$$\begin {aligned} \int _{0}^{1}\frac {\ln (1+x^{2})}{1+x^{2}}dx &=\int _{0}^{1}\frac {\ln \left (\frac {1+x^{2}}{2x}\right )+\ln 2+\ln x}{1+x^2}dx\\ &=\frac {1}2\int _{0}^{\frac {\pi }2}\ln \left (\frac {1}{\sin \theta }\right )d\theta +\frac {\pi \ln 2}4-G \quad \left (x=\tan \frac {\theta }2\right )\\ &=\frac {\pi \ln 2}4+\frac {\pi \ln 2}4-G\\ &=\textcolor {blue}{\frac {\pi \ln 2}2-G}. \end {aligned}$$