Integral of $$$\frac{\ln\left(1 - x\right)}{x}$$$

Find $$$\int \frac{\ln\left(1 - x\right)}{x}\, dx$$$.

This integral (Polylogarithm Function) does not have a closed form:

$${\color{red}{\int{\frac{\ln{\left(1 - x \right)}}{x} d x}}} = {\color{red}{\left(- \operatorname{Li}_{2}\left(x\right)\right)}}$$

Therefore,

$$\int{\frac{\ln{\left(1 - x \right)}}{x} d x} = - \operatorname{Li}_{2}\left(x\right)$$

Add the constant of integration:

$$\int{\frac{\ln{\left(1 - x \right)}}{x} d x} = - \operatorname{Li}_{2}\left(x\right)+C$$

$$$\int \frac{\ln\left(1 - x\right)}{x}\, dx = - \operatorname{Li}_{2}\left(x\right) + C$$$A