LogSine Moments $\int_0^\sigma \theta^k \ln^{n-1-k}\big| 2\sin\frac{\theta}{2}\big|d\theta$

Question

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.