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. There is an error in both branches, for $y<b$ and for $y>b$. For $y<b$ a square root is missing in the argument of the Bessel $Y_0$ function,

$$\int_{0}^\infty \frac{\sin\left(b\sqrt{a^2+x^2}\right)}{a^2+x^2}\cos yx\,dx =\frac{e^{-a y} \,\text{Ei}(a y)-e^ {ay} \,\text{Ei}(-a y)}{2 a}$$
$$\qquad\qquad-\frac{\pi}{2}  \int_y^b Y_0\left(a \sqrt{t^2-y^2}\right) \, dt,\;\;0\leq y<b,$$