Fourier transform of composed functions

We prove an explicit formula for the Fourier transform of $f(u(t))$, given the Fourier transform of $f(t)$, assuming $f\in L^2(-\mathrm{\infty },\mathrm{\infty })$ and $u$ sufficiently well behaved. We illustrate its usefulness by calculating the Fourier transform of $(a^2+\sinh (bt{)}^2{)}^{-1}$.