How to integrate $$\displaystyle{\Large\displaystyle \int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x}$$
My idea
\begin{align*} \int_{0}^{1} \frac{\ln^3 (1+x) \ln^3 x}{x} \, \mathrm{d}x &= -18 \sum_{n=1}^{\infty} \left ( -1 \right )^{n+1} \left ( \mathcal{H}_{n+1}^2 - \mathcal{H}_{n+1}^{(2)} \right ) \frac{1}{\left (n+2 \right )^4} \\ &= 18 \sum_{n=1}^{\infty} (-1)^n \left ( \mathcal{H}_{n+1}^2 - \mathcal{H}_{n+1}^{(2)} \right ) \frac{1}{\left ( n+2 \right )^4} \\ &= 18 \sum_{n=2}^{\infty} (-1)^{n-1} \left ( \mathcal{H}_{n}^2 - \mathcal{H}_{n+1}^{(2)} \right ) \frac{1}{\left ( n+1 \right )^4} \end{align*}