Is there a closed-form expression for the Fourier transform $\widehat{f}$ of $f(x)=\exp(-\sinh(x)^2)$? Can anything nice be said about $\widehat{f}$? (I can see $\widehat{f}$ has exponential decay; see below.)
Motivation: here $f(x)$ is a function of doubly exponential decay; since it can be continued analytically to a strip around the $x$-axis, its Fourier transform $\widehat{f}$ has exponential decay. This is interesting because one can't decay faster than doubly exponentially and also have a Fourier transform that decays exponentially or faster (this is a form of the uncertainty principle). You can find what amounts to this function $f$ in the work of Ramachandra in analytic number theory (for instance).
Your case is a variant of the doubly exponential function. These have no known closed-form FT, if you defined by "closed form" a finite combination of given functions.
What you can do however is expand the "base" exponential into a series of $\sinh{x}$ then into a series of $e^{kx}$ after applying the binomial theorem. Then apply FT on the expanded exponentials. You will end up with a series of Gaussians as a representation of the FT, which some authors assimilate to a "closed form", an ambiguous term over which there is no consensus.