Hello experts,

I would like to know how to derive analytically the skewness of a cumulative distribution function (CDF). This is the CDF I am working with:

P(x)=\Phi\left(\frac{ln(1+x)+\frac{1}{2}l(l-1)\sigma^2}{|l|\sigma}\right)

With the cumulative standard normal distribution \Phi. The corresponding probability density function (PDF) with $p(x)=P'(x)$ and $\varphi=\Phi'$:

p(x)=\varphi\left(\frac{ln(1+x)+\frac{1}{2}l(l-1)\sigma^2}{|l|\sigma}\right)\cdot \frac{1}{(1+x)|l|\sigma}

So far so good I'd say. Now, how I can derive the skewness from that point on? My only idea is to derive the skewness empirically, that is plugging in x values from x>-1 (x is not defined for x=-1 because of ln(1+x)) to infinity and then solving for

\frac{1}{n}\sum_{i=1}^{n}\left(\frac{x_i-\overline{x}}{s}\right)^3

with s being the standard deviation of x.

I hope there is a way to do it analytically. Thanks for hints and help.