On page 140 of Special Functions and Their Applications by N. N. Lebedev, we are asked to derive the following two integral representations:
$$\tag{1}K_{\nu }(x) K_{\nu }(y)=\frac{1}{2} (\pi \csc (\nu \pi )) \int_{\log \left(\frac{y}{x}\right)}^{ \infty } J_0\left(\sqrt{2 x y \cosh (t)-x^2-y^2}\right) \sinh (\nu t) \, dt,$$ $x>0$, $y>0$, $| \Re(\nu )| <\frac{1}{4}$ and
$$\tag{2}I_{\nu }(x) K_{\nu }(y)=\frac{1}{2} \int_{\log \left(\frac{y}{x}\right)}^{ \infty } J_0\left(\sqrt{2 x y \cosh (t)-x^2-y^2}\right) e^{-\nu t} \, dt ,$$ $x>0$, $y>0$, $\Re(\nu )>-\frac{1}{4}$.
Here $K_{\nu }(z)$,$I_{\nu }(z)$ are the Macdonald function and the modified Bessel function of the first kind respectively and $J_0(z)$ is the Bessel function of the first kind of order $0$.
A more general form of these two results (Bessel functions of different order) can be found in the paper: Integrals for the Product of two Bessel Functions (II), Dixon & Ferrar, The Quarterly Journal of Mathematics, Vol. OS-4, Issue 1, 1933, pp. 297-304. Is there a simpler derivation available for the special cases given above?