I am wondering if I can simplify
$${\rm Im} \left[ {\rm Li}_2(i x)\right] \ , $$
in terms of more elementary functions, when $x$ is real (in particular, I am interested in $0<x<1$). I checked the main functional identities involving the dilogarithm - for example here - but couldn't find something useful. There is a similar but more general question here but the answers don't seem useful either.
$$\Im(\text{Li}_2(i x))=\sum_{n=0}^\infty (-1)^n\,\frac{ x^{2 n+1}}{(2 n+1)^2}=\frac{1}{4} x \,\Phi \left(-x^2,2,\frac{1}{2}\right)$$ where appears the Lerch transcendent function which is valid for any $x$.
If you need to compute it for $0\leq x \leq 1$, you may use quite accurate approximation using $$\frac{1}{4} x \,\Phi \left(-x^2,2,\frac{1}{2}\right)\sim x\, P_n(x)$$ where $P_n$ is a $[2n,2n]$ Padé approximant.
For example $$P_1=\frac{56 x^2+225}{81 x^2+225}$$ $$P_2=\frac{1302208 x^4+14806575 x^2+20539575}{225 \left(10575 x^4+75950 x^2+91287\right)}$$
To give you an idea of the accuracy, looking at the norm $$\Phi_n=\int_0^1 \left(\frac{1}{4} x\, \Phi \left(-x^2,2,\frac{1}{2}\right)-x\,P_n\right)^2\,dx$$
$$\Phi_1=4.5\times 10^{-7} \quad \Phi_2=2.5\times 10^{-10} \quad \Phi_3=1.6\times 10^{-13}$$