Table of Integrals

A. Dieckmann, Physikalisches Institut der Uni Bonn

This table contains hundreds of expressions: indefinite and definite integrals of elliptic integrals, of square roots, arcustangents and a few more exotic functions. Most of them are not found in Gradshteyn-Ryzhik.

Results may be valid outside of the given region of parameters, but should always be checked numerically!

Indefinite Integrals:

strange connection between EllipticTheta and Elliptic Integrals…

apparently it is not known how to integrate m[q] = InverseEllipticNomeQ[q] over q,

the following expressions have therefore been calculated backwards, that is by differentiating the right side with respect to q :

With a common term like

the next six integrals are :

Here the result is a sum shown in Mathematica syntax:

Let sl[x] and cl[x] denote the lemniscate sine and cosine functions, ϖ is the lemniscate constant: