Integral Involving Cos and Cosh

Question

I would like to evaluate the following integral $\int\limits_{0}^{\infty} \dfrac{\cos \left(\omega t\right)}{\cosh ^{\sigma}\left(t\right)} dt$ where $\sigma$ and $\omega$ are positive real constants. If you have noticed this integral in any table or handbook, lease let me know. Thanks for your help...

Answer 1

As @Jack D'Aurizio commented, this involves the Gaussian hypergeometric function.

$$I=\int \cosh ^{-\sigma }(t) \cos ( \omega t )\;dt=\Re \int \cosh ^{-\sigma }(t) e^ {i \omega t})\;dt$$ $$J=\int \cosh ^{-\sigma }(t) \,e^ {i \omega t}\;dt$$ $$J=\frac{\left(e^{2 t}+1\right) e^{i t \omega } \cosh ^{-\sigma }(t) }{\sigma +i \omega } \, _2F_1\left(1,\frac{2-\sigma +i \omega }{2} ;\frac{2+\sigma +i \omega }{2};-e^{2 t}\right)$$ $$K=\int_0^\infty \cosh ^{-\sigma }(t)\, e^ {i \omega t}\;dt$$ $$K=\frac{2^{\sigma } }{\sigma -i \omega }\,\,\, _2F_1\left(\sigma ,\frac{\sigma -i \omega}{2} ;\frac{2+\sigma -i \omega }{2};-1\right)$$