How to make this function suitable for Fourier transform

Question

I have a function that I want to Fourier transform, $$f(t) = A e^{-t/\tau} \sin(\omega_{0}(1 - \alpha e^{-t/\tau}) t) \text{,}$$ but it turns out the software I am using (Mathematica) really struggles with this. I was wondering if anyone can see a way of tidying up this expression, perhaps with some integral forms so that I can go ahead and take the FT of this.

Answer 1

The long and short of this is; you can't. The best thing to do is to approximate the exponential component with a series expansion of $e^{x}$, or, use the solutions for the chirped oscillating wave.