Reduction theorems for elliptic integrands with the square root of two quadratic factors

Recent methods of making integral tables and symbolic integration programs for elliptic integrals depend critically on reduction theorems for two particular integrals. These theorems assume that certain variables have positive real part, and the assumption is not always satisfied if the integrand contains the square root of two quadratic polynomials, each with conjugate complex zeros. A new remedy for this difficulty is the use of duplication theorems, leading to the weaker assumption that only sums of two variables must have positive real part. Numerical examples are given. In an Appendix related methods are used to simplify part of an algorithm for numerical computation of a symmetric elliptic integral of the third kind.