How to calculate the following 3D ${\bf k}$-space integral?

Question

I'm struggling to calculate$$ \sum_{a,b=\pm}\int\frac{\text d\mathbf{k}}{(2\pi)^3} \frac{\theta(-\xi_{\mathbf{k}+\mathbf{q}}^a)-\theta(-\xi_{\mathbf{k}}^b)}{\xi_{\mathbf{k}}^b-\xi_{\mathbf{k}+\mathbf{q}}^a}, $$ where $\xi_{\mathbf{k}}^\pm=\xi_{\mathbf{k}}\pm \alpha k_\perp=\frac12k^2-\mu\pm \alpha k_\perp$, for $k_\perp^2=k_x^2+k_y^2$. The integration is over the whole ${\bf k}$-space. $\theta$ is the Heaviside step function. $\mu$ is the chemical potential.