Does a Bessel function count as "elementary" ? The Fourier transform of $J_0(x)$ is $2/\sqrt{1-x^2}$ for $|x|<1$ and zero otherwise. 

Concerning the combination of arctangents, why not just Fourier transform ${\rm arctan}\,x$ itself into $-i\pi e^{-|x|}/x$ ?

Besides $1/\cosh x$, you can Fourier transform $1/\sinh x$ into $-i\pi\tanh(\pi x/2)$.