how to $\int_0^{ +\infty} \frac{\sin(x)}{x+1}\, dx \leq \frac e5 \ln(\pi)$?

Hint : Use the following identity :

1)

$$\frac{e^{-x}}{{1+x²}}-\frac{e}{5}\frac{(e^{-x}-e^{-\pi x})}{x}<0 $$ for all $x>0$

2)Use the Frullani's integral to find :

$$\int_{0}^{\infty}\frac{e}{5}\frac{(e^{-x}-e^{-\pi x})}{x}=\frac{e}{5}ln(\pi)$$