Complex integral with branch cuts.

Question

The problem is the following, $$ \ \int_{-\infty}^{\infty} du e^{-iu w }\bigg( \cos (\theta u) - i \frac{\xi }{\theta} \sin (\theta u)\bigg)^{-1/2} $$ When we go to complex $u$ plane there are branch cuts at, $$u = -i \frac{1}{2\theta}\log \bigg( \frac{\xi + \theta}{\xi-\theta} \bigg) -\frac{\pi k}{\theta}$$where $k$ is an integer. My question is how to evaluate the branch cut discontinuity for such a function. I am confused since it is a square root of a trigonometric function of a complex variable now. What should be the general strategy here?