I am looking for the closed form of the following integral involving inverse Fourier,
$$\int_{-\infty}^\infty \frac{e^{-i\tau\omega}}{(A-\cosh(2\tau))^{\frac{\nu}{2}-1}}\,\mathrm{d}\tau$$ where $\nu>2$ is an integer. If $\nu$ is a odd number there is a branch cut. I am unable to do the integral in that case.
Both $\tau$ and $\omega$ are real and $\omega>0$. A is a real constant and $A>0$. Can someone kindly refer to me any formula or book where this type of integral is discussed. Specific cases such as $\nu=3 ,5$ etc comments are also welcome.