Log-trigonometric integrals and elliptic functions
arXiv:2012.01161
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 \[ \int\limits_0^{π/{2}}\ln\left(\cosh\frac{x}{\sqrt{3}}+\cos\frac{\ln \left(2\cos x\right)}{\sqrt{3}}\right)dx=\frac{π^2}{8\sqrt{3}}-\fracπ{4}\ln\left(1+\sqrt{3}\right)+\frac{13π}{24}\ln 2. \]
10 pages