hyperbolic functions type problem

Question

Here below is a problem which is difficult for me. Show that $$\int_{-\infty}^{\infty}(1-\frac{\tanh(x+y)}{\tanh(x-y)})dx=4y\coth{y}$$

Thank you!

Answer 1

The first thing I should do is to reduce to the same denominator and expand $\tanh(x+y)$ and $\tanh(x-y)$ as functions of $\tanh(x)$ and $\tanh(y).$
Sorry to just give you only this; I hope and wish it could be of some help to you.