Ways to calculate $\int_0^1 \frac{-\log x}{1+x}\ \mathrm dx$ I came across the integral
$$
\int_0^1 \frac{-\log x}{1+x}\ \mathrm dx = \frac{\pi^2}{12},
$$
which can be calculated as $\frac 1 2 \zeta(2)$ using analytic number theory.
I'm interested if this integral can be calculated in any other interesting, possibly more elementary ways?

Integrating by part, you arrive to  
$$-\text{Li}_2(-x)-\log (x) \log (x+1)$$  
and, using your bounds, the result is $\frac{\pi ^2}{12}$
