I hope you find this integral interesting.

> 
> Evaluate
>  $$\int\_0^\infty\frac{\sin\left(\,3x\,\right)\sin\left(\,4x\,\right)
> \sin\left(\,5x\,\right)\cos\left(\,6x\,\right)}{x\,\sin^{2}\left(\,x\,\right)\cosh\left(\,x\,\right)}\,\,\mathrm{d}x\tag1$$
> 
> 
> 

This problem is taken from the PhD graduate entry tests in my college. I've tried to use product-to-sum trigonometric identities
$$2\sin 4x\sin 3x=\cos x-\cos 5x$$
and
$$2\cos 6x\sin 5x=\sin 11x-\sin x$$
I got a bunch of the following form 
$$\int\_0^\infty\frac{\sin \alpha x\cos \beta x}{x\sin^2 x\cosh x}\ dx\quad\Longrightarrow\quad\int\_0^\infty\frac{\sin \gamma x}{x\sin^2 x\cosh x}\ dx\tag2$$
I tried
$$I'(\gamma)=\int\_0^\infty\frac{\cos \gamma x}{\sin^2 x\cosh x}\ dx\tag3$$
but the latter form is not easy to evaluate either. Can anyone here help me to evaluate $(1)$? Thanks in advance.