I've encountered the following integral: $$ \int_{-\pi}^\pi d\omega \exp\{ \sum_{i,j=1}^n R_j H_{ij}^c(\omega) d_i + \sum_{i,j=1}^n I_j H_{ij}^s(\omega) d_i \},$$ where $R_j, I_j, d_i$ are reals and $$ H_{ij}^c = \cos(ij \omega),\\H_{ij}^s = \sin(ij \omega).$$
I don't necessarily expect a closed form but I thought it couldn't hurt to ask here.
The integral is of the form $\int f(\cos \omega, \sin \omega) d\omega$ since we can expand the $H$-matrices in $c_{pq}\cos^p(\omega)\sin^q(\omega)$ form (being multiple-angle relations). I've tried this and the substitutions $u = \sin \omega, z = \tan(\omega/2)$ but this integral seems way above my math level. I would be very grateful for any pointers.