where *R* is a rational function of its two arguments, *P* is the square root of a polynomial of degree 3 or 4 with no repeated roots, and *c* is a constant.

Particular examples include:

- The
**complete elliptic integral of the first kind***K*is defined as

- The
**complete elliptic integral of the second kind***E*is defined as

Historically properties of these integrals were studied in connection with the problem of the arc length of an ellipse, by Fagnano and Leonhard Euler.