I am new to Bessel functions and need to solve the following integral
\begin{equation} \int J_{0}\left(\alpha x\right)J_{0}\left(\beta x\right)\cos(kx)\,\mathrm{d}x \end{equation}
with $J_{0}$ denothing the zeroth-order Bessel functions of the first kind, and real-valued parameters $\alpha$, $\beta$, and $k$. I already checked Wolframalpha, Gradshteyn's book on integrals, and looked on the stackexchange forum, however without success. However, I found on WolframAlpha and some similar related integrals in Integral of squared absolute of Bessel function of first kind, Integral of squared Bessel function, How to derive a Hypergeometric function from the integration of the square of Bessel function. During my search I also came across the orthogonality property of Bessel functions as well as to a product of Bessel functions with a Gaussian kernel. However, it seems to me that my integrals of interest cannot be related to these approaches.
Do you know some strategy how to obtain a solution? Your help is appreciated, thank you :)