View source for Pseudo-elliptic integral

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756701.png" /> be a rational function of two variables and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756702.png" /> a polynomial of degree three or four, without multiple roots. A pseudo-elliptic integral is an integral of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756703.png" /></td> </tr></table>

which can be expressed elementarily, that is, by algebraic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756704.png" /> or in the logarithms of such functions. For example,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075670/p0756705.png" /></td> </tr></table>

is a pseudo-elliptic integral. See [[Elliptic integral|Elliptic integral]].