Dilogarithm integral $\int^x_0 \frac{\operatorname{Li}_2(1-t)\log(1-t)}{t}\, dt$ I am hoping to find a closed form for the following 
$$\tag{1} \sum_{k\geq 1}\frac{H_k}{k^3} x^k $$
Using the generating function 
$$\sum_{k\geq 1}H^{(n)}_k x^k = \frac{\operatorname{Li}_n(x)}{1-x}$$
I could find this by simple integration 
So I am stuck at evaluating 
$$\tag{2}\int^x_0 \frac{\operatorname{Li}_2(1-t)\log(1-t)}{t}\, dt$$
For $x=\pm 1$ the problem can be solved , but what about the general case ?
 A: $$\begin{align}
&\int^x_0 \frac{\operatorname{Li}_2(1-t)\log(1-t)}{t}\,dt=\\
&\frac{\operatorname{Li}_2^2(1-x)}2-2\operatorname{Li}_4\left(1-\frac1x\right)-2\operatorname{Li}_4(1-x)+2\operatorname{Li}_4(x)-\operatorname{Li}_2\left(1-\frac1x\right)\log^2\left(\frac1x-1\right)+\\
&\operatorname{Li}_2(x)\left(\log^2\left(\frac1x-1\right)+\log(1-x)\log(x)\right)+2\operatorname{Li}_3(x)\log\left(\frac1x\right)+\\&2\operatorname{Li}_3\left(1-\frac1x\right)\log\left(\frac1x-1\right)+2\operatorname{Li}_3(1-x)\left(\log\left(\frac1x-1\right)+\log(x)\right)-\\
&\frac14\log^4\left(\frac1x\right)+\frac13\log^3(1-x)\left(2\log(x)-\log\left(\frac1x\right)\right)-\\
&\log(1-x)\left(\log^3\left(\frac1x\right)+\frac13\pi^2\log\left(\frac1x\right)+\frac{\pi^2}6\log(x)\right)+\\&\log^2(1-x)\left(-\log^2\left(\frac1x\right)+\frac12\log^2(x)+\log(x)\log\left(\frac1x\right)-\frac{\pi^2}6\right)-\frac{11\pi^4}{360},\end{align}$$
that can be checked by taking derivatives from both sides.
See the corresponding indefinite integral at WolframAlpha.
A: The form of the integrand suggests the use of the reflection formula might come handy. We have:
\begin{eqnarray}
&&\int\limits_0^x Li_2(1-t) \frac{\log(1-t)}{t} dt =
\int\limits_0^x \left(\frac{\pi^2}{6} - \log(t) \log(1-t) - Li_2(t) \right) \cdot \frac{\log(1-t)}{t} dt \\
&&=-\frac{\pi^2}{6} Li_2(x) + \frac{1}{2} Li_2(x)^2 - \int\limits_0^x \log(t) \frac{\log(1-t)^2}{t} dt \\
&&=-\frac{\pi^2}{6} Li_2(x) + \frac{1}{2} Li_2(x)^2 - \left(-2 S_{2,2}(x) + 2 \log(x) S_{1,2}(x)\right)
\end{eqnarray}
where $S_{p,q}(x)$ are the Nielsen generalized polylogarithms (see http://mathworld.wolfram.com/NielsenGeneralizedPolylogarithm.html   for definition).
A: Start with using the reflection formula of the dilogarithm function,
$$\text{Li}_2(1-x)=\zeta(2)-\ln(x)\ln(1-x)-\text{Li}_2(x)$$
$$\Longrightarrow I=\zeta(2)\underbrace{\int_0^1\frac{\ln(1-x)}{x}\ dx}_{-\zeta(2)}-\underbrace{\int_0^1\frac{\ln(x)\ln^2(1-x)}{x}\ dx}_{\text{Beta function:}\ -\frac12\zeta(4)}-\underbrace{\int_0^1\frac{\ln(1-x)\text{Li}_2(x)}{x}\ dx}_{-\frac12\text{Li}_2^2(x)|_0^1=-\frac54\zeta(4)}=-\frac34\zeta(4)$$
