In attempts to arrive at an inverse Fourier transform for a certain integral, I have reduced it to a real integral of the following type
$$I(t) = \int_{-\infty}^{\infty} \frac{\exp(-s^2 \omega^2 / 2)}{1 + \omega^2 \tau^2} (\cos \omega t + \omega \tau \sin \omega t)\,d\omega$$
It is known that all variables are real $\tau, \sigma >0$, and $t$ can be arbitrary.