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, you can Fourier transform ${\rm arctan}\,x$ itself into $-i\pi e^{-|x|}/x$.

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

$$\widehat{1\over \cosh^2(x)} = {\pi \xi\over \sinh({\pi\xi\over 2})}$$