<s>[INST] Integral $\int_0^1 \frac{\ln(1+x+x^2)\ln(1-x+x^2)}{x}dx$ 
Prove that
  $$\sf I=\int_0^1 \frac{\ln(1+x+x^2)\ln(1-x+x^2)}{x}dx=\frac{\pi}{6\sqrt{3}}\psi_1\left(\frac{1}{3}\right)-\frac{\pi^3}{9\sqrt{3}}-\frac{19}{18}\zeta(3).$$

I have thought about the integral from above after I saw this similar integral and  I believe changing the sign to have $\sf 1+x+x^2$ might get us a nice  closed form.
So I started using the following formula: $$\sf 2ab=(a+b)^2-a^2-b^2$$
$$\sf \Rightarrow 2I=\int_0^1\frac{\ln^2(1+x^2+x^4)}{x}dx-\int_0^1\frac{\ln^2(1+x+x^2)}{x}dx-\int_0^1\frac{\ln^2(1-x+x^2)}{x}dx$$
Using in the first integral the substitution $\sf x^2\rightarrow x $  gets us: 
$$\sf \int_0^1\frac{\ln^2(1+x^2+x^4)}{x}dx=\frac12\int_0^1\frac{\ln^2(1+x+x^2)}{x}dx$$
$$\sf \Rightarrow I=-\frac14\int_0^1\frac{\ln^2(1+x+x^2)}{x}dx-\frac12\int_0^1\frac{\ln^2(1-x+x^2)}{x}dx$$
Well, now we only need to find:
$$\sf I(a)=\int_0^1\frac{\ln^2(1+ax+x^2)}{x}dx $$
Then set $a=1$ and $a=-1$. Of course  I tried to use Feynman's trick:
$$\sf I'(a)=2\int_0^1\frac{\ln(1+ax+x^2)}{1+ax+x^2}dx$$
But quickly gave up as it doesn't look promising. 
Another way might be to let $\sf x+\frac12=\frac{\sqrt 3}{2}t$ in order to get:
$$\sf \int_0^1\frac{\ln^2(1+x+x^2)}{x}dx=\int_\frac{1}{\sqrt 3}^\sqrt 3 \frac{\ln^2\left(\frac34(1+t^2)\right)}{t-\frac{1}{\sqrt 3}}dt$$
But well.. I would appreciate some help!

Update. In the meantime I found something a conjecture:
$$\sf \int_0^1\frac{\ln^2(1+x+x^2)}{x}dx=\frac{2\pi}{9\sqrt3}\psi_1\left(\frac13\right)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$$
 [/INST] As shown in the question we have:
$$\sf  I=-\frac14\int_0^1\frac{\ln^2(1+x+x^2)}{x}dx-\frac12\int_0^1\frac{\ln^2(1-x+x^2)}{x}dx$$
For the first integral we can write:
$$\sf (a-b)^2=a^2-b^2-2b(a-b);\ a=\ln(1-x^3),b=\ln(1-x)$$
$$\sf \Rightarrow  \int_0^1\frac{\ln^2(1+x+x^2)}{x}dx=\int_0^1 \frac{\left(\ln(1-x^3)-\ln(1-x)\right)^2}{x}dx$$
$$\sf =\color{blue}{\int_0^1 \frac{\ln^2(1-x^3)}{x}dx}-\int_0^1 \frac{\ln^2(1-x)}{x}dx-2\int_0^1 \frac{\ln(1-x)\ln(1+x+x^2)}{x}dx$$
$$\sf \overset{\color{blue}{x^3\to x}}=-\frac23\int_0^1 \frac{\ln^2(1-x)}{x}dx-2\int_0^1 \frac{\ln(1-x)\ln(1+x+x^2)}{x}dx=-\frac43\zeta(3)-2J$$
Note also that:
$$\sf \int_0^1 \frac{\ln^2(1-x)}{x}dx=\int_0^1 \frac{\ln^2 x}{1-x}dx=\sum_{n=1}^\infty \int_0^1 x^{n-1}\ln^2 x\, dx=2\sum_{n=1}^\infty \frac{1}{n^3}=2\zeta(3)$$
The latter integral can be found here:
$$\sf J=\int_0^1 \frac{\ln(1-x)\ln(1-x+x^2)}{x}dx=-\frac{\pi}{9\sqrt 3}\psi_1\left(\frac13\right)+\frac{2\pi^3}{27\sqrt 3}-\frac13\zeta(3) $$
$$\Rightarrow \boxed{\sf \int_0^1\frac{\ln^2(1+x+x^2)}{x}dx=\frac{2\pi}{9\sqrt3}\psi_1\left(\frac13\right)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)}$$
The second integral can be found here:
$$\boxed{\sf\int_0^1 \frac{\ln^2(1-x+x^2)}{x}dx=-\frac{4\pi}{9\sqrt{3}}\psi_1\left(\frac{1}{3}\right)+\frac{8\pi^3}{27\sqrt{3}}+\frac{22}{9}\zeta(3)}$$
Combining the boxed results yields:
$$\boxed{\sf  \int_0^1 \frac{\ln(1+x+x^2)\ln(1-x+x^2)}{x}dx=\frac{\pi}{6\sqrt{3}}\psi_1\left(\frac{1}{3}\right)-\frac{\pi^3}{9\sqrt{3}}-\frac{19}{18}\zeta(3)}$$
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