Let
and consider the problem of integrating
and
The classical technique of reducing such integrals to Legendre normal form involves working with the poles of the rational function, converting the rational function to partial fractions, and reducing each "smaller" part. However, the poles of and are seen to be
so all the poles involve nested radicals. Numerically, at the current setting of Digits, these are given by:
Converting say to a complete partial fraction decomposition can be accomplished by
again an expression that involves complicated nested radicals. The elliptic integration algorithm in Maple takes special care to avoid working with unnecessary radicals whenever possible. Integrals are first reduced to ones involving no repeated poles by using a classical reduction technique discovered by Hermite in the last century which introduces no new radicals during the reduction. Thus, you obtain
In addition, whenever possible, the elliptic integration algorithm in Maple works with an implicit rather than with an explicit representation of such roots. After Hermite reduction to eliminate multiple poles, a final answer is reduced to normal form by using the sum over roots functionality available in Maple. Thus, you obtain
You can evaluate this numerically by
If you want to evaluate the integral numerically to higher digits, you would simply do
Sometimes one also uses the roots of the polynomial under the radical in an implicit rather than an explicit form. For example, you obtain (after some waiting and using alias to obtain a simpler form of answer):
Here the answer is expressed in a (complicated) expression involving implicit roots indexed by their proximity to a numerical root. In this case, you can evaluate such an integral numerically to any number of digits by using:
