Dilogarithm integral $\int^x_0 \frac{\operatorname{Li}_2(1-t)\log(1-t)}{t}\, dt$

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 ?