Can the integral \begin{equation} \int_{0}^\pi\frac{d\omega}{(2\sin(\omega/2))^{2\alpha}+2\lambda(2\sin(\omega/2))^\alpha\sin(\alpha\omega/2)+\lambda^2},\quad 0<\alpha<2,\quad \lambda>0 \end{equation} be evaluated?
At first thought, I tried to write $\sin(\alpha\omega/2)$ as function of $\sin(\omega/2)$ such that a variable replacement $t=\sin(\omega/2)$ can be done. However, it seems no such relation exists.
Could you give me some ideas?
Thank you!