Elliptic Integrals

Abstract

An elliptic integral is an integral of the form int(A(x)+B(x)sqrt(S(x)))/(C(x)+D(x)sqrt(S(x)))dx, (1) or int(A(x)dx)/(B(x)sqrt(S(x))), (2) where A(x), B(x), C(x), and D(x) are polynomials in x, and S(x) is a polynomial of degree 3 or 4. Stated more simply, an elliptic integral is an integral of the form intR(w,x)dx, (3) where R(w,x) is a rational function of x and w, w^2 is a function of x that is cubic or quartic in x, R(w,x) contains at least one odd power of w, and w^2 has no...