Question Number 222127 by Nicholas666 last updated on 18/Jun/25
$$ \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{ln}\left(\mathrm{tan}\left(\mathrm{tan}^{−\mathrm{1}} \left({e}^{\frac{\mathrm{1}}{\pi}\:\mathrm{tan}^{−\mathrm{1}} \:{u}} \right)\right)\right)\:}{{u}^{\mathrm{2}} \:+\:\mathrm{2}\pi{u}\:+\:\mathrm{2}\pi^{\mathrm{2}} }\:{du} \\ $$$$ \\ $$