<s>[INST] Evaluation of a dilogarithmic integral 
Problem. Prove that the following dilogarithmic integral has the indicated value:
  $$\int_{0}^{1}\mathrm{d}x \frac{\ln^2{(x)}\operatorname{Li}_2{(x)}}{1-x}\stackrel{?}{=}-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$


My attempt:
I began by using the polylogarithmic expansion in terms of generalized harmonic numbers,
$$\frac{\operatorname{Li}_r{(x)}}{1-x}=\sum_{n=1}^{\infty}H_{n,r}\,x^n;~~r=2.$$
Then I switched the order of summation and integration and used the substitution $u=-\ln{x}$ to evaluate the integral:
$$\begin{align}
\int_{0}^{1}\mathrm{d}x \frac{\ln^2{(x)}\operatorname{Li}_2{(x)}}{1-x}
&=\int_{0}^{1}\mathrm{d}x\ln^2{(x)}\sum_{n=1}^{\infty}H_{n,2}x^n\\
&=\sum_{n=1}^{\infty}H_{n,2}\int_{0}^{1}\mathrm{d}x\,x^n\ln^2{(x)}\\
&=\sum_{n=1}^{\infty}H_{n,2}\int_{0}^{\infty}\mathrm{d}u\,u^2e^{-(n+1)u}\\
&=\sum_{n=1}^{\infty}H_{n,2}\frac{2}{(n+1)^3}\\
&=2\sum_{n=1}^{\infty}\frac{H_{n,2}}{(n+1)^3}.
\end{align}$$
So I've reduced the integral to an Euler sum, but unfortunately I've never quite got the knack for evaluating Euler sums. How to proceed from here?
 [/INST] It is easy to see that
$$2\sum^\infty_{n=1}\frac{H_n^{(2)}}{(n+1)^3}=2\sum^\infty_{n=1}\frac{H_{n+1}^{(2)}}{(n+1)^3}-2\sum^\infty_{n=1}\frac{1}{(n+1)^5}=2\sum^\infty_{n=1}\frac{H_n^{(2)}}{n^3}-2\zeta(5)$$
Consider $\displaystyle f(z)=\frac{\pi\cot{\pi z} \ \Psi^{(1)}(-z)}{z^3}$. We know that
$$\pi\cot{\pi z}=\frac{1}{z-n}-2\zeta(2)(z-n)+O((z-n)^3)$$
and
$$\Psi^{(1)}(-z)=\frac{1}{(z-n)^2}+\left(H_n^{(2)}+\zeta(2)\right)+O(z-n)$$
At the positive integers,
\begin{align}
{\rm Res}(f,n)
&=\operatorname*{Res}_{z=n}\left[\frac{1}{z^3(z-n)^3}+\frac{H_n^{(2)}-\zeta(2)}{z^3(z-n)}\right]\\
&=\frac{H_n^{(2)}}{n^3}-\frac{\zeta(2)}{n^3}+\frac{6}{n^5}\\
\end{align}
At the negative integers,
\begin{align}
{\rm Res}(f,-n)&=-\frac{\Psi^{(1)}(n)}{n^3}\\&=\frac{H_{n-1}^{(2)}-\zeta(2)}{n^3}\\&=\frac{H_{n}^{(2)}}{n^3}-\frac{\zeta(2)}{n^3}-\frac{1}{n^5}\tag1
\end{align}
At $z=0$, 
\begin{align}
{\rm Res}(f,0)&=[z^2]\left(\frac{1}{z}-2\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)-4\zeta(2)\zeta(3)
\end{align}
Since the sum of the residues $=0$, we conclude that
\begin{align}
\color\red{\int^1_0\frac{\log^2{x} \ {\rm Li}_2(x)}{1-x}{\rm d}x}
&=2\sum^\infty_{n=1}\frac{H_n^{(2)}}{n^3}-2\zeta(5)\\
&=\zeta(2)\zeta(3)-6\zeta(5)+\zeta(2)\zeta(3)+\zeta(5)-4\zeta(5)+4\zeta(2)\zeta(3)-2\zeta(5)\\
&\large{\color\red{=6\zeta(2)\zeta(3)-11\zeta(5)}}
\end{align}
Explanation
$(1):$ Use the functional equation $\displaystyle \Psi^{(1)}(z+1)=-\frac{1}{z^2}+\Psi^{(1)}(z)$ which is derived by differentiating the functional equation of the digamma function, as well as the fact that $\displaystyle H_n^{(2)}=\frac{1}{n^2}+H_{n-1}^{(2)}$. 
As for how to obtain the laurent series, the series for $\Psi(z)$ was cleverly derived here by Random Variable. In essence,
$$\color{blue}{\gamma+\Psi(-z)=\frac{1}{z-n}+H_n+\sum^\infty_{k=1}(-1)^k\left(H_n^{(k+1)}+(-1)^{k+1}\zeta(k+1)\right)(z-n)^k}$$
Differentiating yields
$$\color{blue}{\Psi^{(1)}(-z)=\frac{1}{(z-n)^2}+\sum^\infty_{k=1}(-1)^{k+1}k\left(H_n^{(k+1)}+(-1)^{k+1}\zeta(k+1)\right)(z-n)^{k-1}}$$
For $\pi\cot{\pi z}$,
\begin{align}
\color{blue}{\pi\cot{\pi z}}
&=\Psi(1-z)-\Psi(z) \ \ \ \ \ \text{(reflection formula for digamma function)}\\
&=\int^1_0\frac{t^{z-1}-t^{-z}}{1-t}{\rm d}t \ \ \ \ \ \text{(recall that $\Psi(z)=-\gamma+H_{z-1}$)}\\
&=\sum_{k=0}^\infty\int^1_0\left(t^{z+k-1}-t^{-z+k}\right){\rm d}t\\
&=\sum_{k=0}^\infty\left(\frac{1}{z+k}+\frac{1}{z-k-1}\right)\\
&=\frac{1}{z}+\frac{1}{z-1}+\frac{1}{z+1}+\frac{1}{z-2}+\frac{1}{z+2}+\cdots\\
&=\frac{1}{z}+\sum^\infty_{k=1}\left(\frac{1}{z-k}+\frac{1}{z+k}\right)\\
&=\frac{1}{z}+\sum^\infty_{k=1}\frac{2z}{z^2-k^2}\\
&=\frac{1}{z}-2\sum^\infty_{k=1}\sum^\infty_{m=1}\frac{z^{2m-1}}{k^{2m}}\\
&=\color{blue}{\frac{1}{z}-2\sum^\infty_{m=1}\zeta(2m)z^{2m-1}}\\
&=\pi\cot(\pi (z-n)) \ \ \ \ \ \text{(since cotangent has a period of $\pi$)}\\
&=\color{blue}{\frac{1}{z-n}-2\sum^\infty_{m=1}\zeta(2m)(z-n)^{2m-1}}\\
\end{align}
</s>