Recovering cosh(ax) from it's fourier transform

Question

Let's say $f(x)=\cosh(ax)$, where $a$ is a complex number and $x$ is real. Then the fourier transform is $F(\omega)=\sqrt \frac{\pi}{2} \delta(\omega-ia)+\sqrt \frac{\pi}{2} \delta(\omega+ia)$. So it's the sum of two dirac deltas that take their nonzero values at the complex numbers $z=\pm ia$. Now lets say we blindly take the inverse fourier transform. $\frac{1}{\sqrt {2\pi}}\int_{-\infty}^{\infty}F(\omega)e^{i \omega x}d \omega=0$, since we just integrated along the real line and the dirac deltas take their nonzero values somewhere in the complex plane. What went wrong? Are we supposed to use some kind of contour? What contour are we supposed to use?