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!
Integrals kindly contributed by Mariusz Iwaniuk are appended at the bottom of this page .
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 :
Antiderivatives (indefinite integrals) of products of Gauss functions with Error functions (some not found elsewhere):
For integrals with limits like zero and infinity look at https://www-elsa.physik.uni-bonn.de/~dieckman/IntegralsDefinite/DefInt.html
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:
See below Mariusz's integrals that I was able to verify :