\|- x + a + a) - 3a log(\|- x + a - a) + 6a log(x) - 2a +--R (4) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 169 +ee:=complexNormalize dd +--R +--R x +--R (5) 3a log(-------) - a +--R +----+ +--R | 2 +--R \|- x +--R Type: Expression Integer +--E + +--S 170 14:264 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R (6) - 3a log(\|- 1 ) - a +--R Type: Expression Integer +--E + )spool )lisp (bye) @ diff --git a/src/input/schaum2.input.pamphlet b/src/input/schaum2.input.pamphlet index 184122e..dfbaf88 100644 --- a/src/input/schaum2.input.pamphlet +++ b/src/input/schaum2.input.pamphlet @@ -742,58 +742,11 @@ cc:=aa-bb --R 15a --R Type: Expression Integer --E -@ -If we had the terms -<<*>>= ---S 52 -t1:=(3*a*x-2*b) ---R ---R (4) 3a x - 2b ---R Type: Polynomial Integer ---E ---S 53 -t2:=(a*x+b) ---R ---R (5) a x + b ---R Type: Polynomial Integer ---E -@ -We can construct the Axiom result -<<*>>= ---S 54 -2*t1*t2*sqrt(t2)/(15*a^2) ---R ---R 2 2 2 +-------+ ---R (6a x + 2a b x - 4b )\|a x + b ---R (6) -------------------------------- ---R 2 ---R 15a ---R Type: Expression Integer ---E -@ -and we can construct the Spiegel result -<<*>>= ---S 55 -2*t1*sqrt(t2^3)/(15*a^2) +--S 52 14:90 Schaums and Axiom agree +dd:=rootSimp cc --R ---R +----------------------------+ ---R | 3 3 2 2 2 3 ---R (6a x - 4b)\|a x + 3a b x + 3a b x + b ---R (7) ------------------------------------------ ---R 2 ---R 15a ---R Type: Expression Integer ---E -@ -the difference of these two depends on -<<*>>= ---S 56 14:90 Axiom cannot simplify this expression -t2*sqrt(t2)-sqrt(t2^3) ---R ---R +----------------------------+ ---R | 3 3 2 2 2 3 +-------+ ---R (8) - \|a x + 3a b x + 3a b x + b + (a x + b)\|a x + b +--R (4) 0 --R Type: Expression Integer --E @ @@ -807,7 +760,7 @@ Note: the sqrt term is almost certainly \sqrt{(ax+b)} <<*>>= )clear all ---S 57 +--S 53 aa:=integrate(x^2*sqrt(a*x+b),x) --R --R @@ -819,27 +772,26 @@ aa:=integrate(x^2*sqrt(a*x+b),x) --R Type: Union(Expression Integer,...) --E ---S 58 -bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^2)*sqrt((a*x+b)^3) +--S 54 +bb:=(2*(15*a^2*x^2-12*a*b*x+8*b^2))/(105*a^3)*sqrt((a+b*x)^3) --R --R --R +----------------------------+ ---R 2 2 2 | 3 3 2 2 2 3 ---R (30a x - 24a b x + 16b )\|a x + 3a b x + 3a b x + b +--R 2 2 2 | 3 3 2 2 2 3 +--R (30a x - 24a b x + 16b )\|b x + 3a b x + 3a b x + a --R (2) -------------------------------------------------------- ---R 2 +--R 3 --R 105a --R Type: Expression Integer --E ---S 59 14:91 Axiom cannot simplify this expression +--S 55 14:91 Axiom cannot simplify this expression. Schaums typo? cc:=aa-bb ---R --R --R (3) ---R +----------------------------+ ---R 3 2 2 2 | 3 3 2 2 2 3 ---R (- 30a x + 24a b x - 16a b )\|a x + 3a b x + 3a b x + b +--R +----------------------------+ +--R 2 2 2 | 3 3 2 2 2 3 +--R (- 30a x + 24a b x - 16b )\|b x + 3a b x + 3a b x + a --R + --R 3 3 2 2 2 3 +-------+ --R (30a x + 6a b x - 8a b x + 16b )\|a x + b @@ -848,6 +800,20 @@ cc:=aa-bb --R 105a --R Type: Expression Integer --E + +@ +Notice that if we factor the numerator of 'aa' we get an expression that +differs from schaums on by the order of the variables in the square root. +(We can square the term (a*x+b) and drag it under the square root to get +the cubic term). It appears that Schaums has a typo. +<<*>>= +--S 56 +factor numer aa +--R +--R 2 2 2 +-------+ +--R (4) 2(a x + b)(15a x - 12a b x + 8b )\|a x + b +--RType: Factored SparseMultivariatePolynomial(Integer,Kernel Expression Integer) +--E @ \section{\cite{1}:14.92~~~~~\displaystyle @@ -858,7 +824,7 @@$$ <<*>>= )clear all ---S 60 +--S 57 aa:=integrate(sqrt(a*x+b)/x,x) --R --R @@ -875,7 +841,7 @@ aa:=integrate(sqrt(a*x+b)/x,x) --R Type: Union(List Expression Integer,...) --E ---S 61 +--S 58 dd:=integrate(1/(x*sqrt(a*x+b)),x) --R --R @@ -889,7 +855,7 @@ dd:=integrate(1/(x*sqrt(a*x+b)),x) --R Type: Union(List Expression Integer,...) --E ---S 62 +--S 59 bb1:=2*sqrt(a*x+b)+b*dd.1 --R --R @@ -903,7 +869,7 @@ bb1:=2*sqrt(a*x+b)+b*dd.1 --R Type: Expression Integer --E ---S 63 +--S 60 bb2:=2*sqrt(a*x+b)+b*dd.2 --R --R @@ -917,7 +883,7 @@ bb2:=2*sqrt(a*x+b)+b*dd.2 --R Type: Expression Integer --E ---S 64 +--S 61 cc11:=bb1-aa.1 --R --R @@ -932,7 +898,7 @@ cc11:=bb1-aa.1 --R Type: Expression Integer --E ---S 65 +--S 62 cc12:=bb1-aa.2 --R --R @@ -947,7 +913,7 @@ cc12:=bb1-aa.2 --R Type: Expression Integer --E ---S 66 +--S 63 cc21:=bb2-aa.1 --R --R @@ -962,7 +928,7 @@ cc21:=bb2-aa.1 --R Type: Expression Integer --E ---S 67 14:92 Axiom cannot simplify this expression +--S 64 14:92 Axiom cannot simplify this expression cc22:=bb2-aa.2 --R --R @@ -986,7 +952,7 @@ $$<<*>>= )clear all ---S 68 +--S 65 aa:=integrate(sqrt(a*x+b)/x^2,x) --R --R @@ -1008,7 +974,7 @@ aa:=integrate(sqrt(a*x+b)/x^2,x) --R Type: Union(List Expression Integer,...) --E ---S 69 +--S 66 dd:=integrate(1/(x*sqrt(a*x+b)),x) --R --R @@ -1022,7 +988,7 @@ dd:=integrate(1/(x*sqrt(a*x+b)),x) --R Type: Union(List Expression Integer,...) --E ---S 70 +--S 67 bb1:=-sqrt(a*x+b)/x+a/2*dd.1 --R --R @@ -1036,7 +1002,7 @@ bb1:=-sqrt(a*x+b)/x+a/2*dd.1 --R Type: Expression Integer --E ---S 71 +--S 68 bb2:=-sqrt(a*x+b)/x+a/2*dd.2 --R --R @@ -1050,7 +1016,7 @@ bb2:=-sqrt(a*x+b)/x+a/2*dd.2 --R Type: Expression Integer --E ---S 72 +--S 69 cc11:=bb1-aa.1 --R --R @@ -1058,7 +1024,7 @@ cc11:=bb1-aa.1 --R Type: Expression Integer --E ---S 73 +--S 70 cc21:=bb-aa.1 --R --R @@ -1073,7 +1039,7 @@ cc21:=bb-aa.1 --R Type: Expression Integer --E ---S 74 +--S 71 cc12:=bb1-aa.2 --R --R @@ -1088,7 +1054,7 @@ cc12:=bb1-aa.2 --R Type: Expression Integer --E ---S 75 14:93 Schaums and Axiom agree +--S 72 14:93 Schaums and Axiom agree cc22:=bb2-aa.2 --R --R @@ -1106,7 +1072,7 @@$$ <<*>>= )clear all ---S 76 14:94 Axiom cannot do this integral +--S 73 14:94 Axiom cannot do this integral aa:=integrate(x^m/sqrt(a*x+b),x) --R --R @@ -1128,7 +1094,7 @@ $$<<*>>= )clear all ---S 77 14:95 Axiom cannot do this integral +--S 74 14:95 Axiom cannot do this integral aa:=integrate(1/(x^m*sqrt(a*x+b)),x) --R --R @@ -1150,7 +1116,7 @@$$ <<*>>= )clear all ---S 78 14:96 Axiom cannot do this integral +--S 75 14:96 Axiom cannot do this integral aa:=integrate(x^m*sqrt(a*x+b),x) --R --R @@ -1171,7 +1137,7 @@ $$<<*>>= )clear all ---S 79 14:97 Axiom cannot do this integral +--S 76 14:97 Axiom cannot do this integral aa:=integrate(sqrt(a*x+b)/x^m,x) --R --R @@ -1194,7 +1160,7 @@ Note: 14.98 is the same as 14.97 <<*>>= )clear all ---S 80 14:98 Axiom cannot do this integral +--S 77 14:98 Axiom cannot do this integral aa:=integrate(sqrt(a*x+b)/x^m,x) --R --R @@ -1215,7 +1181,7 @@$$ <<*>>= )clear all ---S 81 +--S 78 aa:=integrate((a*x+b)^(m/2),x) --R --R @@ -1228,7 +1194,7 @@ aa:=integrate((a*x+b)^(m/2),x) --R Type: Union(Expression Integer,...) --E ---S 82 +--S 79 bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2)) --R --R @@ -1241,7 +1207,7 @@ bb:=(2*(a*x+b)^((m+2)/2))/(a*(m+2)) --R Type: Expression Integer --E ---S 83 14:99 Axiom cannot simplify this expression +--S 80 cc:=aa-bb --R --R @@ -1253,6 +1219,33 @@ cc:=aa-bb --R a m + 2a --R Type: Expression Integer --E + +--S 81 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 82 +dd:=explog cc +--R +--R m + 2 m +--R ----- - +--R 2 2 +--R - 2(a x + b) + (2a x + 2b)(a x + b) +--R (5) ----------------------------------------- +--R a m + 2a +--R Type: Expression Integer +--E + +--S 83 14:99 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.100~~~~~$\displaystyle
@@ -1292,7 +1285,7 @@
bb:=(2*(a*x+b)^((m+4)/2))/(a^2*(m+4))-(2*b*(a*x+b)^((m+2)/2))/(a^2*(m+2))
--R                                                     Type: Expression
Integer
--E

