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How to evaluate $\int _0^1\frac{\ln ^2\left(1-x\right)\ln ^3\left(1+x\right)}{1+x}\:dx$
I want to evaluate
$$\int _0^1\frac{\ln ^2\left(1-x\right)\ln ^3\left(1+x\right)}{1+x}\:dx$$
Im not sure if this has a closed form, integration by parts is out of the question since there would be ...
user805212
12
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5
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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}$...
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