TITLE: Closed form of a log-trig integral
QUESTION [8 upvotes]: Does there exists a closed form of$$ \displaystyle \int _{ 0 }^{ \pi /2 }{ x\sqrt { \tan { x }  } \log { (\cos { x } ) }\ dx }$$
If exists can someone find a way to tackle this integral and provide a closed-form of it. Many similar integrals have closed form and I believe this one, too.

REPLY [2 votes]: $ \displaystyle \int _{ 0 }^{ \pi /2 }{ x\sqrt { \tan { (x) }  } \log { (\cos { (x) } ) } dx }=-\frac{\pi\sqrt{2}}{8}(\frac{\pi^2}{3}+4G+3\ln^22+{\pi}{\ln{2}}) $