How to calculate integral $$J_n=\int_{-\pi}^\pi \frac{\sin{(nx)}}{(1+2^n) \sin{x}}\,\mathrm{d}x\:?$$
I tried partial integration but did not succeed in finding a recurrent relation? Also, tried Moivre formula for $I_n+iJ_n$, where $I_n=\int_{-\pi}^\pi \frac{\cos{(nx)}}{(1+2^n) \sin{x}} dx$, but also without success.
Any help is welcome. Thanks in advance.