New indefinite integrals of Heun functions

TL;DR

This paper derives numerous new indefinite integrals involving Heun functions, hypergeometric functions, and elliptic functions, which cannot be computed by standard symbolic software, expanding the analytical tools for special functions.

Contribution

It introduces a large set of new indefinite integrals involving Heun functions and related special functions using a Lagrangian approach, not obtainable by common computer algebra systems.

Findings

Derived new indefinite integrals involving Heun functions.

Obtained integrals involving hypergeometric and elliptic functions.

Results are not computable by Maple or Mathematica.

Abstract

We present a conspicuous number of indefinite integrals involving Heun functions and their products obtained by means of the Lagrangian formulation of a general homogeneous linear ordinary differential equation. As a by-product we also derive new indefinite integrals involving the Gauss hypergeometric function and products of hypergeometric functions with elliptic functions of the first kind. All integrals we obtained cannot be computed using Maple and Mathematica.