how to prove this integral

Question

Could you tell me how to prove this,

$$\int_{-\infty}^\infty \tanh \left(\frac{x}{a}\right) e^{2 i b x}\frac{dx}{x^2}=-\frac{a} {2}\sum_{x_n>0} \frac{4 \pi i}{x_n^2} e^{-2 b x_n}$$ where ($x_{n}=\pi a (2n+1))$

Thanks in advance

Answer 1

Hint: $\tanh$ has simple poles on the imaginary axis, and their residue matches your RHS.