---S 86     14:100 Axiom cannot simplify this expression
+--S 86
cc:=aa-bb
--R
--R
@@ -1311,6 +1304,40 @@ cc:=aa-bb
--R     a m  + 6a m + 8a
--R                                                     Type: Expression
Integer
--E
+
+--S 87
+explog:=rule(%e^(n*log(x)) == x^n)
+--R
+--R          n log(x)     n
+--R   (4)  %e         == x
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 88
+dd:=explog cc
+--R
+--R   (5)
+--R                          m + 4                       m + 2
+--R                          -----                       -----
+--R                            2                           2
+--R       (- 2m - 4)(a x + b)      + (2b m + 8b)(a x + b)
+--R     +
+--R                                                 m
+--R                                                 -
+--R           2      2  2                2          2
+--R       ((2a m + 4a )x  + 2a b m x - 4b )(a x + b)
+--R  /
+--R      2 2     2      2
+--R     a m  + 6a m + 8a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 89     14:100 Schaums and Axiom agree
+ee:=complexNormalize dd
+--R
+--R   (6)  0
+--R                                                     Type: Expression
Integer
+--E
@

\section{\cite{1}:14.101~~~~~$\displaystyle @@ -1323,7 +1350,7 @@ $$<<*>>= )clear all ---S 87 +--S 90 aa:=integrate(x^2*(a*x+b)^(m/2),x) --R --R @@ -1341,7 +1368,7 @@ aa:=integrate(x^2*(a*x+b)^(m/2),x) --R Type: Union(Expression Integer,...) --E ---S 88 +--S 91 bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_ (4*b*(a*x+b)^((m+4)/2))/(a^3*(m+4))+_ (2*b^2*(a*x+b)^((m+2)/2))/(a^3*(m+2)) @@ -1363,7 +1390,7 @@ bb:=(2*(a*x+b)^((m+6)/2))/(a^3*(m+6))-_ --R Type: Expression Integer --E ---S 89 14:101 Axiom cannot simplify this expression +--S 92 cc:=aa-bb --R --R @@ -1390,6 +1417,48 @@ cc:=aa-bb --R a m + 12a m + 44a m + 48a --R Type: Expression Integer --E + +--S 93 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 94 +dd:=explog cc +--R +--R (5) +--R m + 6 m + 4 +--R ----- ----- +--R 2 2 2 2 +--R (- 2m - 12m - 16)(a x + b) + (4b m + 32b m + 48b)(a x + b) +--R + +--R m + 2 +--R ----- +--R 2 2 2 2 2 +--R (- 2b m - 20b m - 48b )(a x + b) +--R + +--R 3 2 3 3 3 2 2 2 2 2 3 +--R ((2a m + 12a m + 16a )x + (2a b m + 4a b m)x - 8a b m x + 16b ) +--R * +--R m +--R - +--R 2 +--R (a x + b) +--R / +--R 3 3 3 2 3 3 +--R a m + 12a m + 44a m + 48a +--R Type: Expression Integer +--E + +--S 95 14:101 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.102~~~~~\displaystyle @@ -1401,7 +1470,7 @@$$ <<*>>= )clear all ---S 90 14:102 Axiom cannot do this integral +--S 96 14:102 Axiom cannot do this integral aa:=integrate((a*x+b)^(m/2)/x,x) --R --R @@ -1423,7 +1492,7 @@ $$<<*>>= )clear all ---S 91 14:103 Axiom cannot do this integral +--S 97 14:103 Axiom cannot do this integral aa:=integrate((a*x+b)^(m/2)/x^2,x) --R --R @@ -1446,7 +1515,7 @@$$ <<*>>= )clear all ---S 92 14:104 Axiom cannot do this integral +--S 98 14:104 Axiom cannot do this integral aa:=integrate(1/(x*(a*x+b)^(m/2)),x) --R --R diff --git a/src/input/schaum6.input.pamphlet b/src/input/schaum6.input.pamphlet index 9b6f4f0..854e780 100644 --- a/src/input/schaum6.input.pamphlet +++ b/src/input/schaum6.input.pamphlet @@ -792,7 +792,7 @@ explog:=rule(%e^(n*log(x)) == x^n) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 64 14:140 Schaums and Axiom cannot simplify this expression +--S 64 dd:=explog cc --R --R 2 2 n 2 2 2 2 n - 1 @@ -802,6 +802,13 @@ dd:=explog cc --R (2n - 2)(x + a ) (x + a ) --R Type: Expression Integer --E + +--S 65 14:140 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.141~~~~~$\displaystyle\int{\frac{dx}{x(x^2+a^2)^n}}$} @@ -812,7 +819,7 @@ $$<<*>>= )clear all ---S 65 14:141 Axiom cannot do this integral +--S 66 14:141 Axiom cannot do this integral aa:=integrate(1/(x*(x^2+a^2)^n),x) --R --R @@ -833,7 +840,7 @@$$ <<*>>= )clear all ---S 66 14:142 Axiom cannot do this integral +--S 67 14:142 Axiom cannot do this integral aa:=integrate(x^m/((x^2+a^2)^n),x) --R --R @@ -854,7 +861,7 @@ $$<<*>>= )clear all ---S 67 14:143 Axiom cannot do this integral +--S 68 14:143 Axiom cannot do this integral aa:=integrate(1/(x^m*(x^2+a^2)^n),x) --R --R diff --git a/src/input/schaum7.input.pamphlet b/src/input/schaum7.input.pamphlet index ce37f19..fcdd71c 100644 --- a/src/input/schaum7.input.pamphlet +++ b/src/input/schaum7.input.pamphlet @@ -915,7 +915,7 @@ explog:=rule(%e^(n*log(x)) == x^n) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 76 14:159 Axiom cannot simplify this expression +--S 76 dd:=explog cc --R --R 2 2 n 2 2 2 2 n - 1 @@ -926,6 +926,12 @@ dd:=explog cc --R Type: Expression Integer --E +--S 77 14:159 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.160~~~~~\displaystyle\int{\frac{dx}{x(x^2-a^2)^n}}} @@ -936,7 +942,7 @@$$ <<*>>= )clear all ---S 77 14:160 Axiom cannot compute this integral +--S 78 14:160 Axiom cannot compute this integral aa:=integrate(1/(x*(x^2-a^2)^n),x) --R --R @@ -957,7 +963,7 @@ $$<<*>>= )clear all ---S 78 14:161 Axiom cannot compute this integral +--S 79 14:161 Axiom cannot compute this integral aa:=integrate(x^m/((x^2-a^2)^n),x) --R --R @@ -978,7 +984,7 @@$$ <<*>>= )clear all ---S 79 14:162 Axiom cannot compute this integral +--S 80 14:162 Axiom cannot compute this integral aa:=integrate(1/(x^m*(x^2-a^2)^n),x) --R --R diff --git a/src/input/schaum8.input.pamphlet b/src/input/schaum8.input.pamphlet index 778d135..10d9ef5 100644 --- a/src/input/schaum8.input.pamphlet +++ b/src/input/schaum8.input.pamphlet @@ -1103,7 +1103,7 @@ explog:=rule(%e^(n*log(x)) == x^n) --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 95 14:178 Axiom cannot simplify this expression +--S 95 dd:=explog cc --R --R 2 2 n 2 2 2 2 n - 1 @@ -1114,6 +1114,13 @@ dd:=explog cc --R Type: Expression Integer --E +--S 96 14:178 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.179~~~~~$\displaystyle\int{\frac{dx}{x(a^2-x^2)^n}}$} @@ -1124,7 +1131,7 @@ $$<<*>>= )clear all ---S 96 14:179 Axiom cannot integrate this expression +--S 97 14:179 Axiom cannot integrate this expression aa:=integrate(1/(x*(a^2-x^2)^n),x) --R --R @@ -1145,7 +1152,7 @@$$ <<*>>= )clear all ---S 97 14:180 Axiom cannot integrate this expression +--S 98 14:180 Axiom cannot integrate this expression aa:=integrate(x^m/((a^2-x^2)^n),x) --R --R @@ -1166,7 +1173,7 @@ $$<<*>>= )clear all ---S 98 14:181 Axiom cannot integrate this expression +--S 99 14:181 Axiom cannot integrate this expression aa:=integrate(1/(x^m*(a^2-x^2)^n),x) --R --R diff --git a/src/input/schaum9.input.pamphlet b/src/input/schaum9.input.pamphlet index fceb094..d632de7 100644 --- a/src/input/schaum9.input.pamphlet +++ b/src/input/schaum9.input.pamphlet @@ -35,7 +35,7 @@ bb:=log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 3 14:182 Axiom cannot simplify this expression +--S 3 cc:=aa-bb --R --R +-------+ +-------+ @@ -43,6 +43,15 @@ cc:=aa-bb --R (3) - log(\|x + a + x) - log(\|x + a - x) --R Type: Expression Integer --E + +--S 4 14:182 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(a ) +--R Type: Expression Integer +--E + @ This is equal to -\log(a^2) but Axiom cannot prove it. @@ -51,7 +60,7 @@$$\int{\frac{x}{\sqrt{x^2+a^2}}}=\sqrt{x^2+a^2}$$<<*>>= )clear all ---S 4 +--S 5 aa:=integrate(x/(sqrt(x^2+a^2)),x) --R --R @@ -65,7 +74,7 @@ aa:=integrate(x/(sqrt(x^2+a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 5 +--S 6 bb:=sqrt(x^2+a^2) --R --R +-------+ @@ -74,7 +83,7 @@ bb:=sqrt(x^2+a^2) --R Type: Expression Integer --E ---S 6 14:183 Schaums and Axiom agree +--S 7 14:183 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -90,7 +99,7 @@$$ <<*>>= )clear all ---S 7 +--S 8 aa:=integrate(x^2/sqrt(x^2+a^2),x) --R --R @@ -109,7 +118,7 @@ aa:=integrate(x^2/sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 8 +--S 9 bb:=(x*sqrt(x^2+a^2))/2-a^2/2*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -120,7 +129,7 @@ bb:=(x*sqrt(x^2+a^2))/2-a^2/2*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 9 +--S 10 cc:=aa-bb --R --R +-------+ +-------+ @@ -131,14 +140,14 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 10 +--S 11 logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b)) --R --I (4) c log(b) + c log(a) + %K == c log(a b) + %K --R Type: RewriteRule(Integer,Integer,Expression Integer) --E ---S 11 14:184 Schaums and Axiom differ by a constant +--S 12 14:184 Schaums and Axiom differ by a constant dd:=logmul1 cc --R --R 2 2 @@ -158,7 +167,7 @@ $$<<*>>= )clear all ---S 12 +--S 13 aa:=integrate(x^3/sqrt(x^2+a^2),x) --R --R @@ -172,7 +181,7 @@ aa:=integrate(x^3/sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 13 +--S 14 bb:=(x^2+a^2)^(3/2)/3-a^2*sqrt(x^2+a^2) --R --R +-------+ @@ -183,7 +192,7 @@ bb:=(x^2+a^2)^(3/2)/3-a^2*sqrt(x^2+a^2) --R Type: Expression Integer --E ---S 14 14:185 Schaums and Axiom agree +--S 15 14:185 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -198,7 +207,7 @@$$ <<*>>= )clear all ---S 15 +--S 16 aa:=integrate(1/(x*sqrt(x^2+a^2)),x) --R --R @@ -210,7 +219,7 @@ aa:=integrate(1/(x*sqrt(x^2+a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 16 +--S 17 bb:=-1/a*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -223,7 +232,7 @@ bb:=-1/a*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 17 +--S 18 cc:=aa-bb --R --R (3) @@ -237,19 +246,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 18 -divlog:=rule(log(a/b) == log(a) - log(b)) ---R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 19 14:186 Axiom cannot simplify this expression -dd:=divlog cc +--S 19 +dd:=expandLog cc --R ---R (5) +--R (4) --R +-------+ +-------+ +-------+ --R | 2 2 | 2 2 | 2 2 --R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) @@ -260,6 +260,15 @@ dd:=divlog cc --R Type: Expression Integer --E +--S 20 14:186 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R a +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.187~~~~~$\displaystyle
@@ -270,7 +279,7 @@ $$<<*>>= )clear all ---S 20 +--S 21 aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x) --R --R @@ -282,7 +291,7 @@ aa:=integrate(1/(x^2*sqrt(x^2+a^2)),x) --R Type: Union(Expression Integer,...) --E ---S 21 +--S 22 bb:=-sqrt(x^2+a^2)/(a^2*x) --R --R +-------+ @@ -294,7 +303,7 @@ bb:=-sqrt(x^2+a^2)/(a^2*x) --R Type: Expression Integer --E ---S 22 14:187 Schaums and Axiom differ by a constant +--S 23 14:187 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 1 @@ -314,7 +323,7 @@$$
<<*>>=
)clear all

