I've been trying to compute the following integral with no avail, does anyone have any ideas? All free constants are strictly positive. $K_\nu(x)$ is the usual modified Bessel function of the second kind, also known as Macdonald function.
$$ \int_0^\infty K_{i a_1}(b_1 x) K_{i a_2}(b_2 x) K_{i a_3}(b_3 x) K_{i a_4}(b_4 x)x\mathrm{d}x $$