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 $$\underset{0}{\overset{\pi \mathrm{/}2}{\int }}\ln (\cosh \frac{x}{\sqrt{3}}+\cos \frac{\ln (2\cos x)}{\sqrt{3}})dx=\frac{{\pi }^2}{8\sqrt{3}}-\frac{\pi }{4}\ln (1+\sqrt{3})+\frac{13\pi }{24}\ln 2.$$