I am trying to compute
$$\int_{-\infty}^{+\infty} dx \, e^{i x t a} f(x) f(x-t b)$$
where $f(x) = \frac{1}{\sqrt{1+x^2}}$ and $t, a, b \in \mathbb{R}$.
Mathematica for instance doesn't know how to do it. I have tried to do it in the complex plane. I think the contour should be something like in the picture, with two branch cuts: from $i$ to $-i$ and from $i+tb$ to $-i+tb$ (I forgot the $t$ in the figure).
But the problem is that I don't know how to compute the integral along the vertical segments, next to the branch cuts, basically because it looks very similar to the actual integral. Any idea? Being able to compute the Fourier transform of the integral with respect to $t$ would also work for me.