A note on a modified Bessel function integral

We investigate the integral $${\int }_0^{\mathrm{\infty }}{\cosh }^{\mu }\text{\hspace{0.17em}⁣}t{K}_{\nu }(z\cosh t)dt\mathrm{\Re }(z)>0,$$ where $K$ denotes the modified Bessel function, for non-negative integer values of the parameters $\mu$ and $\nu$. When the integers are of different parity, closed-form expressions are obtained in terms of $z^{-1}{e}^{-z}$ multiplied by a polynomial in $z^{-1}$ of degree dependent on the sign of $\mu -\nu$.