Pseudo-elliptic integral

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Let $R(\cdot ,\cdot)$ be a rational function of two variables and $f(z)$ a polynomial of degree three or four, without multiple roots. A pseudo-elliptic integral is an integral of the form \begin{equation*} \int R(z,\sqrt{f(z)})\,dz, \end{equation*} which can be expressed elementarily, that is, by algebraic functions in $z$ or in the logarithms of such functions. For example, \begin{equation*} \int \frac{z^3\,dz}{\sqrt{z^4-1}} \end{equation*} is a pseudo-elliptic integral. See Elliptic integral.

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Pseudo-elliptic integral. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article