$$ \begin {aligned} \int _{0}^{1}\frac {\ln x\ln (1+x^{2})}{1+x^{2}}dx &=\int _{0}^{1}\sum_{n=1}^\infty \sum _{m=0}^\infty \frac { (-1)^{n+m+1}}nx^{2(n+m)}\ln xdx\\ &=\sum _{n,m=1}^\infty \frac {(-1)^{n+m+1}}{n(2n+2m-1)^2}\\ &=\sum _{k=2}^\infty \frac {(-1)^{k+1}}{(2k-1)^2}\sum _{n=1}^{k-1}\frac {1}n\\ &=\sum _{k=1}^\infty \frac {(-1)^{k}H_k}{(2k+1)^{2}} \end {aligned} $$