Reduction of Elliptic Integrals to Legendre Normal Form

It is well known that any elliptic integral can be transformed into a linear combination of elementary functions and Legendre’s three Elliptic functions. Methods for transforming these integrals to the Legendre form are described in numerous papers and textbooks. However, when it comes to actually designing and implementing such a reduction algorithm the existing methods require significant modification before they can be used in practical problems. As an example, in all cases these algorithms require the need for computing the roots of polynomials of arbitrary degree. Symbolic root-solving either fails or produces expressions for the roots that are unwieldy. In this paper we describe two methods for reducing elliptic integrals to their Legendre normal form in a computer algebra system. In both approaches presented here, the factorization of the polynomials are delayed and an exact, symbolic closed form solution is computed. These exact forms can then be used for numerical evaluation to arbitrary precision using such methods as the AGM algorithm.