TITLE: log sin and log cos integral, maybe relate to fourier series
QUESTION [6 upvotes]: I try to use the method of differentiation under integral sign for the first one
And integrate it back, but I failed to find the constant $c$ ....
Anyone hav other method? 
$$\begin{align}
  & \int_{0}^{\frac{\pi }{2}}{{{x}^{2}}{{\ln }^{2}}\left( 2\cos x \right)\text{d}x} \\ 
 & \int_{0}^{\frac{\pi }{3}}{x{{\ln }^{2}}\left( 2\sin \frac{x}{2} \right)}\text{d}x \\ 
\end{align}$$

REPLY [2 votes]: A related problem. One can manage to get the following  closed form solution for the second integral,
$$\int_{0}^{\frac{\pi }{3}}{x{{\ln }^{2}}\left( 2\sin \frac{x}{2} \right)}\text{d}x = \frac{1}{4}\,{_5F_4\left(1,1,1,1,1;\,\frac{3}{2},2,2,2;\,\frac{1}{4}\right)}\sim 0.2555485412 $$