What's a natural way to compute $\text{Li}_2(x)+\text{Li}_2(1-x)$ in closed form ?
Once you know the answer $\text{Li}_2(x)+\text{Li}_2(1-x)=\frac{\pi^2}{6}-\log(x)\log(1-x)$ , computing the derivative of the function $x\mapsto \text{Li}_2(x)+\text{Li}_2(1-x) + \log(x)\log(1-x)$ gives a fast, out-of-the- blue proof.
I'd like someone to show a computation of $\text{Li}_2(x)+\text{Li}_2(1-x)$ with no prior knowledge of the answer. I'm unable to do that with the integral formula for $\text{Li}_2$
