How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Question

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: $$\begin{align}\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx&=12\zeta(5)-\pi^2\zeta(3)\tag2\\ \int_0^1\frac{\ln^2(1+x)\ln x}x\mathrm dx&=\frac{\pi^4}{24}-\frac16\ln^42+\frac{\pi^2}6\ln^22-\frac72\zeta(3)\ln2-4\operatorname{Li}_4\!\left(\tfrac12\right)\tag3\\ \int_0^1\frac{\ln^3(1+x)\ln x}{x^2}\mathrm dx&=\frac34\zeta(3)-\frac{63}4\zeta(3)\ln2+\frac{23\pi^4}{120}\\&-\frac34\ln^42-2\ln^32+\frac{3\pi^2}4\ln^22-18\operatorname{Li}_4\!\left(\tfrac12\right).\tag4\end{align}$$ Thanks!

Answer 1

Start with integration by parts (IBP) by setting $u=\ln^3(1+x)$ and $dv=\dfrac{\ln x}{x}\ dx$ yields \begin{align} I&=-\frac32\int_0^1\frac{\ln^2(1+x)\ln^2 x}{1+x}\ dx\\ &=-\frac32\int_1^2\frac{\ln^2x\ln^2 (x-1)}{x}\ dx\quad\Rightarrow\quad\color{red}{x\mapsto1+x}\\ &=-\frac32\int_{\large\frac12}^1\left[\frac{\ln^2x\ln^2 (1-x)}{x}-\frac{2\ln^3x\ln(1-x)}{x}+\frac{\ln^4x}{x}\right]\ dx\quad\Rightarrow\quad\color{red}{x\mapsto\frac1x}\\ &=-\frac32\int_{\large\frac12}^1\frac{\ln^2x\ln^2 (1-x)}{x}\ dx+3\int_{\large\frac12}^1\frac{\ln^3x\ln(1-x)}{x}\ dx-\left.\frac3{10}\ln^5x\right|_{\large\frac12}^1\\ &=-\frac32\color{red}{\int_{\large\frac12}^1\frac{\ln^2x\ln^2 (1-x)}{x}\ dx}+3\int_{\large\frac12}^1\frac{\ln^3x\ln(1-x)}{x}\ dx-\frac3{10}\ln^52. \end{align} Applying IBP again to evaluate the red integral by setting $u=\ln^2(1-x)$ and $dv=\dfrac{\ln^2 x}{x}\ dx$ yields \begin{align} \color{red}{\int_{\large\frac12}^1\frac{\ln^2x\ln^2 (1-x)}{x}\ dx}&=\frac13\ln^52+\frac23\color{blue}{\int_{\large\frac12}^1\frac{\ln^3x\ln (1-x)}{1-x}\ dx}. \end{align}

For the simplicity, let $$ \color{blue}{\mathbf{H}_{m}^{(k)}(x)}=\sum_{n=1}^\infty \frac{H_{n}^{(k)}x^n}{n^m}\qquad\Rightarrow\qquad\color{blue}{\mathbf{H}(x)}=\sum_{n=1}^\infty H_{n}x^n, $$ Introduce a generating function for the generalized harmonic numbers for $|x|<1$ $$ \color{blue}{\mathbf{H}^{(k)}(x)}=\sum_{n=1}^\infty H_{n}^{(k)}x^n=\frac{\operatorname{Li}_k(x)}{1-x}\qquad\Rightarrow\qquad\color{blue}{\mathbf{H}(x)}=-\frac{\ln(1-x)}{1-x} $$ and the following identity $$ H_{n+1}^{(k)}-H_{n}^{(k)}=\frac1{(n+1)^k}\qquad\Rightarrow\qquad H_{n+1}-H_{n}=\frac1{n+1} $$

