Could anyone help me with a closed form solution of the following integral?
$$\int_1^\infty \frac{K_0(a\,x)\cos(b\,x)}{x^2}dx=\,\,?\quad \text{with} \quad \{a,b\}\in \Re\,.$$
Here $K$ is the modified Bessel function of the second kind.
I have checked Gradshteyn and Ryzhik's 7th edition, but I could not find this integral (or related ones) because of the integration bound (all of the integrals in the relevant sections I could find were between $0$ and $\infty$ or between $0$ and $1$).
I have also tried looking into the $\texttt{Mathematica}$ package $\texttt{HolonomicFunctions}$, but from what I can understand (I am by no means an expert) these methods seem to be more tuned towards verifying known identities rather than computing unknown integrals.
Any help is appreciated! Thank you!