Definite Integral from 0 to Half Pi of Square of Logarithm of Cosine x

Theorem

 * $\displaystyle \int_0^{\pi/2} \paren {\map \ln {\cos x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$