Let us integrating the indefinite form of the blue integral. \begin{align} \color{blue}{\int\frac{\ln^3x\ln (1-x)}{1-x}\ dx}=&-\int\sum_{n=1}^\infty H_nx^n\ln^3x\ dx\\ =&-\sum_{n=1}^\infty H_n\int x^n\ln^3x\ dx\\ =&-\sum_{n=1}^\infty H_n\frac{\partial^3}{\partial n^3}\left[\int x^n\ dx\right]\\ =&-\sum_{n=1}^\infty H_n\frac{\partial^3}{\partial n^3}\left[\frac{x^{n+1}}{n+1}\right]\\ =&-\sum_{n=1}^\infty H_n\left[\frac{x^{n+1}\ln^3x}{n+1}-\frac{3x^{n+1}\ln^2x}{(n+1)^2}+\frac{6x^{n+1}\ln x}{(n+1)^3}-\frac{6x^{n+1}}{(n+1)^4}\right]\\ =&-\ln^3x\sum_{n=1}^\infty \frac{H_{n+1}x^{n+1}}{n+1}+\ln^3x\sum_{n=1}^\infty \frac{x^{n+1}}{(n+1)^2}+3\ln^2x\sum_{n=1}^\infty \frac{H_{n+1}x^{n+1}}{(n+1)^2}\\&-3\ln^2x\sum_{n=1}^\infty \frac{x^{n+1}}{(n+1)^3}-6\ln x\sum_{n=1}^\infty \frac{H_{n+1}x^{n+1}}{(n+1)^3}+6\ln x\sum_{n=1}^\infty \frac{x^{n+1}}{(n+1)^4}\\&+6\sum_{n=1}^\infty \frac{H_{n+1}x^{n+1}}{(n+1)^4}-6\sum_{n=1}^\infty \frac{x^{n+1}}{(n+1)^5}\\ =&\ -\sum_{n=1}^\infty\left[\frac{H_nx^{n}\ln^3x}{n}-\frac{x^{n}\ln^3x}{n^2}-\frac{3H_nx^{n}\ln^2x}{n^2}+\frac{3x^{n}\ln^2x}{n^3}\right.\\& \left.\ +\frac{6H_nx^{n}\ln x}{n^3}-\frac{6x^{n}\ln x}{n^4}-\frac{6H_nx^{n}}{n^4}+\frac{6x^{n}}{n^5}\right]\\ =&\ -\color{blue}{\mathbf{H}_{1}(x)}\ln^3x+\operatorname{Li}_2(x)\ln^3x+3\color{blue}{\mathbf{H}_{2}(x)}\ln^2x-3\operatorname{Li}_3(x)\ln^2x\\&\ -6\color{blue}{\mathbf{H}_{3}(x)}\ln x+6\operatorname{Li}_4(x)\ln x+6\color{blue}{\mathbf{H}_{4}(x)}-6\operatorname{Li}_5(x). \end{align} Therefore \begin{align} \color{blue}{\int_{\Large\frac12}^1\frac{\ln^3x\ln (1-x)}{1-x}\ dx} =&\ 6\color{blue}{\mathbf{H}_{4}(1)}-6\operatorname{Li}_5(1)-\left[\color{blue}{\mathbf{H}_{1}\left(\frac12\right)}\ln^32-\operatorname{Li}_2\left(\frac12\right)\ln^32\right.\\&\left.\ +3\color{blue}{\mathbf{H}_{2}\left(\frac12\right)}\ln^22-3\operatorname{Li}_3\left(\frac12\right)\ln^22+6\color{blue}{\mathbf{H}_{3}\left(\frac12\right)}\ln 2\right.\\&\ -6\operatorname{Li}_4(x)\ln 2+6\color{blue}{\mathbf{H}_{4}(x)}-6\operatorname{Li}_5(x)\bigg]\\ =&\ 12\zeta(5)-\pi^2\zeta(3)+\frac{3}8\zeta(3)\ln^22-\frac{\pi^4}{120}\ln2-\frac{1} {4}\ln^52\\&\ -6\color{blue}{\mathbf{H}_{4}\left(\frac12\right)}+6\operatorname{Li}_4\left(\frac12\right)\ln 2+6\operatorname{Li}_5\left(\frac12\right). \end{align} Using the similar approach as calculating the blue integral, then \begin{align} \int\frac{\ln^3x\ln (1-x)}{x}\ dx&=-\int\sum_{n=1}^\infty \frac{x^{n-1}}{n}\ln^3x\ dx\\ &=-\sum_{n=1}^\infty \frac{1}{n}\int x^{n-1}\ln^3x\ dx\\ &=-\sum_{n=1}^\infty \frac{1}{n}\frac{\partial^3}{\partial n^3}\left[\int x^{n-1}\ dx\right]\\ &=-\sum_{n=1}^\infty \frac{1}{n}\frac{\partial^3}{\partial n^3}\left[\frac{x^{n}}{n}\right]\\ &=-\sum_{n=1}^\infty \frac{1}{n}\left[\frac{x^{n}\ln^3x}{n}-\frac{3x^{n}\ln^2x}{n^2}+\frac{6x^{n}\ln x}{n^3}-\frac{6x^{n}}{n^4}\right]\\ &=\sum_{n=1}^\infty \left[-\frac{x^{n}\ln^3x}{n^2}+\frac{3x^{n}\ln^2x}{n^3}-\frac{6x^{n}\ln x}{n^4}+\frac{6x^{n}}{n^5}\right]\\ &=6\operatorname{Li}_5(x)-6\operatorname{Li}_4(x)\ln x+3\operatorname{Li}_3(x)\ln^2x-\operatorname{Li}_2(x)\ln^3x. \end{align} Hence $$ \int_{\large\frac{1}{2}}^1\frac{\ln^3x\ln (1-x)}{x}\ dx=\frac{\pi^2}{6}\ln^32-\frac{21}{8}\zeta(3)\ln^22-6\operatorname{Li}_4\left(\frac{1}{2}\right)\ln2-6\operatorname{Li}_5\left(\frac{1}{2}\right)+6\zeta(5). $$ Combining altogether, we have

\begin{align} I=&\ \frac{\pi^4}{120}\ln2-\frac{33}4\zeta(3)\ln^22+\frac{\pi^2}2\ln^32-\frac{11}{20}\ln^52+6\zeta(5)+\pi^2\zeta(3)\\ &\ +6\color{blue}{\mathbf{H}_{4}\left(\frac12\right)}-18\operatorname{Li}_4\left(\frac12\right)\ln2-24\operatorname{Li}_5\left(\frac12\right). \end{align}

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Continuing my answer in: A sum containing harmonic numbers $\displaystyle\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}$, we have \begin{align} \color{blue}{\mathbf{H}_{3}\left(x\right)}=&\frac12\zeta(3)\ln x-\frac18\ln^2x\ln^2(1-x)+\frac12\ln x\left[\color{blue}{\mathbf{H}_{2}\left(x\right)}-\operatorname{Li}_3(x)\right]\\&+\operatorname{Li}_4(x)-\frac{\pi^2}{12}\operatorname{Li}_2(x)-\frac12\operatorname{Li}_3(1-x)\ln x+\frac{\pi^4}{60}.\tag1 \end{align} Dividing $(1)$ by $x$ and then integrating yields $$\small\begin{align} \color{blue}{\mathbf{H}_{4}\left(x\right)}=&\frac14\zeta(3)\ln^2 x-\frac18\int\frac{\ln^2x\ln^2(1-x)}x\ dx+\frac12\int\frac{\ln x}x\bigg[\color{blue}{\mathbf{H}_{2}\left(x\right)}-\operatorname{Li}_3(x)\bigg]\ dx\\&+\operatorname{Li}_5(x)-\frac{\pi^2}{12}\operatorname{Li}_3(x)-\frac12\int\frac{\operatorname{Li}_3(1-x)\ln x}x\ dx+\frac{\pi^4}{60}\ln x\\ =&\frac14\zeta(3)\ln^2 x+\frac{\pi^4}{60}\ln x+\operatorname{Li}_5(x)-\frac{\pi^2}{12}\operatorname{Li}_3(x)-\frac18\color{red}{\int\frac{\ln^2x\ln^2(1-x)}x\ dx}\\&+\frac12\left[\color{purple}{\sum_{n=1}^\infty\frac{H_{n}}{n^2}\int x^{n-1}\ln x\ dx}-\color{green}{\int\frac{\operatorname{Li}_3(x)\ln x}x\ dx}-\color{orange}{\int\frac{\operatorname{Li}_3(1-x)\ln x}x\ dx}\right].\tag2 \end{align}$$ Evaluating the red integral using the same technique as the previous one yields \begin{align} \color{red}{\int\frac{\ln^2x\ln^2(1-x)}x\ dx}&=\frac13\ln^3x\ln^2(1-x)-\frac23\color{blue}{\int\frac{\ln(1-x)\ln^3 x}{1-x}\ dx}. \end{align} Evaluating the purple integral yields \begin{align} \color{purple}{\sum_{n=1}^\infty\frac{H_{n}}{n^2}\int x^{n-1}\ln x\ dx}&=\sum_{n=1}^\infty\frac{H_{n}}{n^2}\frac{\partial}{\partial n}\left[\int x^{n-1}\ dx\right]\\ &=\sum_{n=1}^\infty\frac{H_{n}}{n^2}\left[\frac{x^n\ln x}{n}-\frac{x^n}{n^2}\right]\\ &=\color{blue}{\mathbf{H}_{3}(x)}\ln x-\color{blue}{\mathbf{H}_{4}(x)}. \end{align} Evaluating the green integral using IBP by setting $u=\ln x$ and $dv=\dfrac{\operatorname{Li}_3(x)}{x}\ dx$ yields \begin{align} \color{green}{\int\frac{\operatorname{Li}_3(x)\ln x}x\ dx}&=\operatorname{Li}_4(x)\ln x-\int\frac{\operatorname{Li}_4(x)}x\ dx\\ &=\operatorname{Li}_4(x)\ln x-\operatorname{Li}_5(x). \end{align} Evaluating the orange integral using IBP by setting $u=\operatorname{Li}_3(1-x)$ and $dv=\dfrac{\ln x}{x}\ dx$ yields \begin{align} \color{orange}{\int\frac{\operatorname{Li}_3(1-x)\ln x}x\ dx}&=\frac12\operatorname{Li}_3(1-x)\ln^2 x+\frac12\color{maroon}{\int\frac{\operatorname{Li}_2(1-x)\ln^2 x}{1-x}\ dx}. \end{align} Applying IBP again to evaluate the maroon integral by setting $u=\operatorname{Li}_2(1-x)$ and $$ dv=\dfrac{\ln^2 x}{1-x}\ dx\quad\Rightarrow\quad v=2\operatorname{Li}_3(x)-2\operatorname{Li}_2(x)\ln x-\ln(1-x)\ln^2x, $$ we have $$\small{\begin{align} \color{maroon}{\int\frac{\operatorname{Li}_2(1-x)\ln^2 x}{1-x}\ dx}=&\left[2\operatorname{Li}_3(x)-2\operatorname{Li}_2(x)\ln x-\ln(1-x)\ln^2x\right]\operatorname{Li}_2(1-x)\\ &-2\int\frac{\operatorname{Li}_3(x)\ln x}{1-x}\ dx+2\int\frac{\operatorname{Li}_2(x)\ln x}{1-x}\ dx+\color{blue}{\int\frac{\ln(1-x)\ln^3 x}{1-x}\ dx}. \end{align}}$$

