Log-trigonometric integrals and elliptic functions

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 $$\int\limits_0^{{\pi}/{2}}\ln\left(\cosh\frac{x}{\sqrt{3}}+\cos\frac{\ln \left(2\cos x\right)}{\sqrt{3}}\right)dx=\frac{\pi^2}{8\sqrt{3}}-\frac{\pi}{4}\ln\left(1+\sqrt{3}\right)+\frac{13\pi}{24}\ln 2.$$