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:

With the cumulative standard normal distribution . The corresponding probability density function (PDF) with and :

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

with s being the standard deviation of x.

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