I don't think your expression can be evaluated in closed form.
You can write z= I/(2i+x) in trigonometric form z= re^(i a) and get representation using Euler formula: Re[ Li_2(z)] = Sum_{k=1}^{\infty}r^k cos(k a)/k^2
although I always use Mahematica in my work, I found Maple preferable in dealing with functions of complex variables. Unfortunately, I did not find an answer in both of them.