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How to evaluate the closed form of the following integrals $$\int\limits_0^1 {\frac{{{{\log }^{{p_1}}}\left( {1 - x} \right){{\log }^{{p_2}}}\left( x \right){{\log }^{{p_3}}}\left( {1 + x} \right)}}{x}} dx,$$ where $p_1,p_2,p_3$ are nonnegative integers and ${p_1} + {p_3} \ge 1$.

It can be expressed in terms of multiple zeta values?

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Too long for a comment.

We know that $$\frac{\log^{m}\left(1-x\right)}{m!}=\sum_{n\geq m}\left(-1\right)^{n}\frac{s\left(n,m\right)}{n!}x^{n},\,-1\leq x<1$$ where $s(n,m)$ are the Stirling's number of the first kind, so $$I\left(p_{1},p_{2},p_{3}\right)=\int_{0}^{1}\frac{\log^{p_{1}}\left(1-x\right)\log^{p_{2}}\left(x\right)\log^{p_{3}}\left(1+x\right)}{x}dx$$ $$=p_{1}!p_{3}!\sum_{n\geq p_{1}}\sum_{k\geq p_{3}}\left(-1\right)^{n}\frac{s\left(n,p_{1}\right)s\left(k,p_{3}\right)}{n!k!}\int_{0}^{1}x^{n+k-1}\log^{p_{2}}\left(x\right)dx$$ and, integrating by parts, we have $$I\left(p_{1},p_{2},p_{3}\right)=p_{1}!p_{2}!p_{3}!\sum_{n\geq1}\sum_{k\geq1}\left(-1\right)^{n+p_{2}}\frac{s\left(n,p_{1}\right)s\left(k,p_{3}\right)}{n!k!\left(n+k\right)^{p_{2}+1}}$$ now since $$s\left(n,m\right)=\left(n-1\right)!H_{n-1}^{\left(\left\{ 1\right\} _{m-1}\right)}$$ where $$\left(\left\{ 1\right\} _{m-1}\right)=\left(\overbrace{1,\dots,1}^{{\scriptstyle m-1\textrm{ times}}}\right)$$ and $$H_{n-1}^{\left(s_{1},\dots,s_{k-1}\right)}=\sum_{1\leq n_{k-1}<n_{k-2}<\dots<n_{1}\leq n-1}\frac{1}{n_{1}^{s_{1}}\cdots n_{k-1}^{s_{k-1}}}$$ we get $$I\left(p_{1},p_{2},p_{3}\right)=p_{1}!p_{2}!p_{3}!\left(-1\right)^{p_{2}}\sum_{n\geq1}\sum_{k\geq1}\frac{\left(-1\right)^{n}H_{n-1}^{\left(\left\{ 1\right\} _{p_{1}-1}\right)}H_{k-1}^{\left(\left\{ 1\right\} _{p_{3}-1}\right)}}{nk\left(n+k\right)^{p_{2}+1}}\tag{1}$$ and since $$\zeta\left(p_{2}+1,\left\{ 1\right\} _{p_{3}-1}\right)=\sum_{k\geq1}\frac{H_{k-1}^{\left(\left\{ 1\right\} _{p_{3}-1}\right)}}{k^{p_{2}+1}}$$ we (probably) may write the RHS of $(1)$ as a series invovling the multivariate zeta function.

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