Dilogarithm Identities

Question

Is there a cleaner way to write: $$ f(x) = \operatorname{Li}_2(i x) - \operatorname{Li}_2(-i x) $$ in terms of simpler functions? I don't know enough about dilogarithms, and the basic identities I see on wikipedia are not helping me.

Answer 1

Actually, if you define

$$g(x) = \Im{[\text{Li}_2(i x)]}$$

then you can show using the series definition of

$$\text{Li}_2(x) = \sum_{n=1}^{\infty} \frac{x^n}{n^2}$$

that

$$g(x) = \int_0^x dt \frac{\arctan{t}}{t} $$