This integral is known as the moments for the generalized log-sine integrals. The notation I am using is similar to Lewin and what he used in the 1950's-1980's. $$ Ls^{(k)}_n(\sigma):=-\int_0^\sigma \theta^k \ln^{n-1-k}\big| 2\sin\frac{\theta}{2}\big|d\theta,\quad k\geq 0,n\geq 1. $$ Note that in each case the modulus is not needed for $0\leq \sigma \leq 2\pi$. Thanks. The only progress I had realized was that for $k=0$ $$ Ls_1(\sigma)=-\sigma,\quad Ls^{(0)}_n(\sigma)=Ls_n(\sigma), $$ explicitly leads to $$ Ls_2(\sigma)=Cl_2(\sigma):=\sum_{n=1}^\infty \frac{\sin (n\sigma)}{n^2} $$ which is the Clausen function.

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    $\begingroup$ This is your 3rd question within 20 minutes... $\endgroup$ – Shahar Apr 12 '14 at 23:44
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    $\begingroup$ @Shahar Ok, and what's your point? I was aware of how many questions I have posted. Clearly you have trouble adding, as I posted 3 questions within 16 minutes, not 20. Do you have any mathematical input on this question?? I am well aware of how many questions you can post in 24 hours. $\endgroup$ – Jeff Faraci Apr 12 '14 at 23:48

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