I am trying to calculate the following integral: $$\int_0^\infty \frac{\sin(at)}{2\cosh 2\pi t-1} \text{ d}t$$
The inspiration for it was that a similar integral carries a closed form:
$$\int_0^\infty \frac{\sin at}{e^{2\pi t}-1}\text{ d}t=\frac{1}{4}\coth \frac{a}{2}-\frac{1}{2a}$$
I was wondering how exactly to go about solving the topmost integral.