---S 23
+--S 24
aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x)
--R
--R
@@ -337,7 +346,7 @@ aa:=integrate(1/(x^3*sqrt(x^2+a^2)),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 24
+--S 25
bb:=-sqrt(x^2+a^2)/(2*a^2*x^2)+1/(2*a^3)*log((a+sqrt(x^2+a^2))/x)
--R
--R               +-------+
@@ -351,7 +360,7 @@
bb:=-sqrt(x^2+a^2)/(2*a^2*x^2)+1/(2*a^3)*log((a+sqrt(x^2+a^2))/x)
--R                                                     Type: Expression
Integer
--E

---S 25
+--S 26
cc:=aa-bb
--R
--R   (3)
@@ -366,19 +375,10 @@ cc:=aa-bb
--R                                                     Type: Expression
Integer
--E

---S 26
-divlog:=rule(log(a/b) == log(a) - log(b))
+--S 27
+dd:=expandLog cc
--R
---R            a
---R   (4)  log(-) == - log(b) + log(a)
---R            b
---R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 27     14:188 Axiom cannot simplify this expression
-dd:=divlog cc
---R
---R   (5)
+--R   (4)
--R              +-------+             +-------+                 +-------+
--R              | 2    2              | 2    2                  | 2    2
--R       - log(\|x  + a   + a) + log(\|x  + a   - x + a) - log(\|x  + a   - x
- a)
@@ -389,6 +389,17 @@ dd:=divlog cc
--R     2a
--R                                                     Type: Expression
Integer
--E
+
+--S 28     14:188 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+
+        log(- 1)
+   (5)  --------
+             3
+           2a
+                                                     Type: Expression Integer
+--E
+
@

\section{\cite{1}:14.189~~~~~$\displaystyle\int{\sqrt{x^2+a^2}}~dx$}
@@ -398,7 +409,7 @@ $$<<*>>= )clear all ---S 28 +--S 29 aa:=integrate(sqrt(x^2+a^2),x) --R --R @@ -417,7 +428,7 @@ aa:=integrate(sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 29 +--S 30 bb:=(x*sqrt(x^2+a^2))/2+a^2/2*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -428,7 +439,7 @@ bb:=(x*sqrt(x^2+a^2))/2+a^2/2*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 30 14:189 Axiom cannot simplify this expression +--S 31 cc:=aa-bb --R --R +-------+ +-------+ @@ -438,6 +449,17 @@ cc:=aa-bb --R 2 --R Type: Expression Integer --E + +--S 32 14:189 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(a ) +--R (4) - --------- +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.190~~~~~\displaystyle\int{x\sqrt{x^2+a^2}}~dx} @@ -447,7 +469,7 @@$$
<<*>>=
)clear all

---S 31
+--S 33
aa:=integrate(x*sqrt(x^2+a^2),x)
--R
--R
@@ -461,7 +483,7 @@ aa:=integrate(x*sqrt(x^2+a^2),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 32
+--S 34
bb:=(x^2+a^2)^(3/2)/3
--R
--R                  +-------+
@@ -472,7 +494,7 @@ bb:=(x^2+a^2)^(3/2)/3
--R                                                     Type: Expression
Integer
--E

---S 33     14:190 Schaums and Axiom agree
+--S 35     14:190 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
@@ -489,7 +511,7 @@ $$<<*>>= )clear all ---S 34 +--S 36 aa:=integrate(x^2*sqrt(x^2+a^2),x) --R --R @@ -508,7 +530,7 @@ aa:=integrate(x^2*sqrt(x^2+a^2),x) --R Type: Union(Expression Integer,...) --E ---S 35 +--S 37 bb:=(x*(x^2+a^2)^(3/2))/4-(a^2*x*sqrt(x^2+a^2))/8-a^4/8*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -519,7 +541,7 @@ bb:=(x*(x^2+a^2)^(3/2))/4-(a^2*x*sqrt(x^2+a^2))/8-a^4/8*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 36 14:191 Axiom cannot simplify this expression +--S 38 cc:=aa-bb --R --R +-------+ +-------+ @@ -529,6 +551,16 @@ cc:=aa-bb --R 8 --R Type: Expression Integer --E + +--S 39 14:191 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R a log(a ) +--R (4) --------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.192~~~~~\displaystyle @@ -539,7 +571,7 @@$$
<<*>>=
)clear all

---S 37
+--S 40
aa:=integrate(x^3*sqrt(x^2+a^2),x)
--R
--R
@@ -557,7 +589,7 @@ aa:=integrate(x^3*sqrt(x^2+a^2),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 38
+--S 41
bb:=(x^2+a^2)^(5/2)/5-(a^2*(x^2+a^2)^(3/2))/3
--R
--R                           +-------+
@@ -568,7 +600,7 @@ bb:=(x^2+a^2)^(5/2)/5-(a^2*(x^2+a^2)^(3/2))/3
--R                                                     Type: Expression
Integer
--E

---S 39     14:192 Schaums and Axiom agree
+--S 42     14:192 Schaums and Axiom agree
cc:=aa-bb
--R
--R   (3)  0
@@ -584,7 +616,7 @@ $$<<*>>= )clear all ---S 40 +--S 43 aa:=integrate(sqrt(x^2+a^2)/x,x) --R --R @@ -603,7 +635,7 @@ aa:=integrate(sqrt(x^2+a^2)/x,x) --R Type: Union(Expression Integer,...) --E ---S 41 +--S 44 bb:=sqrt(x^2+a^2)-a*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -614,7 +646,7 @@ bb:=sqrt(x^2+a^2)-a*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 42 14:193 Axiom cannot simplify this expression +--S 45 cc:=aa-bb --R --R (3) @@ -629,6 +661,27 @@ cc:=aa-bb --R x --R Type: Expression Integer --E + +--S 46 +dd:=expandLog cc +--R +--R (4) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R a log(\|x + a + a) - a log(\|x + a - x + a) +--R + +--R +-------+ +--R | 2 2 +--R a log(\|x + a - x - a) - a log(x) +--R Type: Expression Integer +--E + +--S 47 14:193 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R (5) - a log(- 1) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.194~~~~~\displaystyle @@ -639,7 +692,7 @@$$
<<*>>=
)clear all

---S 43
+--S 48
aa:=integrate(sqrt(x^2+a^2)/x^2,x)
--R
--R
@@ -653,7 +706,7 @@ aa:=integrate(sqrt(x^2+a^2)/x^2,x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 44
+--S 49
bb:=-sqrt(x^2+a^2)/x+log(x+sqrt(x^2+a^2))
--R
--R               +-------+         +-------+
@@ -664,7 +717,7 @@ bb:=-sqrt(x^2+a^2)/x+log(x+sqrt(x^2+a^2))
--R                                                     Type: Expression
Integer
--E

---S 45     14:194 Axiom cannot simplify this expression
+--S 50
cc:=aa-bb
--R
--R               +-------+             +-------+
@@ -672,6 +725,15 @@ cc:=aa-bb
--R   (3)  - log(\|x  + a   + x) - log(\|x  + a   - x) - 1
--R                                                     Type: Expression
Integer
--E
+
+--S 51     14:194 Schaums and Axiom differ by a constant
+dd:=complexNormalize cc
+--R
+--R               2
+--R   (4)  - log(a ) - 1
+--R                                                     Type: Expression
Integer
+--E
+
@

