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## Re: [Axiom-developer] 20080426.01.tpd.patch (CATS Schaums-Axiom equivale

 From: Doug Stewart Subject: Re: [Axiom-developer] 20080426.01.tpd.patch (CATS Schaums-Axiom equivalence testing (26-28)) Date: Sun, 27 Apr 2008 05:49:29 -0400 User-agent: Thunderbird 2.0.0.12 (X11/20080227)

```address@hidden wrote:
```
On at least 2 of these problems Schaums and Axiom seem to disagree on the results, namely:
```  14:569 SCHAUMS AND AXIOM DISAGREE?
14:571 SCHAUMS AND AXIOM DISAGREE?
```
as the compute result is not constant but does not seem to have a simplification that eliminates x.
```This needs to be studied further.

```
---S 8 of 28
```+--S 17
aa:=integrate(cosh(a*x)^2,x)
```
--R --R
```@@ -170,6 +248,42 @@ aa:=integrate(cosh(a*x)^2,x)
--R                   2a
--R                                          Type: Union(Expression
Integer,...)
--E
+
+--S 18
+bb:=x/2+(sinh(a*x)*cosh(a*x))/2

```
```
+bb:=x/2+(sinh(a*x)*cosh(a*x))/2

this should be

+bb:=x/2+(sinh(a*x)*cosh(a*x))/2a

I think your Schaums is out of date on this one.

```
```+--R
+--R        cosh(a x)sinh(a x) + x
+--R   (2)  ----------------------
+--R                   2
+--R                                                     Type: Expression
Integer
+--E
+
+--S 19
+cc:=aa-bb
+--R
+--R        (- a + 1)cosh(a x)sinh(a x)
+--R   (3)  ---------------------------
+--R                     2a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 20
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I                             %B sinh(y + x) - %B sinh(y - x)
+--I   (4)  %B cosh(y)sinh(x) == -------------------------------
+--R                                            2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 21     14:569 SCHAUMS AND AXIOM DISAGREE?
+dd:=sinhcoshrule cc
+--R
+--R        (- a + 1)sinh(2a x)
+--R   (5)  -------------------
+--R                 4a
+--R                                                     Type: Expression
Integer
+--E
@
```
```

For this next one Maxima gives

(%i6) integrate(1/cosh(a*x)^2,x);
(%o6) -4/(2*a*%e^(2*a*x)+2*a)

But if I try
(%i7) integrate(1/cos(a*x*i)^2,x);
```
(%o7) (2*sin(2*a*i*x))/(a*i*sin(2*a*i*x)^2+a*i*cos(2*a*i*x)^2+2*a*i*cos(2*a*i*x)+a*i)
```(%i8) trigreduce(%);
(%o8) sin(2*a*i*x)/(a*i*cos(2*a*i*x)+a*i)
(%i9) trigrat(%);
(%o9) sin(a*i*x)/(a*i*cos(a*i*x))
(%i10) trigreduce(%);
(%o10) tan(a*i*x)/(a*i)

Now this can be written as
tanh(ax)  /ai
and this is different than Schaums

I think my logic is OK here But I am not an expert.

