  We investigate the integral \[\int_0^\infty \cosh^\mu\!t\,K_\nu(z\cosh t)\,dt
\qquad \Re(z)>0,\] 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 $\mu-\nu$.
