Re: [Axiom-developer] [Q] How to classify # integrate(log(%i+z^2), z) -> 2 ?

[William Sit]

Bill Page wrote:
> 
> On Friday, February 11, 2005 9:56 AM Vladimir Bondarenko wrote:
> > ...
> > ===> Case 2.
> >
> > -> integrate(log(%i+z^2), z)
> >
> >                   +-----+    2
> >      +-----+    z\|- 4%i  + z  - %i            2
> >     \|- 4%i log(-------------------) + 2z log(z  + %i) - 4z
> >                        2
> >                       z  + %i
> >    [-------------------------------------------------------,
> >                                2
> >                                  +---+
> >            2          +---+     \|4%i
> >     z log(z  + %i) - \|4%i atan(------) - 2z]
> >                                   2z
> > ...
> > But why the user really needs 2 forms of the same answer?
> >
> 
> I think you should report it as a bug.
> 
I'm not sure, because you can use

(4) -> complexIntegrate(log(%i+z^2),z)

   (4)
                    +-----+
        +-----+    \|- 4%i  + 2z            2
       \|- 4%i log(-------------) + 2z log(z  + %i)
                         2
     +
                        +-----+
          +-----+    - \|- 4%i  + 2z
       - \|- 4%i log(---------------) - 4z
                            2
  /
     2
                            Type: Expression Complex Integer

to obtain one form. Using integrate when the integrand involves complex values
and ask for real forms does not sound correct. Axiom may (I am not sure) have
interpreted %i as a parameter with no special property.

A comparison with a parameter to replace %i shows the form of the answers in
both cases (integrate or complexIntegrate) are the same.

(5) -> complexIntegrate(log(a+z^2),z)

   (5)
               +----+                       +----+
    +----+    \|- 4a  + 2z     +----+    - \|- 4a  + 2z            2
   \|- 4a log(------------) - \|- 4a log(--------------) + 2z log(z  + a) - 4z
                    2                           2
   ---------------------------------------------------------------------------
                                        2
                                                     Type: Expression Integer
(6) -> integrate(log(a+z^2),z)

   (6)
                 +---+    2
     +---+    2z\|- a  + z  - a           2
   [\|- a log(-----------------) + z log(z  + a) - 2z,
                     2
                    z  + a
                 +-+
        +-+     \|a            2
    - 2\|a atan(----) + z log(z  + a) - 2z]
                  z
                                     Type: Union(List Expression Integer,...)



William