I've noticed it is not in Erdelyi or Gradsteyn, although the version with the sine replaced by a cosine is in Erdelyi (page 26 eq. 33).
I've tried using the substitution $x=a \sinh z$ to avoid a square root.
I would also like to know the Fourier sine transform of the sine and cosine form.
Fourier cosine transform of $\frac{\sin(b\sqrt{a^2+x^2})}{a^2+x^2}$
H.Davies
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