A challenging logarithmic integral $\int_0^1 \frac{\log(1+x)\log(1-x)}{1+x}dx$

While playing around with Mathematica, I found that

$$\int_0^1 \frac{\log(1+x)\log(1-x)}{1+x}dx = \frac{1}{3}\log^3(2)-\frac{\pi^2}{12}\log(2)+\frac{\zeta(3)}{8}$$

Please help me prove this result.

Ms. Chris's sis asked me exactly same question a few days ago in chatroom & I could answer it.

Here is my answer. Let $I$ be the integral. Using magic substitution $2t=1+x$ we get \begin{align} I&=\int_{\frac{1}{2}}^1 \frac{\log(2t)\log(2-2t)}{t}dt\\ &=\int_{\frac{1}{2}}^1 \frac{\log t\log(1-t)}{t}dt+\ln2\int_{\frac{1}{2}}^1 \frac{\log(1-t)}{t}dt+\ln2\int_{\frac{1}{2}}^1 \frac{\log t}{t}dt+\ln^22\int_{\frac{1}{2}}^1\frac{1}{t}dt\\ &=I_1+I_2+I_3+I_4 \end{align} All of the above integrals are trivial. For instance, $I_1$ & $I_2$ can be solved by using elementary way: using series expansion for $$\frac{\log(1-t)}{t}=\sum_{k=1}^\infty\frac{t^{k-1}}{k}$$ We can also use the reflection formula for dilog function $$\text{Li}_2(x)+\text{Li}_2(1-x)=\frac{\pi^2}{6}-\ln x\ln(1-x)$$for solving $I_1$ and the integral representation of dilog function $$\text{Li}_2(x)=\int_0^x \frac{\log(1-t)}{t}dt$$for solving $I_2$. And for $I_3$ & $I_4$, of course an high school student can easily solved it. So $$I=\frac{1}{3}\ln^3(2)-\frac{\pi^2}{12}\ln(2)+\frac{\zeta(3)}{8}$$ Done! (>‿◠)✌

$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\int_{0}^{1}{\ln\pars{1 + x}\ln\pars{1 - x} \over 1+x}\,\dd x:\ {\large}}$

With $\ds{0 < \mu < 1}$: \begin{align}&\color{#c00000}{\int_{0}^{\mu}% {\ln\pars{1 + x}\ln\pars{1 - x} \over 1+x}\,\dd x} \\[3mm]&=\half\,\ln^{2}\pars{1 + \mu}\ln\pars{1 - \mu} +\color{#00f}{\half\int_{0}^{\mu}{\ln^{2}\pars{1 + x} \over 1 - x}\,\dd x} \tag{1} \end{align}

\begin{align}&\color{#00f}{\half\int_{0}^{\mu}% {\ln^{2}\pars{1 + x} \over 1 - x}\,\dd x} =\half\int_{1}^{1 + \mu}{\ln^{2}\pars{x} \over 2 - x}\,\dd x =\half\int_{1/2}^{\pars{1 + \mu}/2}{\ln^{2}\pars{2x} \over 1 - x}\,\dd x \\[3mm]&=-\,\half\ln\pars{1 - {1 + \mu \over 2}}\ln^{2}\pars{1 + \mu} +\half\int_{1/2}^{\pars{1 + \mu}/2}\ln\pars{1 - x}\,{2\ln\pars{2x} \over x}\,\dd x \\[3mm]&=-\,\half\ln\pars{1 - \mu \over 2}\ln^{2}\pars{1 + \mu} -\int_{1/2}^{\pars{1 + \mu}/2}{\rm Li}_{2}'\pars{x}\ln\pars{2x}\,\dd x \\[3mm]&=-\,\half\,\ln^{2}\pars{1 + \mu}\ln\pars{1 - \mu} +\half\,\ln\pars{2}\ln^{2}\pars{1 + \mu} -{\rm Li}_{2}\pars{1 + \mu \over 2}\ln\pars{1 + \mu} \\[3mm]&+\int_{1/2}^{\pars{1 + \mu}/2}{\rm Li}_{2}\pars{x}\,{1 \over x}\,\dd x \\[3mm]&=-\,\half\,\ln^{2}\pars{1 + \mu}\ln\pars{1 - \mu} +\half\,\ln\pars{2}\ln^{2}\pars{1 + \mu} - {\rm Li}_{2}\pars{1 + \mu \over 2}\ln\pars{1 + \mu} \\[3mm]&+\int_{1/2}^{\pars{1 + \mu}/2}{\rm Li}_{3}'\pars{x}\,\dd x \end{align}

\begin{align} &\color{#00f}{\half\int_{0}^{\mu}% {\ln^{2}\pars{1 + x} \over 1 - x}\,\dd x} \\[3mm]&=-\,\half\,\ln^{2}\pars{1 + \mu}\ln\pars{1 - \mu} +\half\,\ln\pars{2}\ln^{2}\pars{1 + \mu} -{\rm Li}_{2}\pars{1 + \mu \over 2}\ln\pars{1 + \mu} \\[3mm]&+{\rm Li}_{3}\pars{1 + \mu \over 2} - {\rm Li}_{3}\pars{\half} \end{align}

Replacing in $\pars{1}$ and taking the limit $\ds{\mu \to 1^{-}}$: \begin{align}&\color{#c00000}{\int_{0}^{1}% {\ln\pars{1 + x}\ln\pars{1 - x} \over 1+x}\,\dd x} = \half\,\ln^{3}\pars{2} - {\rm Li}_{2}\pars{1}\ln\pars{2} +{\rm Li}_{3}\pars{1} - {\rm Li}_{3}\pars{\half} \end{align}

