Evaluate $$\int_0^1\ln(1-x)\ln x\ln(1+x) \mathrm{dx}$$

Does the following series have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1) \end{equation} where $\Psi_3(x)$ is the polygamma function of order 3. Here is my attempt. Using equation (11) from Mathworld Wolfram: \begin{equation} \Psi_n(z)=(-1)^{n+1} n!\l...

The value of $I$ is a $\mathbb{Q}$-linear combination of values of the multiple polylogarithm at rational arguments. I'll explain how to compute this. Expanding each logarithm in the integrand as an integral, multiplying out, and dividing into regions, and making the substitution $x\leftrightarr...