[Axiom-developer] Re: Axiom' integrator

[Waldek Hebisch]

Martin Rubey wrote:
> > If, we can finish the implementation of this algorithm in Axiom,
> > then would your re-worded statement above be correct?
>
> is no. Furthermore, Axiom only returns "failed" if the Risch procedure didn't
> succeed, which constitutes hardly a proof...
>
> Still, it would be great if we could complete the implementation of the Risch
> "algorithm". (Or find out that it is completely implemented anyway.)

AFAIU the claim is that Axiom integrator will do one of the following:

1) give elementary integral
2) prove that elementary integral does not exist
3) give error message about unimplemented branch of the Risch algorithm
4) run out of resources

This claim is both in detailed Axiom documentation and in Bronstein post
to sci.math.symbolic. 

Risch algorithm in Axiom do have unimplemented branches, the following
is an example:

integrate(D(exp(-a*sqrt(1 - b*cos(theta)))/(a+b+theta), [theta], [1]), theta)

result:

   >> Error detected within library code:
   Function not supported by Risch d.e.

I doubt that just finishing the implementation will give much improvement.
Namely, the unimplemented cases have huge resource requirements, so 
implementing them is likely to lead to case 4 above.

I think that one can gain more by implementing a bunch of shortcuts
(like the reduction I show in my answer to Bob McElrath) to handle
common "textbook" cases.

-- 
                              Waldek Hebisch
address@hidden