I am trying to figure out how the integral identities $$ \int_0^\infty J_0\left(2a\sinh\left(\frac{x}{2}\right)\right)\sin\left(bx\right)dx = \frac{2}{\pi}\sinh\left(\pi b\right)\left[K_{ib}\left(a\right)\right]^2 $$ and $$ \int_0^\infty Y_0\left(2a\sinh\left(\frac{x}{2}\right)\right)\cos\left(bx\right)dx = -\frac{2}{\pi}\cosh\left(\pi b\right)\left[K_{ib}\left(a\right)\right]^2 $$ are derived. (under the assumptions $a>0$, $b>0$)
Focusing for now on the first one, and guided by the symmetry between the expressions, I tried to expand the integrals on the left using the Mehler-Sonine integral representation, getting (for the first identity) $$ \frac{2}{\pi}\int_0^\infty\int_0^\infty \sin\left(2a\sinh\left(\frac{x}{2}\right)\cosh\left(t\right)\right)\sin\left(bx\right)dtdx $$ However that doesn't seem to get me much further as (I think?) these don't satisfy any condition allowing exchange of integrals.
Trying to work backwards from the answer the only real obvious expansion I could think of was $$ \frac{2}{\pi}\sinh\left(\pi b\right)\left[K_{ib}\left(a\right)\right]^2 = \frac{2}{\pi}\sinh\left(\pi b\right)\left[\frac{1}{\sin\left(\frac{1}{2}i\pi b\right)}\int_0^\infty\sin\left(a\sinh\left(t\right)\right)\sinh\left(ibt\right)dt\right]^2\\ =\frac{2}{\pi}\sinh\left(\pi b\right)\left[\frac{1}{\sinh\left(\frac{1}{2}\pi b\right)}\int_0^\infty\sin\left(a\sinh\left(t\right)\right)\sin\left(bt\right)dt\right]^2 $$ But I don't really see a way to cleverly combine the two integrals here to get back to the expression above either. What is a viable approach to derive these identities?
background
I am trying to compute variants of the above integrals which I cannot find in books with tables. In particular, I am trying to compute $$ \lim_{T\rightarrow\infty}\frac{1}{T}\int_0^TxJ_0\left(2a\sinh\left(\frac{x}{2}\right)\right)\sin\left(bx\right)dx $$ and $$ \lim_{T\rightarrow\infty}\frac{1}{T}\int_0^TxY_0\left(2a\sinh\left(\frac{x}{2}\right)\right)\cos\left(bx\right)dx $$ again assuming $a>0$ and $b>0$. My hope is that the techniques for the above somewhat standard integrals perhaps also apply to these, or at least can offer some inspiration for how to approach these.