I was thinking of the generalization of the problem here, that is
$$\int_0^{\large \frac{\pi}{2}}\frac{x^{2n}\cdot\log{{\sin{x}}}}{\sin^{2n}{x}}dx, \space n\ge 1$$ Maybe you recommend me some tools?
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2Long time no see ! – Claude Leibovici Apr 12 '15 at 11:28
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@ClaudeLeibovici Yeah, true. I'm pretty often in chat. :-) – user 1591719 Apr 12 '15 at 11:33
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My first impulse it to try to find a general formula for \begin{equation} \frac{d^{2n}}{dx^{2n}}\left(\log^{k}(\sin{x})\right) \end{equation} for $1 \leq k \leq 2n$, and then try to find a linear combination of such terms that adds to the integrand under consideration. – Mark Hubenthal Apr 17 '15 at 21:45