Re: [Axiom-developer] 20080426.01.tpd.patch (CATS Schaums-Axiom equivalence testing (26-28))

[Doug Stewart]

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I think your Schaums is out of date on this one.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