Fourier inverse transform of $\frac1{(i\omega+a)(\omega^2-b^2)}$

Question

How can I calculate the Inverse Fourier transform of

$$f(\omega)=\frac{1}{(i\omega+a)(\omega^2-b^2)},\;a,b\in\mathbb{R}, a>0.$$

I guess that I cannot use the residual theorem, since the function has $2$ real poles.

Thank you very much for your help.

Best Regards!