About the integral $\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx$
I came across the following Integral and have been completely stumped by it.
$$\large\int_{0}^{1}\dfrac{\log(x)\log^2(1+x)}{x}dx$$
I'm extremely sorry, but the only thing I noticed was that the limits of the Integral were similar to the Beta function. I also got a hint that solving it would require the Polylogarithm, Gamma and the Riemann Zeta Functions. $$$$Would it be possible to solve this without using Complex Methods (I haven't learnt them yet) unless absolutely necessary? Any help on this Integral would be greatly appreciated. Many, many thanks in advance!
EDIT: From the comments given below byDavid H Sir, and Alex S Sir, the Integral becomes: $$\int_0^1 \dfrac{\ln^2(1+x)\ln(x)}{x}dx$$
Just an observation: This is strikingly similar to the Beta Function. Also, if we consider $$\int^1_0 (1+x)^ax^bdx$$ and differentiate twice with respect to $a$ and once with respect to $b$ and set $a=b=0$, we get the above Integral (except the $x$ in the denominator).
I'm not sure, but I think taking series representations of the Integrals would be of help, especially since the closed form includes the Riemann Zeta and the Polylogarithm functions.
Solution 1:
$$I=\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx = -\int_{0}^{+\infty} t \log(1+e^{-t})\,dt$$ is an integral that already appeared on MSE. It can be tackled in many ways, for instance by expanding $\log^2(1+x)$ as its Taylor series:
$$ \log^2(1+x) = 2\sum_{n\geq 1}\frac{(-1)^{n+1} H_n}{n+1} x^{n+1} $$ from which: $$ I = 2\sum_{n\geq 1}\frac{(-1)^{n} H_n}{(n+1)^3} $$ follows. Evaluation of such series is a fashion topic here on MSE: Shobhit mentioned the last series (Euler sum) in his answer to a related question.
Solution 2:
Performing integration by parts by taking $u=\ln^2(1+x)$ and $\mathrm dv=\dfrac{\ln x}{x}\ \mathrm dx$, then \begin{align} I&=\int_0^1\frac{\ln x\ln^2(1+x)}{x}\ \mathrm dx\\ &=\ln x\ln^2(1+x)\Bigg|_0^1-\int_0^1\frac{\ln^2{x}\ln(1+x)}{1+x}\,\mathrm dx\\ &=-\int_0^1\sum_{k=1}^\infty (-1)^{k-1}H_{k}\,x^k\ln^2x\,\mathrm dx\tag{1}\\ &=-\sum_{k=1}^\infty (-1)^{k-1}H_{k}\int_0^1x^k\ln^2x\,\mathrm dx\\ &=-2\sum_{k=1}^\infty (-1)^{k-1}\frac{H_{k}}{(k+1)^3}\tag{2}\\ &=-2\sum_{k=1}^\infty (-1)^{k-1}\left[\frac{H_{k+1}}{(k+1)^3}-\frac{1}{(k+1)^4}\right]\tag{3}\\ &=2\sum_{k=1}^\infty (-1)^{k-1}\left[\frac{H_{k}}{k^3}-\frac{1}{k^4}\right]\\ &=\frac{11\pi^4}{180}-4\text{Li}_4 \left(\frac{1}{2} \right)-\frac{7\zeta(3)\ln 2}{2}+\frac{\pi^2\ln^2 2}{6}-\frac{\ln^4 2}{6}-2\eta(4)\tag{4}\\ &=\frac{11\pi^4}{180}-4\text{Li}_4 \left(\frac{1}{2} \right)-\frac{7\zeta(3)\ln 2}{2}+\frac{\pi^2\ln^2 2}{6}-\frac{\ln^4 2}{6}-\frac{7\zeta(4)}{4}\tag{5}\\ &=\bbox[8pt,border:3px #FF69B4 solid]{\color{red}{\large\frac{\pi^4}{24}-4\text{Li}_4 \left(\frac{1}{2} \right)-\frac{7\zeta(3)\ln 2}{2}+\frac{\pi^2\ln^2 2}{6}-\frac{\ln^4 2}{6}}} \end{align}
Explanation :
$(1)$ Use generating function $\displaystyle\sum_{k=1}^\infty (-1)^{k-1}H_{k}\,x^k=\frac{\ln(1+x)}{1+x}$
$(2)$ Use formula $\displaystyle\int_0^1 x^k \ln^n x\ \mathrm dx=\frac{(-1)^n n!}{(k+1)^{n+1}}\quad,\ n\in\mathbb{Z}_{n\ge0}$
$(3)$ Use property $\displaystyle H_{k}=H_{k+1}-\frac{1}{k+1}$
$(4)$ Use the result $\displaystyle \sum_{k=1}^\infty (-1)^{k-1} \frac{H_k}{k^3} = \frac{11\pi^4}{360}-2\text{Li}_4 \left(\frac{1}{2} \right)-\frac{7\zeta(3)\ln 2}{4}+\frac{\pi^2\ln^2 2}{12}-\frac{\ln^4 2}{12}$
$(5)$ Use property of Dirichlet eta function $\displaystyle \eta(s)=\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^s}=\left(1-2^{1-s}\right)\zeta(s)$
Solution 3:
$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} \ln^{3}\pars{x \over x + 1} & = \ln^{3}\pars{x} - 3\ln^{2}\pars{x}\ln\pars{x + 1} + 3\color{#66f}{\ln\pars{x}\ln^{2}\pars{x + 1}} - \ln^{3}\pars{x + 1} \end{align} such that \begin{align} \color{#66f}{\ln\pars{x}\ln^{2}\pars{x + 1}} & = \ln^{2}\pars{x}\ln\pars{x + 1} + {1 \over 3}\,\ln^{3}\pars{x + 1} + {1 \over 3}\bracks{\ln^{3}\pars{x \over x + 1} - \ln^{3}\pars{x}} \end{align}
Then, \begin{align} &\int_{0}^{1}{\color{#66f}{\ln\pars{x}\ln^{2}\pars{1 + x}} \over x}\,\dd x = \sum_{i = 1}^{3}\mc{I}_{i} \\[5mm] &\mbox{where}\quad \left\{\begin{array}{rcl} \ds{\mc{I}_{1}} & \ds{\equiv} & \ds{\phantom{1 \over 3}\int_{0}^{1}{\ln^{2}\pars{x}\ln\pars{1 + x} \over x} \,\dd x} \\[3mm] \ds{\mc{I}_{2}} & \ds{\equiv} & \ds{{1 \over 3}\int_{0}^{1}{\ln^{3}\pars{1 + x} \over x}\,\dd x} \\[3mm] \ds{\mc{I}_{3}} & \ds{\equiv} & \ds{{1 \over 3}\int_{0}^{1}\bracks{\ln^{3}\pars{x \over x + 1} - \ln^{3}\pars{x}}{\dd x \over x}} \end{array}\right.\label{1}\tag{1} \end{align}
