Let $\alpha>0$ and $\beta\in\mathbb{R}$. I am looking for an explicit formula for the integral

$$\int_{-\infty}^{\infty} (1+x^2)^{-1/2}e^{-\alpha x^2}e^{-i \beta x}dx.$$

I tried several changes of variables, and contour integration doesn't seem to work.

Motivation comes from the following closely related kernel $$K(s,t)=e^{-\frac{(s-t)^2}{4}}\int_{-\infty}^{\infty}(1+x^2)^{-1/2}e^{-\frac{(s-t)^2}{4} x^2}e^{-i (s^2-t^2) x}dx,$$ which provides an example of a compact integral operator on $L^2(\mathbb{R})$ that is not Hilbert-Schmidt. I would like to check the details.

Thank you!