Let $\alpha>2$, $\ell\in\mathbb{N}$, and $i=\sqrt{-1}$. Consider the following integral $$ I=\int_{-1/2}^{1/2} \int_{-1/2}^{1/2} e^{i2\pi \ell (x+y)}\frac{\sin^2(2\pi x)\sin^2(2\pi y)}{\alpha-\cos(2\pi x)-\cos(2\pi y)}\,\mathrm{d}x\,\mathrm{d}y. $$
Does there exist a closed-form expression for $I$?
Usual contour integral trick of $z=e^{2\pi i x}$ and $w=e^{2\pi i y}$. Then you have a rational function of $z,w$ integrated $dz\,dw$ over $C\times C$, with $C$ being the unit circle. Fix $w$ and do the $z$ integral; then do the $w$ integral.