Could we possibly prove this result without using the polylogarithm? I know how to do it by polylogarithm means, but I want a different way. Is that possible?
$$\int_0^1\frac{x \log^2(1-x)}{1+x^2} \ dx = \frac{35}{32}\zeta(3)+\frac{1}{24}\log^3(2) -\frac{5}{96} \pi^2 \log(2)$$
$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ $\ds{\int_{0}^{1}{x\ln^{2}\pars{1 - x} \over 1 + x^{2}}\,\dd x ={35 \over 32}\,\zeta\pars{3} + {1 \over 24}\,\ln^{3}\pars{2} -{5 \over 96}\,\pi^{2}\ln\pars{2}:\ {\large ?}}$.
\begin{align}&\color{#c00000}{% \int_{0}^{1}{x\ln^{2}\pars{1 - x} \over 1 + x^{2}}\,\dd x} =\Re\int_{0}^{1}{\ln^{2}\pars{1 - x} \over \ic + x}\,\dd x =\Re\int_{0}^{1}{\ln^{2}\pars{x} \over \ic + 1 - x}\,\dd x \\[3mm]&=\Re\int_{0}^{1/\pars{1 + \ic}} {\ln^{2}\pars{\bracks{1 + \ic}x} \over 1 - x}\,\dd x =\Re\int_{0}^{\pars{1 - \ic}/2}\ln\pars{1 - x} \bracks{2\ln\pars{\bracks{1 + \ic}x}\,{1 \over x}}\,\dd x \\[3mm]&=-2\Re\int_{0}^{\pars{1 - \ic}/2}{{\rm Li}_{1}\pars{x} \over x}\, \ln\pars{\bracks{1 + \ic}x}\,\dd x =-2\Re\int_{0}^{\pars{1 - \ic}/2}{\rm Li}_{2}'\pars{x} \ln\pars{\bracks{1 + \ic}x}\,\dd x \\[3mm]&=2\Re\int_{0}^{\pars{1 - \ic}/2}{{\rm Li}_{2}\pars{x} \over x}\,\dd x =2\Re\int_{0}^{\pars{1 - \ic}/2}{\rm Li}_{3}'\pars{x}\,\dd x \end{align}
$$ \color{#c00000}{% \int_{0}^{1}{x\ln^{2}\pars{1 - x} \over 1 + x^{2}}\,\dd x} =2\,\Re{\rm Li}_{3}\pars{1 - \ic \over 2} $$
With one of the MW formulas in group $\pars{1}$: \begin{align} &\overbrace{{\rm Li}_{3}\pars{\half - {\ic \over 2}} +{\rm Li}_{3}\pars{\half + {\ic \over 2}}} ^{\ds{2\,\Re{\rm Li}_{3}\pars{1 - \ic \over 2}}} +{\rm Li}_{3}\pars{1 - {2 \over 1 - \ic}} \\[3mm]&=\zeta\pars{3} + {1 \over 6}\,\ln^{3}\pars{1 - \ic \over 2} + {1 \over 6}\,\pi^{2}\ln\pars{1 - \ic \over 2} -\half\,\ln^{2}\pars{1 - \ic \over 2}\ln\pars{1 + \ic \over 2} \end{align}
I trust you can take from here.
My thoughts on the problem (too long for comment):
Prove that the definite integral $I$ has the stated closed form value: $$I:=\int_{0}^{1}\frac{x\log^2{(1-x)}}{1+x^2}\mathrm{d}x;\\ I=\frac{35}{32}\zeta{(3)}-\frac{5}{16}\zeta{(2)}\log{(2)}+\frac{1}{24}\log^3{(2)}.$$
Define the auxiliary function $f(\mu)$ for all $\mu\in\mathbb{R}^+$ by the definite integral,
$$f(\mu):=\int_{0}^{1}\frac{x(1-x)^{\mu-1}}{1+x^2}\mathrm{d}x.$$
Differentiating with respect to $\mu$,
$$\begin{align} \frac{d^2}{d\mu^2}f(\mu) &=\int_{0}^{1}\frac{\partial^2}{\partial\mu^2}\frac{x(1-x)^{\mu-1}}{1+x^2}\mathrm{d}x\\ &=\int_{0}^{1}\frac{x(1-x)^{\mu-1}\log^2{(1-x)}}{1+x^2}\mathrm{d}x. \end{align}$$
Evaluating the second derivative of $f(\mu)$ at $\mu=1$ then yields the integral $I$:
$$\frac{d^2}{d\mu^2}f(\mu)\bigg{|}_{\mu=1}=\int_{0}^{1}\frac{x(1-x)^{0}\log^2{(1-x)}}{1+x^2}\mathrm{d}x=\int_{0}^{1}\frac{x\log^2{(1-x)}}{1+x^2}\mathrm{d}x=:I.$$
The function $f(\mu)$ itself can be represented as an alternating series of beta functions:
$$\begin{align} f(\mu)&=\int_{0}^{1}\frac{x(1-x)^{\mu-1}}{1+x^2}\mathrm{d}x\\ &=\int_{0}^{1}x(1-x)^{\mu-1}\sum_{n=0}^{\infty}(-1)^nx^{2n}\,\mathrm{d}x\\ &=\sum_{n=0}^{\infty}(-1)^n\int_{0}^{1}x^{2n+1}(1-x)^{\mu-1}\,\mathrm{d}x\\ &=\sum_{n=0}^{\infty}(-1)^n\operatorname{B}{(2n+2,\mu)}\\ &=-\sum_{n=1}^{\infty}(-1)^n\operatorname{B}{(2n,\mu)}. \end{align}$$
The second derivative of $\operatorname{B}{(2n,\mu)}$ with respect to $\mu$ at $\mu=1$ is (courtesy of WolframAlpha),
$$\frac{\partial^2}{\partial\mu^2}\operatorname{B}{(2n,\mu)}\bigg{|}_{\mu=1}=\frac{(H_{2n})^2}{2n}+\frac{\zeta{(2)}}{2n}-\frac{\psi_1{(2n+1)}}{2n}.$$
Hence,
$$\begin{align} I &=\frac{d^2}{d\mu^2}f(\mu)\bigg{|}_{\mu=1}\\ &=-\sum_{n=1}^{\infty}(-1)^n\frac{\partial^2}{\partial\mu^2}\operatorname{B}{(2n,\mu)}\bigg{|}_{\mu=1}\\ &=-\sum_{n=1}^{\infty}(-1)^n\left[\frac{(H_{2n})^2}{2n}+\frac{\zeta{(2)}}{2n}-\frac{\psi_1{(2n+1)}}{2n}\right]\\ &=\sum_{n=1}^{\infty}(-1)^n\left[\frac{\psi_1{(2n+1)}}{2n}-\frac{(H_{2n})^2}{2n}-\frac{\zeta{(2)}}{2n}\right]\\ &=\sum_{n=1}^{\infty}(-1)^n\left[\frac{\psi_1{(2n+1)}}{2n}-\frac{(H_{2n})^2}{2n}\right]-\frac{1}{2}\zeta{(2)}\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\\ &=\sum_{n=1}^{\infty}(-1)^n\frac{\psi_1{(2n+1)}}{2n}-\sum_{n=1}^{\infty}(-1)^n\frac{(H_{2n})^2}{2n}+\frac{1}{2}\zeta{(2)}\log{(2)}. \end{align}$$
...Aaand that's as far as I've gotten.
