Table of Integrals, Series, and Products, 8th Edition

8

Special Functions

8.1 Elliptic Integrals and Functions

8.11 Elliptic integrals

8.110

1. Every integral of the form $\int R\left(x,\sqrt{P(x)}\right)\text{d}x$, where P(x) is a third- or fourth-degree polynomial, can be reduced to a linear combination of integrals leading to elementary functions and the following three integrals:

$\begin{array}{ccc}{\displaystyle \int \frac{\text{d}x}{\sqrt{(1-x^2)(1-k^2{x}^2)}},}& {\displaystyle \int \frac{\sqrt{(1-k^2{x}^2)}}{(1-x^2)}}\text{d}x,& {\displaystyle \int \frac{\text{d}x}{(1-n{x}^2)\sqrt{(1-x^2)(1-k^2{x}^2)}},}\end{array}$

which are called respectively elliptic integrals of the first, second, and third kind in the Legendre normal form. The results of this reduction for the more frequently encountered ...