I want to know how to prove that
$$\int \_0^{+\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x=\frac{4G}{\pi }$$
Here $G$ denotes Catalan's constant, I obtained such result with the help of mathematica.

I also found that the integral equals a certain infinite series
$$\int \_0^{+\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x=\sum \_{n=0}^{+\infty }\frac{\binom{2n}{n}^2}{16^n\left(2n+1\right)}=\frac{4G}{\pi }$$
which can also be found in this [link](https://math.stackexchange.com/questions/3053715/relationship-between-catalans-constant-and-pi/3053744#3053744).

So I've $2$ questions

$1)$$¿$How can we transform the integral into the mentioned series?

$2)$$¿$Is there a simple way to evaluate the main integral without resorting to series expansion?

What I did for question $\#2$ is to employ the substitution $x=\ln\left(t\right)$
$$\int \_0^{+\infty }\frac{\sinh \left(x\right)}{x\cosh ^2\left(x\right)}\:\mathrm{d}x=-2\int \_1^{\infty }\frac{1-t^2}{\ln \left(t\right)\left(1+t^2\right)^2}\:\mathrm{d}t$$
But I'm not sure how to proceed.