We use the generating function for the generalized harmonic numbers evaluate the above integrals involving polylogarithm.

\begin{align} \int\frac{\operatorname{Li}_k(x)\ln x}{1-x}\ dx&=\sum_{n=1}^\infty H_{n}^{(k)}\int x^n\ln x\ dx\\ &=\sum_{n=1}^\infty H_{n}^{(k)}\frac{\partial}{\partial n}\left[\int x^n\ dx\right]\\ &=\sum_{n=1}^\infty H_{n}^{(k)}\left[\frac{x^{n+1}\ln x}{n+1}-\frac{x^{n+1}}{(n+1)^2}\right]\\ &=\sum_{n=1}^\infty\left[\frac{H_{n+1}^{(k)}x^{n+1}\ln x}{n+1}-\frac{x^{n+1}\ln x}{(n+1)^{k+1}}-\frac{H_{n+1}^{(k)}x^{n+1}}{(n+1)^2}+\frac{x^{n+1}}{(n+1)^{k+2}}\right]\\ &=\sum_{n=1}^\infty\left[\frac{H_{n}^{(k)}x^{n}\ln x}{n}-\frac{x^{n}\ln x}{n^{k+1}}-\frac{H_{n}^{(k)}x^{n}}{n^2}+\frac{x^{n}}{n^{k+2}}\right]\\ &=\color{blue}{\mathbf{H}_{1}^{(k)}(x)}\ln x-\operatorname{Li}_{k+1}(x)\ln x-\color{blue}{\mathbf{H}_{2}^{(k)}(x)}+\operatorname{Li}_{k+2}(x). \end{align}

Dividing generating function of $\color{blue}{\mathbf{H}^{(k)}(x)}$ by $x$ and then integrating yields

\begin{align} \sum_{n=1}^\infty \frac{H_{n}^{(k)}x^n}{n}&=\int\frac{\operatorname{Li}_k(x)}{x(1-x)}\ dx\\ \color{blue}{\mathbf{H}_{1}^{(k)}(x)}&=\int\frac{\operatorname{Li}_k(x)}{x}\ dx+\int\frac{\operatorname{Li}_k(x)}{1-x}\ dx\\ &=\operatorname{Li}_{k+1}(x)+\int\frac{\operatorname{Li}_k(x)}{1-x}\ dx. \end{align}

Repeating the process above yields

\begin{align} \sum_{n=1}^\infty \frac{H_{n}^{(k)}x^n}{n^2} &=\int\frac{\operatorname{Li}_{k+1}(x)}{x}\ dx+\int\frac{\operatorname{Li}_k(x)}{x(1-x)}\ dx\\ \color{blue}{\mathbf{H}_{2}^{(k)}(x)}&=\operatorname{Li}_{k+2}(x)+\operatorname{Li}_{k+1}(x)+\int\frac{\operatorname{Li}_k(x)}{1-x}\ dx, \end{align}

where it is easy to show by using IBP that

\begin{align} \int\frac{\operatorname{Li}_2(x)}{1-x}\ dx&=-\int\frac{\operatorname{Li}_2(1-x)}{x}\ dx\\ &=2\operatorname{Li}_3(x)-2\operatorname{Li}_2(x)\ln(x)-\operatorname{Li}_2(1-x)\ln x-\ln (1-x)\ln^2x \end{align}

and

$$ \int\frac{\operatorname{Li}_3(x)}{1-x}\ dx=-\int\frac{\operatorname{Li}_3(1-x)}{x}\ dx=-\frac12\operatorname{Li}_2^2(1-x)-\operatorname{Li}_3(1-x)\ln x. $$