$\bbox[15px,#ffe,border:1px dotted navy]{\ds{\mc{I}_{1} =\ {\large ?}}}$. \begin{align} \mc{I}_{1} & \equiv \int_{0}^{1}{\ln^{2}\pars{x}\ln\pars{1 + x} \over x}\,\dd x = \int_{0}^{-1}{\ln^{2}\pars{-x}\ln\pars{1 - x} \over x}\,\dd x \\[5mm] & = -\int_{0}^{-1}\mrm{Li}_{2}'\pars{x}\ln^{2}\pars{-x}\,\dd x = 2\int_{0}^{-1}{\mrm{Li}_{2}\pars{x} \over x}\,\ln\pars{-x}\,\dd x \\[5mm] & = 2\int_{0}^{-1}\mrm{Li}_{3}'\pars{x}\,\ln\pars{-x}\,\dd x = -2\int_{0}^{-1}{\mrm{Li}_{3}\pars{x} \over x}\,\dd x = -2\int_{0}^{-1}\mrm{Li}_{4}'\pars{x}\,\dd x \\[5mm] & = -2\,\mrm{Li}_{4}\pars{-1} = \bbox[15px,#ffe,border:1px dotted navy]{\ds{7\pi^{4} \over 360}} \label{I1}\tag{I1} \end{align}
$\bbox[15px,#ffe,border:1px dotted navy]{\ds{\mc{I}_{2} =\ {\large ?}}}$. \begin{align} \mc{I}_{2} & \equiv {1 \over 3}\int_{0}^{1}{\ln^{3}\pars{1 + x} \over x}\,\dd x = {1 \over 3}\int_{1}^{2}{\ln^{3}\pars{x} \over x - 1}\,\dd x = {1 \over 3}\int_{1}^{1/2}{\ln^{3}\pars{1/x} \over 1/x - 1} \pars{-\,{1 \over x^{2}}}\,\dd x \\[5mm] = &\ -\,{1 \over 3}\int_{1/2}^{1}{\ln^{3}\pars{x} \over x\pars{1 - x}}\,\dd x = -\,{1 \over 3}\int_{1/2}^{1}{\ln^{3}\pars{x} \over x}\,\dd x - {1 \over 3}\int_{1/2}^{1}{\ln^{3}\pars{x} \over 1 - x}\,\dd x \\[5mm] & = {1 \over 12}\,\ln^{4}\pars{2} - {1 \over 3}\bracks{\ln^{4}\pars{2} + 3\int_{1/2}^{1}{\ln\pars{1 - x} \over x}\,\ln^{2}\pars{x}\,\dd x} \\[5mm] & = -\,{1 \over 4}\,\ln^{4}\pars{2} + \int_{1/2}^{1}\mrm{Li}_{2}'\pars{x}\,\ln^{2}\pars{x}\,\dd x \\[5mm] & = -\,{1 \over 4}\,\ln^{4}\pars{2} - \,\mrm{Li}_{2}\pars{1 \over 2}\ln^{2}\pars{2} -2\int_{1/2}^{1}{\mrm{Li}_{2}\pars{x} \over x}\,\ln\pars{x}\,\dd x \\[5mm] & = -\,{1 \over 4}\,\ln^{4}\pars{2} - \,\mrm{Li}_{2}\pars{1 \over 2}\ln^{2}\pars{2} -2\int_{1/2}^{1}\mrm{Li}_{3}'\pars{x}\,\ln\pars{x}\,\dd x \\[5mm] & = -\,{1 \over 4}\,\ln^{4}\pars{2} - \,\mrm{Li}_{2}\pars{1 \over 2}\ln^{2}\pars{2} - 2\,\mrm{Li}_{3}\pars{1 \over 2}\ln\pars{2} + 2\int_{1/2}^{1}{\mrm{Li}_{3}\pars{x} \over x}\,\dd x \\[5mm] & = -\,{1 \over 4}\,\ln^{4}\pars{2} - \,\mrm{Li}_{2}\pars{1 \over 2}\ln^{2}\pars{2} - 2\,\mrm{Li}_{3}\pars{1 \over 2}\ln\pars{2} + 2\int_{1/2}^{1}\mrm{Li}_{4}'\pars{x}\,\dd x \\[5mm] & = -\,{1 \over 4}\,\ln^{4}\pars{2} - \,\mrm{Li}_{2}\pars{1 \over 2}\ln^{2}\pars{2} - 2\,\mrm{Li}_{3}\pars{1 \over 2}\ln\pars{2} + 2\mrm{Li}_{4}\pars{1} - 2\mrm{Li}_{4}\pars{1 \over 2} \end{align} By using the well know values of $\ds{\,\mrm{Li}_{2}\pars{1/2}}$ and $\ds{\,\mrm{Li}_{3}\pars{1/2}}$ and $\ds{\,\mrm{Li}_{4}\pars{1} = \zeta\pars{4} = \pi^{4}/90}$: \begin{equation} \mc{I}_{2} = \bbox[15px,#ffe,border:1px dotted navy]{\ds{{\pi^{4} \over 45} + {\pi^{2}\ln^{2}\pars{2} \over 12} - {\ln^{4}\pars{2} \over 12} - {7\ln\pars{2} \over 4}\,\zeta\pars{3} - 2\,\mrm{Li}_{4}\pars{1 \over 2}}}\label{I2}\tag{I2} \end{equation}
