Introduction

Abstract

Integrals of the form $\int {R\left[ {t,\sqrt {{P\left( t \right)}} } \right]} dt$ , where P(t) is a polynomial of the third or fourth degree and R is a rational function, have the simplest algebraic integrands that can lead to nonelementary¹ integrals. Equivalent integrals occur in trigonometric and other forms, in pure and applied mathematics. Such integrals are known as elliptic integrals because a special example of this type arose in the rectification of the arc of an ellipse. Although some early work on them was done by Fragnano, Euler, Lagrange and Landen, they were first treated systematically by Legendre, who showed that any elliptic integral may be made to depend on three fundamental integrals which he denoted by F(φ, k), E(φ, k) and Π(φ, n, k). These three integrals are called Legendre’s canonical elliptic integrals of the first, second and third kind respectively. Legendre’s normal forms are not the only standard forms possible, but they have retained their usefulness for over a century.