Evaluation of some integrals involving associated Legendre polynomials and the arctanh function

Résumé

A method for computing a definite integrals is developed. This method consists of representing the product of two associated  Legendre polynomials (ALPs) by a finite series in ALPs with constant coefficients. Each of the two integrals in question is  then transformed in a finite series in simple integrals, each involving a single ALP. Recursive formulas for computing the  latter integrals are proposed. Our method is tested through the computation of the two integrals in question for some values of  degrees and orders of the ALPs. We compare the method with the numerical computation by means of the Simpson method.