See
*A study of alternative methods for the symbolic calculation of elliptic
integrals
by Ed Ng, (1976),
http://portal.acm.org/citation.cfm?id=800205.806359
which describes several ways to do the symbolic reductions, and how to
represent
them canonically, in Macsyma. I think that a method by Carlson is
advocated, but
there are open issues about the best way..
RJF
*Michel Talon wrote:
> .....
> Well, speaking of elliptic functions, i think a worthy addition would be
> recognize the fact that any integral of the form
> integrate(R(y(x),x),x)
> where R(y,x) is a rational function of x and y, and y(x) is the square root
> of a polynomial in x of degree 3 or 4 without double roots is elliptic.
> The traditional case is
> integrate(1/sqrt((1-x^2)*(1-k^2*x^2)),x)
> but the general case boils down to the same after systematic reductions. Of
> course in case of a double root the integral becomes trigonometric. If the
> relation between x and y is more complicated (meaning the curve R(y,x)=0 is
> not elliptic) then it is certain that the integral cannot be computed on
> elementary or elliptic functions.
>
> --
> Michel Talon
>
> _______________________________________________
> Maxima mailing list
> Maxima at math.utexas.edu
> http://www.math.utexas.edu/mailman/listinfo/maxima
>