Integral (6.679.4) in Gradshteyn and Ryzhik claims that $$ \int_0^\infty dx\ J_0\left( 2 a \sinh\left( \frac{x}{2} \right) \right) \sin(b x) \ = \ \frac{2}{\pi} \sinh(\pi b) \left[ K_{ib}(a) \right]^2 $$ I think that this is not correct. Here is a screenshot of some numerics in Mathematica:
For random values $a=2.34$ and $b=3$, it seems that the LHS of the above evaluates to $0.408$ while the RHS evaluates to $0.653$, and so they disagree.
I think there are three options:
1. I am making a mistake somehow in the above.
2. Mathematica is defining Bessel functions $J$ or $K$ differently than G&R, which I think is not likely.
2. This is an error in G&R.
In the last case, I have seen that Equation (58) on Page 115 of Erdelyi: Volume 1 Tables of Integral Transforms is the source for G&R (6.679.4). This integral simply stated there without a source, so I have no clue how this integral is derived.
What is $\int_0^\infty dx\ J_0\left( 2 a \sinh\left( \frac{x}{2} \right) \right) \sin(b x)$?