Now, all unknown terms have been obtained. Putting altogether to $(2)$, we have $$\small{\begin{align} \color{blue}{\mathbf{H}_{4}(x)} =&\ \frac1{10}\zeta(3)\ln^2 x+\frac{\pi^4}{150}\ln x-\frac{\pi^2}{30}\operatorname{Li}_3(x)-\frac1{60}\ln^3x\ln^2(1-x)+\frac65\operatorname{Li}_5(x)\\&-\frac15\left[\operatorname{Li}_3(x)-\operatorname{Li}_2(x)\ln x-\frac12\ln(1-x)\ln^2x\right]\operatorname{Li}_2(1-x)-\frac15\operatorname{Li}_4(x)\\&-\frac35\operatorname{Li}_4(x)\ln x+\frac15\operatorname{Li}_3(x)\ln x+\frac15\operatorname{Li}_3(x)\ln^2x-\frac1{10}\operatorname{Li}_3(1-x)\ln^2 x\\&-\frac1{15}\operatorname{Li}_2(x)\ln^3x-\frac15\color{blue}{\mathbf{H}_{2}^{(3)}(x)}+\frac15\color{blue}{\mathbf{H}_{2}^{(2)}(x)} +\frac15\color{blue}{\mathbf{H}_{1}^{(3)}(x)}\ln x\\&-\frac15\color{blue}{\mathbf{H}_{1}^{(2)}(x)}\ln x+\frac25\color{blue}{\mathbf{H}_{3}(x)}\ln x-\frac15\color{blue}{\mathbf{H}_{2}(x)}\ln^2x+\frac1{15}\color{blue}{\mathbf{H}_{1}(x)}\ln^3x+C.\tag3 \end{align}}$$ The next step is finding the constant of integration. Setting $x=1$ to $(3)$ yields $$\small{\begin{align} \color{blue}{\mathbf{H}_{4}(1)} &=-\frac{\pi^2}{30}\operatorname{Li}_3(1)+\frac65\operatorname{Li}_5(1)-\frac15\operatorname{Li}_4(1)-\frac15\color{blue}{\mathbf{H}_{2}^{(3)}(1)}+\frac15\color{blue}{\mathbf{H}_{2}^{(2)}(1)}+C\\ 3\zeta(5)+\zeta(2)\zeta(3)&=-\frac{\pi^2}{30}\operatorname{Li}_3(1)+\frac{19}{30}\operatorname{Li}_5(1)+\frac{3}{5}\operatorname{Li}_3(1)+C\\ C&=\frac{\pi^4}{450}+\frac{\pi^2}{5}\zeta(3)-\frac35\zeta(3)+3\zeta(5). \end{align}}$$ Thus $$\small{\begin{align} \color{blue}{\mathbf{H}_{4}(x)} =&\ \frac1{10}\zeta(3)\ln^2 x+\frac{\pi^4}{150}\ln x-\frac{\pi^2}{30}\operatorname{Li}_3(x)-\frac1{60}\ln^3x\ln^2(1-x)+\frac65\operatorname{Li}_5(x)\\&-\frac15\left[\operatorname{Li}_3(x)-\operatorname{Li}_2(x)\ln x-\frac12\ln(1-x)\ln^2x\right]\operatorname{Li}_2(1-x)-\frac15\operatorname{Li}_4(x)\\&-\frac35\operatorname{Li}_4(x)\ln x+\frac15\operatorname{Li}_3(x)\ln x+\frac15\operatorname{Li}_3(x)\ln^2x-\frac1{10}\operatorname{Li}_3(1-x)\ln^2 x\\&-\frac1{15}\operatorname{Li}_2(x)\ln^3x-\frac15\color{blue}{\mathbf{H}_{2}^{(3)}(x)}+\frac15\color{blue}{\mathbf{H}_{2}^{(2)}(x)} +\frac15\color{blue}{\mathbf{H}_{1}^{(3)}(x)}\ln x\\&-\frac15\color{blue}{\mathbf{H}_{1}^{(2)}(x)}\ln x+\frac25\color{blue}{\mathbf{H}_{3}(x)}\ln x-\frac15\color{blue}{\mathbf{H}_{2}(x)}\ln^2x+\frac1{15}\color{blue}{\mathbf{H}_{1}(x)}\ln^3x\\&+\frac{\pi^4}{450}+\frac{\pi^2}{5}\zeta(3)-\frac35\zeta(3)+3\zeta(5)\tag4 \end{align}}$$ and setting $x=\frac12$ to $(4)$ yields \begin{align} \color{blue}{\mathbf{H}_{4}\left(\frac12\right)}=&\ \frac{\ln^52}{40}-\frac{\pi^2}{36}\ln^32+\frac{\zeta(3)}{2}\ln^22-\frac{\pi^2}{12}\zeta(3)\\&+\frac{\zeta(5)}{32}-\frac{\pi^4}{720}\ln2+\operatorname{Li}_4\left(\frac12\right)\ln2+2\operatorname{Li}_5\left(\frac12\right).\tag5 \end{align}

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Finally, we obtain

\begin{align} \int_0^1\frac{\ln^3(1+x)\ln x}x\ dx=&\ \color{blue}{\frac{\pi^2}2\zeta(3)+\frac{99}{16}\zeta(5)-\frac25\ln^52+\frac{\pi^2}3\ln^32-\frac{21}4\zeta(3)\ln^22}\\&\color{blue}{-12\operatorname{Li}_4\left(\frac12\right)\ln2-12\operatorname{Li}_5\left(\frac12\right)}, \end{align}

which again matches @Cleo's answer.

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References :

$[1]\ $ Harmonic number

$[2]\ $ Polylogarithm

Answer 2

Just a partial answer for now.

We have: $$ I = -\frac{3}{2}\int_{0}^{1}\frac{\log^2(1+x)\log^2 x}{1+x}\,dx$$ and since: $$\log(1+z)=\sum_{n=1}^{+\infty}\frac{(-1)^{n+1}}{n}z^n$$ it follows that: $$ [z^N]\log^2(1+z)=(-1)^{N+1}\sum_{n=1}^{N-1}\frac{1}{n(N-n)}=(-1)^{N+1}\frac{2H_{N-1}}{N},$$ $$\log^2(1+z)=\sum_{n=1}^{+\infty}\frac{2(-1)^{n+1} H_{n-1}}{n}z^{n}.\tag{1}$$ Let we focus now on: $$J_n = \int_{0}^{1}\frac{x^n\log^2 x}{1+x}\,dx=\frac{\partial^2}{\partial n^2}\int_{0}^{1}\frac{x^n}{1+x}\,dx.$$ We have: $$ J_n = \frac{1}{4}\left(H_{n/2}^{(3)}-H_{(n-1)/2}^{(3)}\right),$$ hence: $$ \color{blue}{I = -\frac{3}{4}\sum_{n=1}^{+\infty}\frac{(-1)^{n+1}H_{n-1}\left(H_{n/2}^{(3)}-H_{(n-1)/2}^{(3)}\right)}{n}}.\tag{2}$$ or, by partial summation: $$ \color{purple}{I=-\frac{3}{4}\sum_{n=1}^{+\infty}H_{n/2}^{(3)}(-1)^n\left(\frac{H_n}{n+1}+\frac{H_{n-1}}{n}\right).}\tag{3}$$ Another identity that follows from the Taylor series of $\log^3(1-z)$ is: $$\color{red}{I=3\sum_{n=1}^{+\infty}\frac{(-1)^{n+1}\left(H_n^2-H_n^{(2)}\right)}{(n+1)^3}.}\tag{4}$$

Answer 3

Indeed, there is a closed form for this integral: $$I=\frac{\pi^2}3\ln^32-\frac25\ln^52+\frac{\pi^2}2\zeta(3)+\frac{99}{16}\zeta(5)-\frac{21}4\zeta(3)\ln^22\\-12\operatorname{Li}_4\left(\frac12\right)\ln2-12\operatorname{Li}_5\left(\frac12\right).$$

Answer 4

This is an updated partial answer that is rather similar to Jack D'Aurizio's approach. (I really hope he doesn't mind.)

