Integral involving modified Bessel function of the second kind $K_{\alpha}(z)$

Question

Given $a,b\geq0 $. What is the closed form of this integral? $$\int_{0}^{\infty}\ln\left(K_{a-b}(2x)\right) K_{a-b}(2x)x^{a+b-1}\mathrm{d}x$$ Where $K_{\alpha}(z)$ is the modified Bessel function of the 2th kind.

I think it can ben useful to consider this following: $K^{(1,0)}_{\alpha}(z):=\dfrac{\mathrm{d}}{\mathrm{d}\alpha}K_{\alpha}(z)$
Since, $$K_{\alpha}(z)=\int_{0}^{\infty}e^{-z\cosh(t)}\cosh(\alpha t)\mathrm{d}t\\K^{(1,0)}_{\alpha}(z)=\int_{0}^{\infty}e^{-z\cosh(t)}\sinh(\alpha t)t\mathrm{d}t$$