\section{\cite{1}:14.195~~~~~$\displaystyle @@ -683,7 +745,7 @@ $$<<*>>= )clear all ---S 46 +--S 52 aa:=integrate(sqrt(x^2+a^2)/x^3,x) --R --R @@ -706,7 +768,7 @@ aa:=integrate(sqrt(x^2+a^2)/x^3,x) --R Type: Union(Expression Integer,...) --E ---S 47 +--S 53 bb:=-sqrt(x^2+a^2)/(2*x^2)-1/(2*a)*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -720,7 +782,7 @@ bb:=-sqrt(x^2+a^2)/(2*x^2)-1/(2*a)*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 48 +--S 54 cc:=aa-bb --R --R (3) @@ -734,19 +796,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 49 -divlog:=rule(log(a/b) == log(a) - log(b)) ---R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 50 14:195 Axiom cannot simplify this expression -dd:=divlog cc +--S 55 +dd:=expandLog cc --R ---R (5) +--R (4) --R +-------+ +-------+ +-------+ --R | 2 2 | 2 2 | 2 2 --R log(\|x + a + a) - log(\|x + a - x + a) + log(\|x + a - x - a) @@ -756,6 +809,15 @@ dd:=divlog cc --R 2a --R Type: Expression Integer --E + +--S 56 14:195 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R log(- 1) +--R (5) - -------- +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.196~~~~~\displaystyle\int{\frac{dx}{(x^2+a^2)^{3/2}}}} @@ -765,7 +827,7 @@$$ <<*>>= )clear all ---S 51 +--S 57 aa:=integrate(1/(x^2+a^2)^(3/2),x) --R --R @@ -777,7 +839,7 @@ aa:=integrate(1/(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 52 +--S 58 bb:=x/(a^2*sqrt(x^2+a^2)) --R --R x @@ -788,7 +850,7 @@ bb:=x/(a^2*sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 53 14:196 Schaums and Axiom differ by a constant +--S 59 14:196 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 1 @@ -807,7 +869,7 @@ $$<<*>>= )clear all ---S 54 +--S 60 aa:=integrate(x/(x^2+a^2)^(3/2),x) --R --R @@ -821,7 +883,7 @@ aa:=integrate(x/(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 55 +--S 61 bb:=-1/sqrt(x^2+a^2) --R --R 1 @@ -832,7 +894,7 @@ bb:=-1/sqrt(x^2+a^2) --R Type: Expression Integer --E ---S 56 14:197 Schaums and Axiom agree +--S 62 14:197 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -849,7 +911,7 @@$$ <<*>>= )clear all ---S 57 +--S 63 aa:=integrate(x^2/(x^2+a^2)^(3/2),x) --R --R @@ -863,7 +925,7 @@ aa:=integrate(x^2/(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 58 +--S 64 bb:=-x/sqrt(x^2+a^2)+log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -876,7 +938,7 @@ bb:=-x/sqrt(x^2+a^2)+log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 59 14:198 Axiom cannot simplify this expression +--S 65 cc:=aa-bb --R --R +-------+ +-------+ @@ -884,6 +946,14 @@ cc:=aa-bb --R (3) - log(\|x + a + x) - log(\|x + a - x) - 1 --R Type: Expression Integer --E + +--S 66 14:198 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(a ) - 1 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.199~~~~~$\displaystyle
@@ -894,7 +964,7 @@ $$<<*>>= )clear all ---S 60 +--S 67 aa:=integrate(x^3/(x^2+a^2)^(3/2),x) --R --R @@ -908,7 +978,7 @@ aa:=integrate(x^3/(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 61 +--S 68 bb:=sqrt(x^2+a^2)+a^2/sqrt(x^2+a^2) --R --R 2 2 @@ -920,7 +990,7 @@ bb:=sqrt(x^2+a^2)+a^2/sqrt(x^2+a^2) --R Type: Expression Integer --E ---S 62 14:199 Schaums and Axiom agree +--S 69 14:199 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -937,7 +1007,7 @@$$
<<*>>=
)clear all

---S 63
+--S 70
aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x)
--R
--R
@@ -956,7 +1026,7 @@ aa:=integrate(1/(x*(x^2+a^2)^(3/2)),x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 64
+--S 71
bb:=1/(a^2*sqrt(x^2+a^2))-1/a^3*log((a+sqrt(x^2+a^2))/x)
--R
--R                         +-------+
@@ -971,7 +1041,7 @@ bb:=1/(a^2*sqrt(x^2+a^2))-1/a^3*log((a+sqrt(x^2+a^2))/x)
--R                                                     Type: Expression
Integer
--E

---S 65
+--S 72
cc:=aa-bb
--R
--R   (3)
@@ -986,19 +1056,10 @@ cc:=aa-bb
--R                                                     Type: Expression
Integer
--E

---S 66
-divlog:=rule(log(a/b) == log(a) - log(b))
---R
---R            a
---R   (4)  log(-) == - log(b) + log(a)
---R            b
---R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
---E
-
---S 67     14:200 Axiom cannot simplify this expression
-dd:=divlog cc
+--S 73
+dd:=expandLog cc
--R
---R   (5)
+--R   (4)
--R            +-------+             +-------+                 +-------+
--R            | 2    2              | 2    2                  | 2    2
--R       log(\|x  + a   + a) - log(\|x  + a   - x + a) + log(\|x  + a   - x -
a)
@@ -1009,6 +1070,17 @@ dd:=divlog cc
--R     a
--R                                                     Type: Expression
Integer
--E
+
+--S 74     14:200 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R          log(- 1)
+--R   (5)  - --------
+--R              3
+--R             a
+--R                                                     Type: Expression
Integer
+--E
+
@

