# Fourier transform of a damped cosine wave with a linear frequency chirp

I want to take the Fourier transform of the following transient signal, $$f(t) = e^{-t/\tau} \cos((\omega_0 + m t)t)$$, where $$m$$ is some gradient parameter in units of $$\rm{Hz}/s$$. I thought this would be quite straight forward -- although most of my approaches have been made using Mathematica -- which struggles to provide anything useful.

I would have assumed the resultant function would have a Lorentzian-like peak profilein a similar way to if one takes the trivial Fourier transform of $$f(t) = e^{-t/\tau} \cos(\omega_0t)$$. Does anyone have any ideas on how I can approach this, or, an alternative to Fourier transforming a damped sinusoidal with a linear (or even non linear) frequency, i.e. a chirp.