Computing $\int_0^1\frac{\ln^3(1-x)\ln(1+x)}{x}dx$ or $\sum_{n=1}^\infty\frac{H_n^{(4)}}{n2^n}$

sequences-and-series integration closed-form harmonic-numbers polylogarithm

This integral was solved by @Song here along with a related integral using a magical algebraic identity

$$2a^3b = -{b^4 \over 2} -{b^4 + 6a^2b^2\over 2} + 3(a^3b+ab^3) - (a-b)^3b$$

with $a=\ln(1-x)$ and $b=\ln(1+x)$

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