Log-trigonometric integrals and elliptic functions

TL;DR

This paper evaluates a class of log-trigonometric integrals using elliptic functions, providing closed-form solutions and demonstrating the connection between integrals and elliptic function theory.

Contribution

It introduces a method to evaluate log-trigonometric integrals in terms of elliptic functions and derives explicit closed-form evaluations for specific integrals.

Findings

Log-trigonometric integrals can be expressed using elliptic functions.

Closed-form solutions for specific integrals are obtained.

Elliptic integral singular values facilitate integral evaluations.

Abstract

A class of log-trigonometric integrals are evaluated in terms of elliptic functions. From this, by using the elliptic integral singular values, one can obtain closed form evaluations of integrals such as \[ \int\limits_0^{{\pi}/{2}}\ln\left(\cosh\frac{x}{\sqrt{3}}+\cos\frac{\ln \left(2\cos x\right)}{\sqrt{3}}\right)dx=\frac{\pi^2}{8\sqrt{3}}-\frac{\pi}{4}\ln\left(1+\sqrt{3}\right)+\frac{13\pi}{24}\ln 2. \]