I am trying to solve the following integral in spherical 3D coordinates $$\mathcal{I}(\theta,\phi)= \int_0^{2\pi}d\phi_{\Delta}\int_{\delta}^{\Delta/2}\sin\theta_{\Delta}d\theta_{\Delta} \frac{1}{[1-\sin\theta_{\Delta}\sin\theta\cos(\phi_{\Delta}-\phi)-\cos\theta_{\Delta}\cos\theta]}$$ where $(\theta_{\Delta},\phi_{\Delta})$ are the spherical angular coordinates that I wish to integrate with respect to, whereas $(\theta,\phi)$ are simply some free angular spherical coordinates (of some other vector).
I would like to learn some hints on computing it. I have tried switching the integration order, writing integrand as another integral, namely $$\frac{1}{[1-\sin\theta_{\Delta}\sin\theta\cos(\phi_{\Delta}-\phi)-\cos\theta_{\Delta}\cos\theta]}= \int_{-\infty}^0d\alpha \exp\Big(\alpha [1-\sin\theta_{\Delta}\sin\theta\cos(\phi_{\Delta}-\phi)-\cos\theta_{\Delta}\cos\theta]\Big)$$ But nothing seems to work. Can someone provide me with a hint? Also, what happens in the $\delta\rightarrow0$ limit?
Thanks