I've been trying to calculate this Kontorovich-Lebedev transform which involves an integral with a product of two modified Bessel functions of the second kind with respect to their order. Does anyone know how to go about doing these?
$$ \int_0^\infty K_{i\xi}(x_1) K_{i\xi}(x_2) \sinh(\pi\xi)\cos(a\xi)\coth(b \xi)\mathrm{d}\xi $$ $a$ is real, $b,~x_1$, and $x_2$ are strictly positive.