This problem comes from a physics model, and finally boils down to solving the following integration: \begin{equation} \int_{-\infty}^{\infty}\frac{e^{\mathrm{i}\sqrt{x^2+C^2}}}{x^2+B^2}dx, \end{equation} where $C,B$ are positive constants satisfying $B>C$. Clearly the integral converges. However, I want to obtain an explicit formula with respect to the parameters $B$ and $C$. I was considering using residue theorem in complex analysis, but then I don't know how to deal with the branch points caused by term $\sqrt{x^2+c^2}$.
So my question is: Can we obtain an analytic expression with respect to $B,C$ for this integral?