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Step 1: Expressing the integral as a sum.

It is easy to derive the formula $$\left(\sum^{\infty}_{n=1}a_nx^n\right)\left(\sum^{\infty}_{n=1}b_nx^n\right)=\sum^\infty_{n=1}\sum^{n}_{k=1}a_kb_{n-k+1}x^{n+1}$$ We apply this formula to derive the Taylor series of $\ln^2(1+x)$. \begin{align} \ln^2(1+x) &=\left(\sum^{\infty}_{n=1}\frac{(-1)^{n-1}}{n}x^n\right)\left(\sum^{\infty}_{n=1}\frac{(-1)^{n-1}}{n}x^n\right)\\ &=\sum^\infty_{n=1}\sum^n_{k=1}\frac{(-1)^{k-1}(-1)^{n-k}}{k(n-k+1)}x^{n+1}\\ &=\sum^\infty_{n=1}\frac{(-1)^{n+1}}{n+1}\sum^n_{k=1}\left(\frac{1}{k}+\frac{1}{n-k+1}\right)x^{n+1}\\ &=\sum^\infty_{n=1}\frac{(-1)^{n+1}2H_n}{n+1}x^{n+1} \end{align} Apply this formula again to obtain the Taylor series of $\displaystyle\frac{\ln^2(1+x)}{1+x}$. \begin{align} \frac{\ln^2(1+x)}{1+x} &=\left(\sum^\infty_{n=1}\frac{(-1)^{n+1}2H_n}{n+1}x^{n+1}\right)\left(\sum^{\infty}_{n=1}(-1)^{n-1}x^{n-1}\right)\\ &=\sum^\infty_{n=1}\sum^n_{k=1}\frac{(-1)^{k+1}(-1)^{n-k}2H_k}{k+1}x^{n+1}\\ &=\sum^\infty_{n=1}2(-1)^{n+1}\sum^n_{k=1}\frac{H_k}{k+1}x^{n+1}\\ \end{align} The inner sum is \begin{align} \sum^n_{k=1}\frac{H_k}{k+1} &=\sum^n_{k=1}\frac{H_{k+1}}{k+1}-\sum^n_{k=1}\frac{1}{(k+1)^2}\\ &=\sum^{n+1}_{k=1}\frac{H_k}{k}-H_{n+1}^{(2)}\\ &=\sum^{n+1}_{k=1}\frac{1}{k}\sum^k_{j=1}\frac{1}{j}-H_{n+1}^{(2)}\\ &=\sum^{n+1}_{j=1}\frac{1}{j}\left(\sum^{n+1}_{k=1}\frac{1}{k}-\sum^{j-1}_{k=1}\frac{1}{k}\right)-H_{n+1}^{(2)}\\ &=H_{n+1}^2-\sum^{n+1}_{j=1}\frac{H_j}{j}\\ &=\frac{H_{n+1}^2-H_{n+1}^{(2)}}{2} \end{align} Hence $$\frac{\ln^2(1+x)}{1+x}=\sum^\infty_{n=1}(-1)^{n+1}\left(H_{n+1}^2-H_{n+1}^{(2)}\right)x^{n+1}$$ Pluck this into the integral. \begin{align} \int^1_0\frac{\ln^3(1+x)\ln{x}}{x}{\rm d}x &=-\frac{3}{2}\int^1_0\frac{\ln^2(1+x)\ln^2{x}}{1+x}{\rm d}x\\ &=-\frac{3}{2}\sum^\infty_{n=1}(-1)^{n+1}\left(H_{n+1}^2-H_{n+1}^{(2)}\right)\int^1_0x^{n+1}\ln^2{x} \ {\rm d}x\\ &=-3\sum^\infty_{n=1}\frac{(-1)^{n+1}\left(H_{n+1}^2-H_{n+1}^{(2)}\right)}{(n+2)^3}\\ &=3\sum^\infty_{n=1}\frac{(-1)^{n}\left(H_{n}^{(2)}-H_{n}^2\right)}{(n+1)^3}\\ \end{align}

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Step 2: Evaluation of $\displaystyle\sum^\infty_{n=1}\frac{(-1)^nH_n^{(2)}}{(n+1)^3}$

We begin with some simple manipulations of the sum. \begin{align} \sum^\infty_{n=1}\frac{(-1)^nH_n^{(2)}}{(n+1)^3} &=\sum^\infty_{n=1}\frac{(-1)^nH_{n+1}^{(2)}}{(n+1)^3}-\sum^\infty_{n=1}\frac{(-1)^n}{(n+1)^5}\\ &=-\frac{15}{16}\zeta(5)-\underbrace{\sum^\infty_{n=1}\frac{(-1)^nH_n^{(2)}}{n^3}}_{S} \end{align} Consider the function $\displaystyle f(z)=\frac{\pi\csc(\pi z)\psi_1(-z)}{z^3}$. At the positive integers, \begin{align} {\rm Res}(f,n) &=\operatorname*{Res}_{z=n}\left[\frac{(-1)^n}{z^3(z-n)^3}+\frac{(-1)^n(H_n^{(2)}+2\zeta(2))}{z^3(z-n)}\right]\\ &=\frac{6(-1)^n}{n^5}+\frac{(-1)^nH_n^{(2)}}{n^3}+\frac{2(-1)^n\zeta(2)}{n^3} \end{align} Summing them up gives $$\sum^\infty_{n=1} {\rm Res}(f,n)=-\frac{45}{8}\zeta(5)+S-\frac{3}{2}\zeta(2)\zeta(3)$$ At the negative integers, \begin{align} {\rm Res}(f,-n) &=-\frac{(-1)^n\psi_1(n)}{n^3}\\ &=\frac{(-1)^nH_n^{(2)}}{n^3}-\frac{(-1)^n\zeta(2)}{n^3}-\frac{(-1)^n}{n^5} \end{align} Summing them up gives $$\sum^\infty_{n=1} {\rm Res}(f,-n)=S+\frac{3}{4}\zeta(2)\zeta(3)+\frac{15}{16}\zeta(5)$$ At $z=0$, \begin{align} {\rm Res}(f,0) &=[z^2]\left(\frac{1}{z}+\zeta(2)z\right)\left(\frac{1}{z^2}+\zeta(2)+2\zeta(3)z+3\zeta(4)z^2+4\zeta(5)z^3\right)\\ &=4\zeta(5)+2\zeta(2)\zeta(3) \end{align} Since the sum of the reisudes $=0$, $$\sum^\infty_{n=1}\frac{(-1)^nH_n^{(2)}}{(n+1)^3}=-\frac{41}{32}\zeta(5)+\frac{5}{8}\zeta(2)\zeta(3)$$