```
\section{\cite{1}:14.571~~~~~\$\displaystyle
```@@ -201,7 +396,7 @@ \$\$
<<*>>=
)clear all
```
---S 10 of 28
```+--S 31
aa:=integrate(1/cosh(a*x)^2,x)
```
--R --R
```@@ -211,6 +406,176 @@ aa:=integrate(1/cosh(a*x)^2,x)
--R          a sinh(a x)  + 2a cosh(a x)sinh(a x) + a cosh(a x)  + a
--R                                          Type: Union(Expression
Integer,...)
```
--E +
```+--S 32
+bb:=tanh(a*x)/a
+--R
+--R        tanh(a x)
+--R   (2)  ---------
+--R            a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 33
+cc:=aa-bb
+--R
+--R                    2                                  2
+--R        (- sinh(a x)  - 2cosh(a x)sinh(a x) - cosh(a x)  - 1)tanh(a x) - 2
+--R   (3)  ------------------------------------------------------------------
+--R                         2                                      2
+--R              a sinh(a x)  + 2a cosh(a x)sinh(a x) + a cosh(a x)  + a
+--R                                                     Type: Expression
Integer
+--E
+
+--S 34
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R                   sinh(x)
+--R   (4)  tanh(x) == -------
+--R                   cosh(x)
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35
+dd:=tanhrule cc
+--R
+--R   (5)
+--R                  3                      2               2
+--R       - sinh(a x)  - 2cosh(a x)sinh(a x)  + (- cosh(a x)  - 1)sinh(a x)
```
+--R + +--R - 2cosh(a x)
```+--R  /
+--R                         2               2                       3
+--R     a cosh(a x)sinh(a x)  + 2a cosh(a x) sinh(a x) + a cosh(a x)  + a
cosh(a x)
+--R                                                     Type: Expression
Integer
+--E
+
+--S 36
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R               2    cosh(2x) - 1
+--R   (6)  sinh(x)  == ------------
+--R                          2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 37
+ee:=sinhsqrrule dd
+--R
+--R   (7)
+--R                   3                2
+--R       - 2sinh(a x)  + (- 2cosh(a x)  - 2)sinh(a x) - 2cosh(a x)cosh(2a x)
```
+--R + +--R - 2cosh(a x)
```+--R  /
+--R                   2                                                3
+--R       4a cosh(a x) sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
```
+--R + +--R a cosh(a x)
```+--R                                                     Type: Expression
Integer
+--E
+
+--S 38
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R               2    cosh(2x) + 1
+--R   (8)  cosh(x)  == ------------
+--R                          2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 39
+ff:=coshsqrrule ee
+--R
+--R   (9)
+--R                   3
+--R       - 2sinh(a x)  + (- cosh(2a x) - 3)sinh(a x) - 2cosh(a x)cosh(2a x)
```
+--R + +--R - 2cosh(a x)
```+--R  /
+--R                                                                           3
+--R       (2a cosh(2a x) + 2a)sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
```
+--R + +--R a cosh(a x)
```+--R                                                     Type: Expression
Integer
+--E
+
+--S 40
+sinhcuberule:=rule(sinh(x)^3 == 1/4*sinh(3*x)-3/4*sinh(x))
+--R
+--R                3    sinh(3x) - 3sinh(x)
+--R   (10)  sinh(x)  == -------------------
+--R                              4
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 41
+gg:=sinhcuberule ff
+--R
+--R   (11)
+--R       - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
```
+--R + +--R - 4cosh(a x)
```+--R  /
+--R
3
+--R       (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(a x)cosh(2a x) + 4a cosh(a x)
```
+--R + +--R 2a cosh(a x)
```+--R                                                     Type: Expression
Integer
+--E
+
+--S 42
+coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
+--R
+--R                3    cosh(3x) - 3cosh(x)
+--R   (12)  cosh(x)  == -------------------
+--R                              4
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 43
+hh:=coshcuberule gg
+--R
+--R   (13)
+--R       - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
```
+--R + +--R - 4cosh(a x)
```+--R  /
+--R       (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) + 2a cosh(a x)cosh(2a x)
```
+--R + +--R - a cosh(a x)
```+--R                                                     Type: Expression
Integer
+--E
+
+--S 44
+coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
+--R
+--I                              %U cosh(y + x) + %U cosh(y - x)
+--I   (14)  %U cosh(x)cosh(y) == -------------------------------
+--R                                             2
+--R                        Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 45
+ii:=coshcoshrule hh
+--R
+--R   (15)
+--R   - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 2cosh(3a x) - 6cosh(a x)
+--R   ----------------------------------------------------------------------
+--R                (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(3a x)
+--R                                                     Type: Expression
Integer
+--E
+
+--S 46     14:571 SCHAUMS AND AXIOM DISAGREE?
+jj:=complexNormalize ii
+--R
+--R               a x 2
+--R         - 2(%e   )  - 2
+--R   (16)  ---------------
+--R               a x 2
+--R         2a (%e   )  - a
+--R                                                     Type: Expression
Integer
+--E
```
```

```