\section{\cite{1}:14.201~~~~~$\displaystyle @@ -1019,7 +1091,7 @@ $$<<*>>= )clear all ---S 68 +--S 75 aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x) --R --R @@ -1031,7 +1103,7 @@ aa:=integrate(1/(x^2*(x^2+a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 69 +--S 76 bb:=-sqrt(x^2+a^2)/(a^4*x)-x/(a^4*sqrt(x^2+a^2)) --R --R 2 2 @@ -1043,7 +1115,7 @@ bb:=-sqrt(x^2+a^2)/(a^4*x)-x/(a^4*sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 70 14:201 Schaums and Axiom differ by a constant +--S 77 14:201 Schaums and Axiom differ by a constant cc:=aa-bb --R --R 2 @@ -1064,7 +1136,7 @@$$ <<*>>= )clear all ---S 71 +--S 78 aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x) --R --R @@ -1091,7 +1163,7 @@ aa:=integrate(1/(x^3*(x^2+a^2)^(3/2)),x) --R Type: Union(Expression Integer,...) --E ---S 72 +--S 79 bb:=-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2))+3/(2*a^5)*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -1106,7 +1178,7 @@ bb:=-1/(2*a^2*x^2*sqrt(x^2+a^2))-3/(2*a^4*sqrt(x^2+a^2))+3/(2*a^5)*log((a+sqrt(x --R Type: Expression Integer --E ---S 73 +--S 80 cc:=aa-bb --R --R (3) @@ -1121,19 +1193,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 74 -divlog:=rule(log(a/b) == log(a) - log(b)) ---R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 75 14:202 Axiom cannot simplify this expression -dd:=divlog cc +--S 81 +dd:=expandLog cc --R ---R (5) +--R (4) --R +-------+ +-------+ --R | 2 2 | 2 2 --R - 3log(\|x + a + a) + 3log(\|x + a - x + a) @@ -1146,6 +1209,16 @@ dd:=divlog cc --R 2a --R Type: Expression Integer --E + +--S 82 14:202 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3log(- 1) +--R (5) --------- +--R 5 +--R 2a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.203~~~~~$\displaystyle\int{(x^2+a^2)^{3/2}}~dx$} @@ -1156,7 +1229,7 @@ $$<<*>>= )clear all ---S 76 +--S 83 aa:=integrate((x^2+a^2)^(3/2),x) --R --R (1) @@ -1177,7 +1250,7 @@ aa:=integrate((x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 77 +--S 84 bb:=(x*(x^2+a^2)^(3/2))/4+(3*a^2*x*sqrt(x^2+a^2))/8+3/8*a^4*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -1188,7 +1261,7 @@ bb:=(x*(x^2+a^2)^(3/2))/4+(3*a^2*x*sqrt(x^2+a^2))/8+3/8*a^4*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 78 14:203 Axiom cannot simplify this expression +--S 85 cc:=aa-bb --R --R +-------+ +-------+ @@ -1198,6 +1271,16 @@ cc:=aa-bb --R 8 --R Type: Expression Integer --E + +--S 86 14:203 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R 3a log(a ) +--R (4) - ---------- +--R 8 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.204~~~~~\displaystyle\int{x(x^2+a^2)^{3/2}}~dx} @@ -1205,7 +1288,7 @@$$\int{x(x^2+a^2)^{3/2}}=\frac{(x^2+a^2)^{5/2}}{5}$$<<*>>= )clear all ---S 79 +--S 87 aa:=integrate(x*(x^2+a^2)^(3/2),x) --R --R @@ -1223,7 +1306,7 @@ aa:=integrate(x*(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 80 +--S 88 bb:=(x^2+a^2)^(5/2)/5 --R --R +-------+ @@ -1234,7 +1317,7 @@ bb:=(x^2+a^2)^(5/2)/5 --R Type: Expression Integer --E ---S 81 14:204 Schaums and Axiom agree +--S 89 14:204 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -1251,7 +1334,7 @@$$ <<*>>= )clear all ---S 82 +--S 90 aa:=integrate(x^2*(x^2+a^2)^(3/2),x) --R --R @@ -1280,7 +1363,7 @@ aa:=integrate(x^2*(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 83 +--S 91 bb:=(x*(x^2+a^2)^(5/2))/6-(a^2*x*(x^2+a^2)^(3/2))/24-(a^4*x*sqrt(x^2+a^2))/16-a^6/16*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -1291,7 +1374,7 @@ bb:=(x*(x^2+a^2)^(5/2))/6-(a^2*x*(x^2+a^2)^(3/2))/24-(a^4*x*sqrt(x^2+a^2))/16-a^ --R Type: Expression Integer --E ---S 84 14:205 Axiom cannot simplify this expression +--S 92 cc:=aa-bb --R --R +-------+ +-------+ @@ -1301,6 +1384,16 @@ cc:=aa-bb --R 16 --R Type: Expression Integer --E + +--S 93 14:205 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 6 2 +--R a log(a ) +--R (4) --------- +--R 16 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.206~~~~~$\displaystyle\int{x^3(x^2+a^2)^{3/2}}~dx$} @@ -1310,7 +1403,7 @@ $$<<*>>= )clear all ---S 85 +--S 94 aa:=integrate(x^3*(x^2+a^2)^(3/2),x) --R --R @@ -1340,7 +1433,7 @@ aa:=integrate(x^3*(x^2+a^2)^(3/2),x) --R Type: Union(Expression Integer,...) --E ---S 86 +--S 95 bb:=(x^2+a^2)^(7/2)/7-(a^2*(x^2+a^2)^(5/2))/5 --R --R +-------+ @@ -1351,7 +1444,7 @@ bb:=(x^2+a^2)^(7/2)/7-(a^2*(x^2+a^2)^(5/2))/5 --R Type: Expression Integer --E ---S 87 14:206 Schaums and Axiom agree +--S 96 14:206 Schaums and Axiom agree cc:=aa-bb --R --R (3) 0 @@ -1368,7 +1461,7 @@$$ <<*>>= )clear all ---S 88 +--S 97 aa:=integrate((x^2+a^2)^(3/2)/x,x) --R --R @@ -1391,7 +1484,7 @@ aa:=integrate((x^2+a^2)^(3/2)/x,x) --R Type: Union(Expression Integer,...) --E ---S 89 +--S 98 bb:=(x^2+a^2)^(3/2)/3+a^2*sqrt(x^2+a^2)-a^3*log((a+sqrt(x^2+a^2))/x) --R --R +-------+ @@ -1404,7 +1497,7 @@ bb:=(x^2+a^2)^(3/2)/3+a^2*sqrt(x^2+a^2)-a^3*log((a+sqrt(x^2+a^2))/x) --R Type: Expression Integer --E ---S 90 +--S 99 cc:=aa-bb --R --R (3) @@ -1420,19 +1513,10 @@ cc:=aa-bb --R Type: Expression Integer --E ---S 91 -divlog:=rule(log(a/b) == log(a) - log(b)) +--S 100 +dd:=expandLog cc --R ---R a ---R (4) log(-) == - log(b) + log(a) ---R b ---R Type: RewriteRule(Integer,Integer,Expression Integer) ---E - ---S 92 14:207 Axiom cannot simplify this expression -dd:=divlog cc ---R ---R (5) +--R (4) --R +-------+ +-------+ --R 3 | 2 2 3 | 2 2 --R a log(\|x + a + a) - a log(\|x + a - x + a) @@ -1442,6 +1526,14 @@ dd:=divlog cc --R a log(\|x + a - x - a) - a log(x) --R Type: Expression Integer --E + +--S 101 14:207 Schaums and Axiom differ by a constant +ee:=complexNormalize dd +--R +--R 3 +--R (5) - a log(- 1) +--R Type: Expression Integer +--E @ \section{\cite{1}:14.208~~~~~$\displaystyle
@@ -1453,7 +1545,7 @@ $$<<*>>= )clear all ---S 93 +--S 102 aa:=integrate((x^2+a^2)^{3/2}/x^2,x) --R --R @@ -1472,7 +1564,7 @@ aa:=integrate((x^2+a^2)^{3/2}/x^2,x) --R Type: Union(Expression Integer,...) --E ---S 94 +--S 103 bb:=-(x^2+a^2)^(3/2)/x+(3*x*sqrt(x^2+a^2))/2+3/2*a^2*log(x+sqrt(x^2+a^2)) --R --R +-------+ +-------+ @@ -1483,7 +1575,7 @@ bb:=-(x^2+a^2)^(3/2)/x+(3*x*sqrt(x^2+a^2))/2+3/2*a^2*log(x+sqrt(x^2+a^2)) --R Type: Expression Integer --E ---S 95 14:208 Axiom cannot simplify this expression +--S 104 cc:=aa-bb --R --R +-------+ +-------+ @@ -1493,6 +1585,17 @@ cc:=aa-bb --R 2 --R Type: Expression Integer --E + +--S 105 14:208 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 2 +--R - 3a log(a ) - 2a +--R (4) ------------------ +--R 2 +--R Type: Expression Integer +--E + @ \section{\cite{1}:14.209~~~~~\displaystyle @@ -1504,7 +1607,7 @@$$
<<*>>=
)clear all

