Pseudo-elliptic integral

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.