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Step 3: Evaluation of $\displaystyle\sum^\infty_{n=1}\frac{(-1)^nH_n^{2}}{(n+1)^3}$

Formula $(45)$ in this page states that this sum is equal to $$4{\rm Li}_5\left(\frac{1}{2}\right)+4{\rm Li}_4\left(\frac{1}{2}\right)\ln{2}+\frac{2}{15}\ln^5{2}-\frac{107}{32}\zeta(5)+\frac{7}{4}\zeta(3)\ln^2{2}-\frac{2}{3}\zeta(2)\ln^2{2}-\frac{3}{8}\zeta(2)\zeta(3)$$ Using a previously derived result is really unsatisfactory for me. Nevertheless, I have not been able to derive this result, as contour integration fails here due to the power of the denominator being odd (which implies that the sum will vanish when I add the residues at the positive and negative integers up). It seems that Tunk-Fey's brilliant approach would be the most viable method to crack this last sum.

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Step 4: Obtaining the final result

Combining our previous results, we get \begin{align} &\ \ \ \ \ \small{\int^1_0\frac{\ln^3(1+x)\ln{x}}{x}{\rm d}x}\\ &=\small{3\sum^\infty_{n=1}\frac{(-1)^n\left(H_{n}^{(2)}-H_n^2\right)}{(n+1)^3}}\\ &=\small{3\left(\frac{33}{16}\zeta(5)+\zeta(2)\zeta(3)-4{\rm Li}_5\left(\frac{1}{2}\right)-4{\rm Li}_4\left(\frac{1}{2}\right)\ln{2}-\frac{2}{15}\ln^5{2}-\frac{7}{4}\zeta(3)\ln^2{2}+\frac{2}{3}\zeta(2)\ln^3{2}\right)}\\ &=\small{\frac{99}{16}\zeta(5)+\frac{\pi^2}{2}\zeta(3)-12{\rm Li}_5\left(\frac{1}{2}\right)-12{\rm Li}_4\left(\frac{1}{2}\right)\ln{2}-\frac{2}{5}\ln^5{2}-\frac{21}{4}\zeta(3)\ln^2{2}+\frac{\pi^2}{3}\ln^3{2}} \end{align}

Answer 5

Related problems and techniques: (I), (II). Here is a different form of solution

$$ I = -3\sum_{n=0}^{\infty} \sum_{k=0}^{n}\frac{(-1)^k{ n\brack k}k(k-1) }{(n+1)^3n!} ,$$

where $ {n \brack k} $ is the Stirling numbers of the first kind.

Answer 6

An alternate form of the answers given by @Cleo and @Tunk-Fey as sum of $1$ and $1/2$ argumented polylogarithm-products with rational coefficients:

$$I = \frac{99}{16}\operatorname{Li}_5(1)-12\operatorname{Li}_5\left(\frac{1}{2}\right) + 15\operatorname{Li}_1\left( \frac{1}{2} \right)\operatorname{Li}_4(1) - 12\operatorname{Li}_1\left(\frac{1}{2}\right)\operatorname{Li}_4\left(\frac{1}{2}\right) - 15\operatorname{Li}_2\left( \frac{1}{2} \right)\operatorname{Li}_3(1)-\frac{51}{4}\operatorname{Li}_1^2\left( \frac{1}{2} \right)\operatorname{Li}_3(1)+12\operatorname{Li}_2(1)\operatorname{Li}_3\left( \frac{1}{2} \right) - \frac{2}{5}\operatorname{Li}_1^5\left(\frac{1}{2}\right),$$

where $\operatorname{Li}_n$ is the polylogarithm function, and specifically

$$\begin{align} & \operatorname{Li}_5(1) \ \ \ = \zeta(5) \\ & \operatorname{Li}_5\left(\textstyle\frac{1}{2}\right) = \textstyle \sum_{k=1}^\infty {2^{-k} \over k^5} \\ & \operatorname{Li}_4(1) \ \ \ = \zeta(4) = \frac{\pi^4}{90} \\ & \operatorname{Li}_4\left(\textstyle\frac{1}{2}\right) = \textstyle \sum_{k=1}^\infty {2^{-k} \over k^4} \\ & \operatorname{Li}_3(1) \ \ \ = \zeta(3) \\ & \operatorname{Li}_3\left(\textstyle\frac{1}{2}\right) = \frac{7}{8} \zeta(3) - \frac{\pi^2}{12} \ln 2 + \frac{1}{6} \ln^3 2 \\ & \operatorname{Li}_2(1) \ \ \ = \zeta(2) = \frac{\pi^2}{6} \\ & \operatorname{Li}_2\left(\textstyle\frac{1}{2}\right) = \frac{\pi^2}{12} - \frac{1}{2} \ln^2 2 \\ & \operatorname{Li}_1\left(\textstyle\frac{1}{2}\right) = \ln2, \end{align}$$ where $\zeta$ is the Riemann zeta function.

Answer 7

UPDATE: The way below may be found in the preprint, A new perspective on the evaluation of the logarithmic integral, $\int_0^1\frac{\log(x)\log^3(1+x)}{x}\textrm{d}x$ by C.I.Valean.

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A magical way proposed by Cornel Ioan Valean

We use the powerful form of the Beta function presented in the book, (Almost) Impossible Integrals, Sums, and Series, $\displaystyle \int_0^1 \frac{x^{a-1}+x^{b-1}}{(1+x)^{a+b}} \textrm{d}x = \operatorname{B}(a,b)$, (see pages $72$-$73$).

Here is the magic ...

By cleverly differentiating in two different ways to get rid of a nasty integral, we simply get the wonderful result

$$4\lim_{\substack{a\to0 \\ b \to 0}}\frac{\partial^{4}}{\partial a^3 \partial b}\operatorname{B}(a,b)-6\lim_{\substack{a\to0 \\ b \to 0}}\frac{\partial^{4}}{\partial a^2 \partial b^2}\operatorname{B}(a,b)$$ $$=8\int_0^1 \frac{\log(x)\log^3(1+x)}{x}\textrm{d}x-4\int_0^1 \frac{\log^3(x)\log(1+x)}{x}\textrm{d}x-4\int_0^1 \frac{\log^4(1+x)}{x}\textrm{d}x.$$ ... and we're wonderfully done!

