The command of Mathematica 11.1.0 

    Integrate[Log[1 - x]*Log[1 + x]/x, {x, 0, 2}]

performs
$$-2 \text{Li}_3\left(\frac{1}{3}\right)+\text{Li}_3\left(-\frac{1}{3}\right)+i \pi  \left(-3 \text{Li}_2\left(\frac{1}{3}\right)+\text{Li}_2\left(-\frac{1}{3}\right)+\log ^2(3)+\log (2) \log (3)-\log (9) \log (3)\right)+\text{Li}_2\left(-\frac{1}{3}\right) \log (3)-\text{Li}_2\left(\frac{1}{3}\right) \log (9)+\zeta (3)+\frac{i \pi ^3}{4}-\frac{1}{3} \log ^3(3) .
 $$
The result is confirmed numerically through

    N[%]
$-1.4024+1.92982 i$

and

  

    NIntegrate[Log[1 - x]*Log[1 + x]/x, {x, 0, 2}] 

 

$-1.40238+1.92982 i $

supplied the warning  "NIntegrate failed to converge to prescribed accuracy after 9 
recursive bisections in x near {x} = {1.00388}. NIntegrate obtained 
-1.40238+1.92982 I and 0.0000965261928395521` for the integral and 
error estimate".

To be sure I do that in Maple by

    int(ln(1-x)*ln(1+x)/x, x = 0 .. 2, numeric);

$ - 1.402399720+ 1.929815441\,i$