With the values: $$ {\rm Li}_{2}\pars{1} = {\pi^{2} \over 6}\,,\quad {\rm Li}_{3}\pars{1} = \zeta\pars{3}\,,\quad {\rm Li}_{3}\pars{\half} = {\ln^{3}\pars{2} \over 6} -{\ln\pars{2} \over 12}\,\pi^{2} + {7 \over 8}\,\zeta\pars{3} $$ we find

$$ \color{#66f}{\large\int_{0}^{1}% {\ln\pars{1 + x}\ln\pars{1 - x} \over 1+x}\,\dd x = {\ln^{3}\pars{2} \over 3} - {\ln\pars{2} \over 12}\,\pi^{2} +{1 \over 8}\,\zeta\pars{3}} \approx {\tt -0.3088} $$

Use your favorite program to compute the indefinite integral in terms of polylogarithms $$\int\frac{\ln(1+x)\ln(1-x)\,dx}{1+x}=\frac{\ln2}{2}\ln^2(1+x)-\ln(1+x)\,\mathrm{Li}_2\left(\frac{1+x}{2}\right)+\mathrm{Li}_3\left(\frac{1+x}{2}\right).$$ [This can be verified by straightforward differentiation].

To compute the definite integral, it suffices to know $\mathrm{Li}_{2,3}\left(\frac12\right)$ and $\mathrm{Li}_{2,3}(1)$. However, the definition of polylogarithm immediately implies $\mathrm{Li}_s(1)=\zeta(s)$. Also, the values $\mathrm{Li}_{2,3}\left(\frac12\right)$ can be found here (formulas (16), (17)).

Following Shobhit's comment, some preliminary lemma.
Lemma 1. $$ \sum_{n\geq 1} H_n x^n = \frac{\log(1-x)}{1-x}. $$ Lemma 2. By Lemma $1$, $$ \sum_{n\geq 1}\frac{H_n}{n+1} x^{n+1} = \frac{1}{2}\log^2(1-x),\qquad \sum_{n\geq 1}\frac{H_n+H_{n+1}}{n+1}x^{n}=\frac{-x+\log^2(1-x)+\text{Li}_2(x)}{x}.$$ Lemma 3. Since $H_{n+1}^2-H_n^2 = \frac{H_n+H_{n+1}}{n+1}$, $$ \sum_{n\geq 1}H_{n}^2 x^n = \frac{\log^2(1-x)+\text{Li}_2(x)}{1-x}.$$ Lemma 4. By Lemma 3, $$ \sum_{n\geq 1}\frac{(-1)^{n+1} H_{n}^2}{n+1} = -\int_{0}^{1}\frac{\log^2(1+x)+\text{Li}_2(-x)}{1+x}\,dx=-\frac{\log^3(2)}{3}-\color{red}{\int_{0}^{1}\frac{\text{Li}_{2}(-x)}{1+x}\,dx}.$$ The problem boils down to the evaluation of the last integral. By integration by parts, it is: $$ \color{red}{\int_{0}^{1}\frac{\text{Li}_{2}(-x)}{1+x}\,dx}=-\frac{\pi^2}{12}\log(2)+\color{blue}{\int_{0}^{1}\frac{\log^2(1+x)}{x}\,dx}\tag{1} $$ but: $$ \color{blue}{\int_{0}^{1}\frac{\log^2(1+x)}{x}\,dx} = -2\int_{0}^{1}\frac{\log(1+x)\log(x)}{1+x}\,dx=\color{blue}{\frac{\zeta(3)}{4}}\tag{2}$$ and the proof is complete. $(2)$ ultimately depends on a reflection formula for $\text{Li}_3$. In our case:

Lemma 5. $$\sum_{n\geq 1}\frac{H_n^2}{n+1}\,x^{n+1} = -\frac{\log(1-x)}{3}\left[\frac{\pi^2}{2}+\log^2(1-x)+3\text{Li}_2(1-x)\right]+2\left[\text{Li}_3(1-x)-\zeta(3)\right]$$

is straightforward to prove through differentiation, leading back to Lemma 3.

We can take $\ 2ab= a^2+b^2-(a-b)^2$ to get: $$I= \frac12 \int_0^1 \frac{\ln^2 (1-x)}{1+x}dx+\frac12\int_0^1\frac{\ln^2(1+x)}{1+x}dx-\frac12 \int_0^1 \frac{\ln^2\left(\frac{1-x}{1+x}\right)}{1+x}dx$$ By letting $\frac{1-x}{1+x}=t$ and expanding into power series the last one we get: $$I=\frac12 J +\frac{\ln^3(1+x)}{6}\bigg|_0^1 -\frac12 \int_0^1 \frac{\ln^2 t}{1+t}dt=\frac12 J +\frac{\ln^3 2}{6}-\frac34\zeta(3) $$ $$J=\int_0^1 \frac{\ln^2(1-x)}{1+x}dx\overset{1-x\to x}=\frac12\int_0^1 \frac{\ln^2 x}{1-\frac{x}{2}}dx=\frac12 \sum_{n=0}^\infty \frac{1}{2^n} \int_0^1 x^{n}\ln^2 xdx$$ $$=\sum_{n=0}^\infty \frac{1}{2^n}\frac{1}{(n+1)^3}=2\operatorname{Li}_3 \left(\frac12\right)\Rightarrow \boxed{I=\operatorname{Li}_3 \left(\frac12\right)+\frac{\ln^3 2}{6}-\frac34\zeta(3)}$$ Of course one can rewrite the trilogarithm's value as seen from $(17)$ in this link, but this form is also valid.