$\displaystyle\int\frac{\ln^3x\ln(1-x)}{1-x}dx=-H_1(x)\ln^3x+\mathrm{Li}_2(x)\ln^3x+3H_2(x)\ln^2x-3\mathrm{Li}_3(x)\ln^2x-6\mathrm{H_3}(x)\ln x+6\mathrm{Li}_4(x)\ln x+6H_4(x)-6\mathrm{Li_5}(x)$
\begin{align} \displaystyle\int\frac{\ln^3x\ln(1-x)}{1-x}dx&=-\int\sum_{k\geq1}H_kx^k\ln^3k\,dx\\ &=-\sum_{k\geq1}H_k\int x^k\ln^3k\,dx\\ &=-\sum_{k\geq1}H_k\frac{\partial^3}{\partial k^3}\left[\int x^k dx\right]\\ &=-\sum_{k\geq1}H_k\frac{\partial^3}{\partial k^3}\left[\frac{x^{k+1}}{k+1}\right]\\ &=-\sum_{k\geq1}H_k\left[\frac{x^{k+1}\ln^3x}{k+1}-\frac{3x^{k+1}\ln^2x}{(k+1)^2}+\frac{6x^{k+1}\ln x}{(k+1)^3}-\frac{6x^{k+1}}{(k+1)^4}\right]\\ &=-\ln^3x\sum_{k\geq1}\frac{H_{k+1}x^{k+1}}{k+1}+\ln^3x\sum_{k\geq1}\frac{x^{k+1}}{(k+1)^2}+3\ln^2x\sum_{k\geq1}\frac{H_{k+1}x^{k+1}}{(k+1)^2} -3\ln^2x\sum_{k\geq1}\frac{x^{k+1}}{(k+1)^3} -6\ln x\sum_{k\geq1}\frac{H_{k+1}x^{k+1}}{(k+1)^3} +6\ln x\sum_{k\geq1}\frac{x^{k+1}}{(k+1)^4} +6\sum_{k\geq1}\frac{H_{k+1}x^{k+1}}{(k+1)^4} -6\ln x\sum_{k\geq1}\frac{x^{k+1}}{(k+1)^4}\\ &=-H_1(x)\ln^3x+\mathrm{Li}_2(x)\ln^3x+3H_2(x)\ln^2x-3\mathrm{Li}_3(x)\ln^2x-6\mathrm{H_3}(x)\ln x+6\mathrm{Li}_4(x)\ln x+6H_4(x)-6\mathrm{Li_5}(x) \end{align}
$\displaystyle\int_0^1\frac{\ln x\ln(1+x)}{1-x}dx=-\frac{3}{2}\zeta(2)\ln2+\zeta(3)$
\begin{align} \displaystyle\int_0^1\frac{\ln x\ln(1+x)}{1-x}dx&=\left[\int_0^x\frac{\ln y}{1-y}dy\ln(1+x)\right]_0^1-\int_0^1\frac{J(x)}{1+x}dx\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left(J(x)=\int_0^x\frac{\ln y}{1-y}dy\,\,\,\,x\in[0,1]\right)\\ &=-\zeta(2)\ln2-\int_0^1\int_0^1\frac{x\ln xy}{(1-xy)(1+x)}dydx\\ &=-\zeta(2)\ln2-\int_0^1\left(\int_0^1\frac{x\ln y}{(1-xy)(1+x)}dx\right)dy-\int_0^1\left(\int_0^1\frac{x\ln x}{(1-xy)(1+x)}dy\right)dx\\ &=-\zeta(2)\ln2+\int_0^1\left[\frac{\ln(1-xy)}{y(1+y)}+\frac{\ln(1+x)}{1+y}\right]_{x=0}^{x=1}\ln y\,dy+\int_0^1\left[\frac{\ln(1-xy)}{1+x}\right]_{y=0}^{y=1}\ln x\,dx\\ &=-\zeta(2)\ln2+\int_0^1\frac{\ln(1-y)\ln y}{y(1+y)}dy+\ln2\int_0^1\frac{\ln y}{1+y}dy+\int_0^1\frac{\ln(1-x)\ln x}{1+x}dx\\ &=-\zeta(2)\ln2+\int_0^1\frac{\ln(1-y)\ln y}{y}dy+\ln2\int_0^1\frac{\ln y}{1+y}dy\\ &=-\zeta(2)\ln2+\frac{1}{2}\left(\ln^2x\ln(1-x)+\int_0^1\frac{\ln^2 y}{1-y}dy\right)+\ln2\left(\int_0^1\frac{\ln y}{1-y}dy-\int_0^1\frac{2y\ln y}{1-y^2}dy\right)\\ &=-\zeta(2)\ln2+\frac{1}{2}\int_0^1\frac{\ln^2y}{1-y}dy+\ln2\left(\int_0^1\frac{\ln y}{1-y}dy-\frac{1}{2}\int_0^1\frac{\ln y}{1-y}dy\right)\\ &=-\frac{3}{2}\zeta(2)\ln2+\zeta(3) \end{align}
$\displaystyle\int_0^1\left(\frac{\ln^2(1-x)\ln^2(1+x)}{1-x}-\frac{\ln^22\ln^2(1-x)}{1-x}\right)dx=\frac{1}{8}\zeta(5)-\frac{1}{2}\ln2\zeta(4)+2\ln^22\zeta(3)-\frac{2}{3}\ln^32\zeta(2)-2\zeta(2)\zeta(3)+\frac{1}{10}\ln^52+4\mathrm{Li}_5\left(\frac{1}{2}\right)$
\begin{align}
\displaystyle I=\int_0^1\left(\frac{\ln^2(1-x)\ln^2(1+x)}{1-x}-\frac{\ln^22\ln^2(1-x)}{1-x}\right)dx&=\int_0^1\frac{\ln^2(1-x)}{1-x}(\ln^2(1+x)-\ln^22)dx\\
