Question Number 132301 by mnjuly1970 last updated on 13/Feb/21
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..\:{nice}…….{calculus}…. \\ $$$$\:\:\:\:\:\:\:\:{prove}\:\:\:{that}\::: \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left(\mathrm{2}{arctan}\left(\frac{{x}}{\mathrm{2}}\right)\right)}{\left({x}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} \right){sinh}\left(\pi{x}\right)}{dx}=\frac{\mathrm{7}}{\mathrm{8}}\:−\frac{\pi^{\mathrm{2}} }{\mathrm{12}} \\ $$$$ \\ $$