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While evaluating the weight $7$ integral $\displaystyle \int_0^1\frac{\ln^3\left(1-x\right)\ln^3\left(1+x\right)}{1+x}\:dx$

I managed to prove that

$$\int_0^1\frac{\ln^3\left(1-x\right)\ln^3\left(1+x\right)}{1+x}\:dx=\frac{3465}{32}\zeta \left(7\right)-\frac{423}{8}\ln \left(2\right)\zeta \left(6\right)-\frac{81}{2}\zeta \left(3\right)\zeta \left(4\right)-36\sum _{k=1}^{\infty }\frac{H_k}{k^6\:2^k}$$ $$+72\operatorname{Li}_7\left(\frac{1}{2}\right)+36\ln \left(2\right)\operatorname{Li}_6\left(\frac{1}{2}\right)-54\zeta \left(2\right)\zeta \left(5\right)-36\ln ^2\left(2\right)\zeta \left(2\right)\zeta \left(3\right)+90\ln ^2\left(2\right)\zeta \left(5\right)$$ $$-21\ln ^3\left(2\right)\zeta \left(4\right)+\frac{21}{2}\ln ^4\left(2\right)\zeta \left(3\right)+36\ln \left(2\right)\zeta ^2\left(3\right)-\frac{21}{10}\ln ^5\left(2\right)\zeta \left(2\right)+\frac{5}{28}\ln ^7\left(2\right)$$ While using the Mathemathica package written by Pisco found here to calculate that integral I saw that the result had MZV which is an indicator that the sum might not have a nice closed form but im quite stubborn and i'd like to know if this sum can be computed without these.

StubbornAtom
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Dennis Orton
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  • I am hopeful that, much like the rest of the terms, it may be something of the form $\frac{n_1}{n_2}\ln^{n_3}(2)\zeta(n_5)\zeta(n_6)$ where $n_i \in \mathbb{Z}$. If true, perhaps there is gap of sorts in a pattern of the other terms that could be neatly filled in by such. – Graviton Aug 09 '20 at 12:49
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    related: https://math.stackexchange.com/q/3247329 – Noam Shalev - nospoon Aug 09 '20 at 14:01
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    The answer by Steven Charlton in the question linked by @nospoon is very informative. Steven proved, via some advanced theories, that $\sum H_n/(n^52^n)$ can be expressed using $\text{Li}_6(1/4), \text{Li}_6(1/8)$. On the other hand, the same theory predicts non-existence of closed form for $\sum H_n/(n^62^n)$ (in terms of ordinary polylog). So the answer of your question is pessimistic, your sum almost surely involves MZV. – pisco Aug 09 '20 at 14:16

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