I am looking at the following function:
$$f(x_1,x_2):=\text{Li}_2 \left( \frac{x_1}{x_2} \right). \tag{1}$$
I would like to know whether this function can be expressed as a sum of harmonic polylogs (HPLs) for which the variables can only be $x_1$, $x_2$, $1-x_1$, $1-x_2$ and $x_2-x_1$. In other words, I am asking whether the following holds:
$$f(x_1,x_2) = r_0(x_1,x_2) \text{Li}_2 (x_1) + r_1(x_1,x_2) \text{Li}_2 (x_2) + r_2(x_1,x_2) \text{Li}_2 (1-x_1) + r_3(x_1,x_2) \text{Li}_2 (1-x_2) + r_4(x_1,x_2) \text{Li}_2 (x_2 - x_1) + \text{products of logs}, \tag{2}$$
where the $r$-functions are expected to be rational functions (maybe even just numerical coefficients?).
Does there exist such an identity for eq. (1)?