Fourier transform of $f(t)=e^{-X\cosh^2(t)}\cosh(t)$

Question

I need to get the Fourier transform of

$$f(t)=e^{-X\cosh^2(t)}\cosh(t)$$

with $X>0$. Any ideas?

Answer 1

Hint

Im not even sure how to evaluate the result but I imagine this isn't the worst starting point. To "reduce the complexity" of the integral use: $$ \cosh(t) = \frac{e^{t}+e^{-t}}{2} \\ \cosh^2(t) = \frac{\cosh(2t)+1}{2}$$ Together the hyperbolic cosines can be replaced with exponentials which may be easier to handle if you use appropriate contour integral black magic.