A first note: a similar strategy has been used in this answer https://math.stackexchange.com/q/3531878.

A BIG BONUS (the extraction of the series $\displaystyle \sum_{n=1}^{\infty}(-1)^{n-1}\frac{H_n}{n^4}$):

The extraction of the series $\displaystyle \sum_{n=1}^{\infty}(-1)^{n-1}\frac{H_n}{n^4}$ is achieved immediately by observing that using the same Beta function limits, we arrive at

$$\lim_{\substack{a\to0 \\ b \to 0}}\frac{\partial^{4}}{\partial a^3 \partial b}\operatorname{B}(a,b)-\lim_{\substack{a\to0 \\ b \to 0}}\frac{\partial^{4}}{\partial a^2 \partial b^2}\operatorname{B}(a,b)$$ $$=\underbrace{\int_0^1 \frac{\log^2(x)\log^2(1+x)}{x}\textrm{d}x}_{\displaystyle 15/4\zeta(5)-4\sum_{n=1}^{\infty} (-1)^{n-1} H_n/n^4}-\int_0^1 \frac{\log^3(x)\log(1+x)}{x}\textrm{d}x,$$ which assures the desired extraction after turning the second integral into the series we want to calculate.

Answer 8

A solution, no harmonic sums, no Beta function involved.

For $m,n$ positive integers, \begin{align}W_{m,n}&=\int_0^{\frac{1}{2}} \frac{\ln^m(1-x)\ln^n x}{1-x}dx\\ K&=\int_0^1 \frac{\ln(1-x)\ln^3 x}{1-x}dx\\ W_{2,2}&=\int_0^{\frac{1}{2}} \frac{\ln^2(1-x)\ln^2 x}{1-x}dx\overset{\text{IBP}}=\frac{\ln^4 2}{3}+\frac{ 2}{3}\underbrace{\int_0^{\frac{1}{2}} \frac{\ln^3(1-x)\ln x}{x}dx}_{u=1-x}\\ &=\boxed{\frac{\ln^5 2}{3}+\frac{2K}{3}-\frac{2W_{1,3}}{3}}\\ W_{3,1}&=\int_0^{\frac{1}{2}} \frac{\ln^3(1-x)\ln x}{1-x}dx\overset{\text{IBP}}=\frac{\ln^5 2}{4}+\underbrace{\int_0^{\frac{1}{2}} \frac{\ln^4(1-x)}{x}dx}_{u=1-x}\\ &=\boxed{\frac{\ln^5 2}{4}+6\zeta(5)-\frac{W_{0,4}}{4}}\\ W_{4,0}&=\int_0^{\frac{1}{2}}\frac{\ln^4(1-x)}{1-x}dx=\boxed{\frac{\ln^5 2}{5}} \end{align}

\begin{align}J&={\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\\ &\overset{u=\frac{x}{1+x}}=-\int_0^{\frac{1}{2}}\frac{\ln\left(\frac{u}{1-u}\right)\ln^3(1-u)}{u(1-u)}du\\ &=\int_0^{\frac{1}{2}}\frac{\ln^4(1-u)}{u(1-u)}du-\int_0^{\frac{1}{2}}\frac{\ln^3(1-u)\ln u}{u(1-u)}du\\ &=W_{4,0}-W_{3,1}+\int_0^{\frac{1}{2}}\underbrace{\frac{\ln^4(1-u)}{u}du}_{z=1-u}-\underbrace{\int_0^{\frac{1}{2}}\frac{\ln^3(1-u)\ln u}{u}du}_{z=1-u}\\ &=W_{4,0}-W_{3,1}+24\zeta(5)-W_{0,4}-K+W_{1,3}\\ &=\boxed{-\frac{\ln^5 2}{20}-K+W_{1,3}+18\zeta(5)-\frac{3W_{0,4}}{4}}\\ \int_0^{\frac{1}{2}}\frac{\ln^4\left(\frac{x}{1-x}\right)}{1-x}dx&=W_{0,4}-4W_{1,3}+6W_{2,2}-4W_{3,1}+W_{4,0}\\ &\boxed{\frac{6\ln^2 5}{5}+4K-8W_{1,3}-24\zeta(5)+2W_{0,4}}\\ \end{align} Therefore, \begin{align}8J+\int_0^{\frac{1}{2}}\frac{\ln^4\left(\frac{x}{1-x}\right)}{1-x}dx&=\frac{4\ln^5 2}{5}-4K+120\zeta(5)-4W_{0,4}\\ J&=\frac{\ln^5 2}{10}-4K+15\zeta(5)-\frac{W_{0,4}}{8}-\frac{1}{8}\underbrace{\int_0^{\frac{1}{2}}\frac{\ln^4\left(\frac{x}{1-x}\right)}{1-x}dx}_{u=\frac{x}{1-x}}\\ &=\frac{\ln^5 2}{10}-\frac{K}{2}+15\zeta(5)-\frac{W_{0,4}}{2}-\frac{1}{8}\times\frac{45\zeta(5)}{2}\\ &\boxed{J=\frac{\ln^5 2}{10}-\frac{K}{2}-\frac{W_{0,4}}{2}+\frac{195\zeta(5)}{16}} \end{align}

For $n\geq 1$, integer, \begin{align}W_{0,n}&=\int_0^{\frac{1}{2}} \frac{\ln^n x}{1-x}dx\\ &\overset{z=2x}=\frac{1}{2}\int_0^1\frac{\ln^n\left(\frac{z}{2}\right)}{1-\frac{z}{2}}dz=\frac{1}{2}\int_0^1\frac{\sum_{k=0}^n\binom{n}{k}(-1)^k(\ln 2)^k(\ln z)^{n-k}}{1-\frac{z}{2}}dz\\ &=(-1)^n(\ln 2)^{n+1}+(-1)^n\underbrace{\sum_{k=0}^{n-1}\binom{n}{k}(n-k)!(\ln 2)^k\text{Li}_{n+1-k}\left(\frac{1}{2}\right)}_{l=n-k}\\&=\boxed{(-1)^n(\ln 2)^{n+1}+(-1)^n\sum_{l=1}^n\binom{n}{l}l!(\ln 2)^{n-l}\text{Li}_{l+1}\left(\frac{1}{2}\right)}\\ \end{align} Therefore, \begin{align}W_{0,4}=\boxed{\ln^5 2-\frac{2\pi^2\ln^3 2}{3}+\frac{21\zeta(3)\ln^2 2}{2} 2+24\text{Li}_4\left(\frac{1}{2}\right)\ln 2+24\text{Li}_5\left(\frac{1}{2}\right)}\end{align}

