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 $ 0.653and 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) $?