$\bbox[15px,#ffe,border:1px dotted navy]{\ds{\mc{I}_{3} =\ {\large ?}}}$. \begin{align} \mc{I}_{3} & \equiv {1 \over 3}\int_{0}^{1}\bracks{\ln^{3}\pars{x \over x + 1} - \ln^{3}\pars{x}}{\dd x \over x} = {1 \over 3}\,\lim_{\epsilon \to 0^{+}}\bracks{% \int_{\epsilon}^{1}\ln^{3}\pars{x \over x + 1}\,{\dd x \over x} - \int_{\epsilon}^{1}{\ln^{3}\pars{x} \over x}\,\dd x} \end{align} In the RHS first integral I'll make the change of variables $\ds{x/\pars{1 + x} \mapsto x}$ such that \begin{align} \mc{I}_{3} & = {1 \over 3}\,\lim_{\epsilon \to 0^{+}}\bracks{% \int_{\epsilon/\pars{1 + \epsilon}}^{1/2}{\ln^{3}\pars{x} \over x - x^{2}} \,\dd x - \int_{\epsilon}^{1}{\ln^{3}\pars{x} \over x}\,\dd x} \\[5mm] & = {1 \over 3}\,\ \underbrace{\lim_{\epsilon \to 0^{+}} \int_{\epsilon/\pars{1 + \epsilon}}^{\epsilon} {\ln^{3}\pars{x} \over x - x^{2}}\,\dd x}_{\ds{=\ 0}}\ +\ {1 \over 3}\int_{0}^{1/2}\bracks{% {\ln^{3}\pars{x} \over x - x^{2}} - {\ln^{3}\pars{x} \over x}}\dd x - {1 \over 3}\int_{1/2}^{1}{\ln^{3}\pars{x} \over x}\,\dd x \\[5mm] & = {1 \over 3}\int_{0}^{1/2}{\ln^{3}\pars{x} \over 1 - x}\,\dd x\ -\ \underbrace{{1 \over 3}\int_{1/2}^{1}{\ln^{3}\pars{x} \over x}\,\dd x} _{\ds{-\,{1 \over 12}\,\ln^{4}\pars{2}}} \end{align} The remaining integral can be evaluated by successive integration by parts: A procedure I already exploited in the $\ds{\,\mc{I}_{2}}$-evaluation $\pars{~\mbox{see}\ \eqref{I2}~}$. Indeed, they are rather similar. Therefore, \begin{equation} \mc{I}_{3} = \bbox[15px,#ffe,border:1px dotted navy]{\ds{% {\pi^{2}\ln^{2}\pars{2} \over 12} - {\ln^{4}\pars{2} \over 12} - {7\ln\pars{2} \over 4}\,\zeta\pars{3} - 2\,\mrm{Li}_{4}\pars{1 \over 2}}}\label{I3}\tag{I3} \end{equation}
By replacing $\ds{\,\mc{I}_{1},\,\mc{I}_{2}\ \mbox{and}\ \,\mc{I}_{3}}$ $\pars{~\mbox{see expressions}\ \eqref{I1},\eqref{I2}\ \mbox{and}\ \eqref{I3}~}$ in \eqref{1}: $$ \int_{0}^{1}{\ln\pars{x}\ln^{2}\pars{1 + x} \over x}\,\dd x = \bbox[15px,#ffe,border:1px dotted navy]{\ds{{\pi^{4} \over 24} + {\pi^{2}\ln^{2}\pars{2} \over 6} - {\ln^{4}\pars{2} \over 6} - {7\ln\pars{2} \over 2}\,\zeta\pars{3} - 4\,\mrm{Li}_{4}\pars{1 \over 2}}} $$