---S 96
+--S 106
aa:=integrate((x^2+a^2)^(3/2)/x^3,x)
--R
--R
@@ -1527,48 +1630,60 @@ aa:=integrate((x^2+a^2)^(3/2)/x^3,x)
--R                                          Type: Union(Expression
Integer,...)
--E

---S 97
-bb:=-(x^2+a^2)^(3/2)/(2*x^3)+3/2*sqrt(x^2+a^2)-3/2*a*log((a+sqrt(x^2+a^2))/x)
+--S 107
+bb:=-(x^2+a^2)^(3/2)/(2*x^2)+3/2*sqrt(x^2+a^2)-3/2*a*log((a+sqrt(x^2+a^2))/x)
--R
--R                    +-------+
---R                    | 2    2                         +-------+
---R              3    \|x  + a   + a       3    2    2  | 2    2
---R        - 3a x log(--------------) + (3x  - x  - a )\|x  + a
+--R                    | 2    2                    +-------+
+--R              2    \|x  + a   + a       2    2  | 2    2
+--R        - 3a x log(--------------) + (2x  - a )\|x  + a
--R                          x
---R   (2)  ------------------------------------------------------
---R                                    3
---R                                  2x
+--R   (2)  -------------------------------------------------
+--R                                 2
+--R                               2x
--R                                                     Type: Expression
Integer
--E

---S 98     14:209 Axiom cannot simplify this expression
+--S 108
cc:=aa-bb
--R
--R   (3)
---R                           +-------+                       +-------+
---R                5     3 3  | 2    2         6     3 4      | 2    2
---R       ((- 12a x  - 3a x )\|x  + a   + 12a x  + 9a x )log(\|x  + a   - x +
a)
---R     +
---R                         +-------+                       +-------+
---R              5     3 3  | 2    2         6     3 4      | 2    2
---R       ((12a x  + 3a x )\|x  + a   - 12a x  - 9a x )log(\|x  + a   - x - a)
---R     +
---R                                                         +-------+
---R                         +-------+                       | 2    2
---R              5     3 3  | 2    2         6     3 4     \|x  + a   + a
---R       ((12a x  + 3a x )\|x  + a   - 12a x  - 9a x )log(--------------)
---R                                                               x
+--R                 +-------+                    +-------+
+--R                 | 2    2                     | 2    2
+--R       - 3a log(\|x  + a   - x + a) + 3a log(\|x  + a   - x - a)
--R     +
---R                                                  +-------+
---R          6     5     2 4     2 3     4 2     4   | 2    2      7     6
2 5
---R       (4x  - 4x  + 7a x  - 7a x  + 3a x  - 3a x)\|x  + a   - 4x  + 4x  -
9a x
+--R               +-------+
+--R               | 2    2
+--R              \|x  + a   + a
+--R       3a log(--------------)
+--R                     x
+--R  /
+--R     2
+--R                                                     Type: Expression
Integer
+--E
+
+--S 109
+dd:=expandLog cc
+--R
+--R   (4)
+--R               +-------+                +-------+
+--R               | 2    2                 | 2    2
+--R       3a log(\|x  + a   + a) - 3a log(\|x  + a   - x + a)
--R     +
---R         2 4     4 3     4 2    6     6
---R       9a x  - 6a x  + 6a x  - a x + a
+--R               +-------+
+--R               | 2    2
+--R       3a log(\|x  + a   - x - a) - 3a log(x)
--R  /
---R                   +-------+
---R        5     2 3  | 2    2      6     2 4
---R     (8x  + 2a x )\|x  + a   - 8x  - 6a x
+--R     2
+--R                                                     Type: Expression
Integer
+--E
+
+--S 110    14:209 Schaums and Axiom differ by a constant
+ee:=complexNormalize dd
+--R
+--R          3a log(- 1)
+--R   (5)  - -----------
+--R               2
--R                                                     Type: Expression
Integer
--E