\begin{align}K&\overset{\text{IBP}}=\left[\ln(1-x)\left(\int_0^x\frac{\ln^3 t}{1-t}dt-\int_0^1\frac{\ln^3 t}{1-t}dt\right)\right]_0^1+\\&\int_0^1\int_0^1\left(\frac{x\ln^3(tx)}{(1-x)(1-tx)}- \frac{\ln^3 t}{(1-x)(1-t)}\right)dtdx\\ &=\int_0^1\int_0^1\left(\frac{\ln^3 x+3\ln^2x\ln t+3\ln x\ln^2 t}{(1-x)(1-t)}-\frac{\ln^3(tx)}{(1-t)(1-tx)}\right)dtdx\\ &\overset{\text{Fubini}}=-2\pi^2\zeta(3)+\int_0^1\int_0^1\left(\frac{\ln^3 x}{(1-x)(1-t)}-\frac{\ln^3(tx)}{(1-t)(1-tx)}\right)dtdx\\ &=-2\pi^2\zeta(3)+\int_0^1\left(\frac{1}{1-t}\left(\int_0^1 \frac{\ln^3 x}{1-x}dx\right)-\frac{1}{t(1-t)}\left(\int_0^t\frac{\ln^3 u}{1-u}du\right)\right)dt\\ &=-2\pi^2\zeta(3)+\int_0^1\left(\frac{1}{1-t}\left(\int_t^1 \frac{\ln^3 x}{1-x}dx\right)-\frac{1}{t}\left(\int_0^t\frac{\ln^3 u}{1-u}du\right)\right)dt\\ &\overset{\text{IBP}}=-2\pi^2\zeta(3)-K+\underbrace{\int_0^1\frac{\ln^4 t}{1-t}dt}_{=24\zeta(5)}\\ &\boxed{K=\int_0^1 \frac{\ln(1-t)\ln^3 t}{1-t}dt=12\zeta(5)-\pi^2\zeta(3)}\\ \end{align}

Therefore, \begin{align}\boxed{J=\frac{\pi^2\ln^3 2}{3}-\frac{2\ln^5 2}{5}-\frac{21\zeta(3)\ln^2 2}{4}-12\text{Li}_4\left(\frac{1}{2}\right)\ln 2+\frac{99\zeta(5)}{16}+\frac{\pi^2\zeta(3)}{2}\!-\!12\text{Li}_5\left(\frac{1}{2}\right)} \end{align}

NB: The idea is to find an expression of $J$ as a linear rational combination of $W_{i,j}$ and $K$, and we have, here, enough equations to determine $J$, solving the linear system of equations obtained.

Answer 9

Here is a proof for $\left(4\right)$ since i couldn't find one: $$\int _0^1\frac{\ln ^3\left(1+x\right)\ln \left(x\right)}{x^2}\:dx$$ $$\overset{\operatorname{IBP}}=-\ln ^3\left(2\right)+3\int _0^1\frac{\ln ^2\left(1+x\right)}{x\left(1+x\right)}\:dx+3\int _0^1\frac{\ln \left(x\right)\ln ^2\left(1+x\right)}{x\left(1+x\right)}\:dx$$

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$$3\underbrace{\int _0^1\frac{\ln ^2\left(1+x\right)}{x\left(1+x\right)}\:dx}_{x=\frac{1}{1+x}}=3\int _0^1\frac{\ln ^2\left(x\right)}{1-x}\:dx-3\int _0^{\frac{1}{2}}\frac{\ln ^2\left(x\right)}{1-x}\:dx$$ $$=6\sum _{k=1}^{\infty }\frac{1}{k^3}-6\sum _{k=1}^{\infty }\frac{1}{k^3\:2^k}-6\ln \left(2\right)\sum _{k=1}^{\infty }\frac{1}{k^2\:2^k}-3\ln ^3\left(2\right)$$ $$=6\zeta \left(3\right)-6\operatorname{Li}_3\left(\frac{1}{2}\right)-6\ln \left(2\right)\operatorname{Li}_2\left(\frac{1}{2}\right)-3\ln ^3\left(2\right)$$ $$=\frac{3}{4}\zeta \left(3\right)-\ln ^3\left(2\right)$$

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$$3\underbrace{\int _0^1\frac{\ln \left(x\right)\ln ^2\left(1+x\right)}{x\left(1+x\right)}\:dx}_{x=\frac{1}{1+x}}$$ $$=3\int _0^{\frac{1}{2}}\frac{\ln \left(x\right)\ln ^2\left(1-x\right)}{x}\:dx-3\int _{\frac{1}{2}}^1\frac{\ln ^3\left(x\right)}{1-x}\:dx$$ $$=-6\sum _{k=1}^{\infty }\frac{H_k}{k^3\:2^k}-6\ln \left(2\right)\sum _{k=1}^{\infty }\frac{H_k}{k^2\:2^k}+6\sum _{k=1}^{\infty }\frac{1}{k^4\:2^k}+6\ln \left(2\right)\sum _{k=1}^{\infty }\frac{1}{k^3\:2^k}+18\sum _{k=1}^{\infty }\frac{1}{k^4}$$ $$-18\sum _{k=1}^{\infty }\frac{1}{k^4\:2^k}-18\ln \left(2\right)\sum _{k=1}^{\infty }\frac{1}{k^3\:2^k}-9\ln ^2\left(2\right)\sum _{k=1}^{\infty }\frac{1}{k^2\:2^k}-3\ln ^4\left(2\right)$$ $$=\frac{69}{4}\zeta \left(4\right)-18\operatorname{Li}_4\left(\frac{1}{2}\right)-\frac{63}{4}\ln \left(2\right)\zeta \left(3\right)+\frac{9}{2}\ln ^2\left(2\right)\zeta \left(2\right)-\frac{3}{4}\ln ^4\left(2\right)$$ Where $\ln ^2\left(1-x\right)=2\sum _{k=1}^{\infty }\left(\frac{H_k}{k}-\frac{1}{k^2}\right)x^k$ is used on the $2$nd line.

See here and here for the $1$st and $2$nd sum.

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Collecting the results yields: $$\int _0^1\frac{\ln ^3\left(1+x\right)\ln \left(x\right)}{x^2}\:dx=\frac{69}{4}\zeta \left(4\right)+\frac{3}{4}\zeta \left(3\right)-18\operatorname{Li}_4\left(\frac{1}{2}\right)-\frac{63}{4}\ln \left(2\right)\zeta \left(3\right)$$ $$+\frac{9}{2}\ln ^2\left(2\right)\zeta \left(2\right)-2\ln ^3\left(2\right)-\frac{3}{4}\ln ^4\left(2\right)$$