Elliptic integral

Summary

An elliptic integral is an integral of a rational function of $t$ and $\sqrt{P(t)}$. Here $P$ is a polynomial of degree $3$ or $4$ without repeated roots. Such integrals arose from the arc length of the ellipse and in general cannot be written with elementary functions. Legendre showed that every elliptic integral reduces to elementary functions and three standard integrals, of the first, second and third kind. Inverting the integral of the first kind leads to the Jacobi elliptic functions.