Summing Legendre polynomials $P_{l}(\cos\theta)$ often leads to expressions containing $\frac{1}{\sqrt{t^2-2t\cos\theta+1}}$, as this is the generating function for the Legendre polynomials. I want to evaluate the following integral: \begin{equation} I(\gamma,s,\theta,\lambda)=\int_0^1dt\frac{t^{s+i\gamma}(1-t)^{-2i\gamma+1}}{\sqrt{(e^{i\lambda}t)^2-2(te^{i\lambda})\cos\theta+1}} \end{equation}
which arose from summing Legendre polynomials. Is there a general approach for handling integrals of this kind? Is there any specific reference I should consult?
All parameters are $\gamma,s,\theta,\lambda$ are assumed to be in a region where this integral converges.