This is an integral in <A HREF="https://books.google.nl/books?id=X7HtCAAAQBAJ">Oberhettinger's Table of Fourier Transforms</A> (page 27, with $r=\sqrt{a^2+x^2}$). It is not a very helpful result, but it does suggest that a fully closed-form expression will not be forthcoming.

<IMG SRC="https://ilorentz.org/beenakker/MO/cosineintegral_Oberhettinger.png"/>

There also seems to be some typo in the formula, a numerical check does not match.