# integration – What is the correct value of \$ int_0 ^ infty dx J_0 left (2 a sinh left ( frac {x} {2} right) right) sin (b x) \$?

Integral (6.679.4) in Gradshteyn and Ryzhik states that
$$int_0 ^ infty dx J_0 left (2 to sinh left ( frac {x} {2} right) right) sin (bx) = frac {2} { pi} sinh ( pi b) left[ K_{ib}(a) right]^ 2$$
I think it's do not correct. Here is a screenshot of some numeric values ​​in Mathematica:

For random values $$a = 2.34$$ and $$b = 3$$, it seems that the LHS from the above evaluates to $$0.408$$ while the ERS evaluates to $$0.653$$and they therefore do not agree.

I think there are three options:

1. I am wrong in a way in the above.

2 Mathematica defines the functions of Bessel $$J$$ or $$K$$ differently from G & R, which I do not think is likely.

2 That's an error in G & R.

In the latter case, I saw that Equation (58) on page 115 of Erdelyi: Volume 1, Integral Transformation Tables, is the source of G & R (6.679.4) . This integral being simply indicated here without source, I do not see how this integral is derived.

What is $$int_0 ^ infty dx J_0 left (2 to sinh left ( frac {x} {2} right) right) sin (b x)$$?