TITLE: Evaluating $\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx$
QUESTION [36 upvotes]: What starting point would you recommend me for the one below?
$$\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx $$
EDIT
Thanks to Felix Marin, we know the integral evaluates to  
$$\displaystyle{\large{\ln^{2}\left(\, 2\,\right) \over 2\pi}}$$

REPLY [3 votes]: We can get the serials Ana calculated.