Question Number 138171 by mnjuly1970 last updated on 11/Apr/21
$$\:\:\:\:\:\:\:\:\:\:\:…….{mathematical}….{analysis}……. \\ $$$$\:\:\:\:\:{if}\::\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{arctan}\left({x}\right).{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{48}}\left({a}\pi^{\mathrm{2}} −{b}\pi{ln}\left(\mathrm{2}\right)+{c}\:{ln}^{\mathrm{2}} \left(\mathrm{2}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:……{then}\:\:….\:{a}^{\mathrm{2}} +\left({b}−{c}+\mathrm{1}\right)^{\mathrm{2}} =??? \\ $$