&\overset{IBP}{=}\frac{2}{3}\int_0^1\frac{\ln^3(1-x)\ln(1+x)}{1+x}dx
\end{align}
ここで
\begin{align}
a^3b=\frac{(a+b)^4}{8}-\frac{(a-b)^4}{8}-ab^3\xrightarrow[b=\ln(1+x)]{a=\ln(1-x)}(\ln(1-x))^3\ln(1+x)=\frac{(\ln(1-x)+\ln(1+x))^4}{8}-\frac{(\ln(1-x)-\ln(1+x))^4}{8}-(\ln(1-x))(\ln(1+x))^3
\end{align}
であるから
\begin{align}
\displaystyle I&=\frac{1}{12}\int_0^1\frac{\ln^4(1-x^2)}{1+x}dx
-\frac{1}{12}\int_0^1\frac{\ln^4(1-x)-\ln^4(1+x)}{1+x}dx-\frac{2}{3}\int_0^1\frac{\ln(1-x)\ln^3(1+x)}{1+x}dx\\
&=\left(\frac{16}5\ln^52-16\ln^32\zeta(2)+48\ln^22\zeta(3)-54\ln2\zeta(4)-24\zeta(2)\zeta(3)+72\zeta(5)\right)-\left(\frac{45}2\zeta(5)\right)-\left(-6\operatorname{Li}_5\left(\frac12\right)+6\zeta(5)-6\ln2\zeta(4)+3\ln^22\zeta(3)-\ln^32\zeta(2)+\frac14\ln^52\right)\\
&=\frac{1}{8}\zeta(5)-\frac{1}{2}\ln2\zeta(4)+2\ln^22\zeta(3)-\frac{2}{3}\ln^32\zeta(2)-2\zeta(2)\zeta(3)+\frac{1}{10}\ln^52+4\mathrm{Li}_5\left(\frac{1}{2}\right)
\end{align}
\begin{align} \int_0^1\frac{\ln^4(1-x^2)}{1+x}\ dx&=\int_0^1(1-x)\frac{\ln^4(1-x^2)}{1-x^2}\ dx\\&\overset{x^2=y}{=}\frac12\int_0^1\frac{(1-\sqrt{y})(\ln^4(1-y))}{(1-y)\sqrt{y}}\ dy\\ &\overset{IBP}{=}-\frac1{20}\int_0^1\frac{\ln^5(1-y)}{y^{3/2}}\ dy=-\frac{1}{20}\lim_{\substack{x\mapsto-\frac{1} {2}\\y\mapsto1}} \frac{\partial^5}{\partial y^5}\text{B}(x,y)\\ &=\frac{16}5\ln^52-16\ln^32\zeta(2)+48\ln^22\zeta(3)-54\ln2\zeta(4)-24\zeta(2)\zeta(3)+72\zeta(5) \end{align}
\begin{align} \int_0^1\frac{\ln^4(1-x)-\ln^4(1+x)}{1+x}\ dx\overset{\frac{1-x}{1+x}=y}{=}\int_0^1\frac{\ln^4x}{1+x}\ dx=\frac{45}2\zeta(5) \end{align}
\begin{align}
\int_0^1\frac{\ln(1-x)\ln^3(1+x)}{1+x}\ dx&\overset{\frac1{1+x}=y}{=}-\int_{\frac{1}{2}}^1 \frac{\ln\left(\frac{2x-1}{x}\right)\ln^3x}{x}\ dx\\
&=\int_{\frac{1}{2}}^1\frac{\ln^4x}{x}\ dx-\int_{\frac{1}{2}}^1\frac{\ln(2x-1)\ln^3x}{x}\ dx\\
&=\frac15\ln^52-\int_{\frac{1}{2}}^1\frac{\ln(1-2x)\ln^3x}{x}\ dx-i\frac{\pi}{4}\ln^42\,\,\,\,\,\,\because\ln(2x-1)=\ln(1-2x)-i\pi\\
&=\frac15\ln^52+\sum_{n\geq1}\frac{2^n}{n}\int_{1/2}^1x^{n-1}\ln^3x\ dx-i\frac{\pi}{4}\ln^42\\
&=\frac15\ln^52+\sum_{n\geq1}\frac{2^n}{n}\left(\frac{\ln^32}{n2^n}+\frac{3\ln^22}{n^22^n}+\frac{6\ln2}{n^32^n}+\frac{6}{n^42^n}-\frac{6}{n^4}\right)-i\frac{\pi}{4}\ln^42\\
&=\frac15\ln^52+\ln^32\zeta(2)+3\ln^22\zeta(3)+6\ln2\zeta(4)+6\zeta(5)-6\operatorname{Li}_5(2)-i\frac{\pi}{4}\ln^42
\end{align}
ここで
$$\operatorname{Li}_5(x)=-\frac74\zeta(4)\ln(-x)-\frac16\zeta(2)\ln^3(-x)-\frac1{120}\ln^5(-x)+\operatorname{Li}_5\left(\frac{1}{x}\right)$$
$x=2$を代入して
$$\operatorname{Li}_5(2)=2\ln2\zeta(4)+\frac13\ln^32\zeta(2)-\frac1{120}\ln^52+\operatorname{Li}_5\left(\frac12\right)-i\frac{\pi}{24}\ln^42$$
これを先の結果に代入して
$$\int_0^1\frac{\ln(1-x)\ln^3(1+x)}{1+x}\ dx=-6\operatorname{Li}_5\left(\frac12\right)+6\zeta(5)-6\ln2\zeta(4)+3\ln^22\zeta(3)-\ln^32\zeta(2)+\frac14\ln^52$$