I have the mixture density of two normal distributions:
\begin{align} f(l)=\pi \phi(l;\mu_1,\sigma^2_1)+(1-\pi)\phi(l;\mu_2,\sigma^2_2) \end{align}
The skewness is given by
\begin{align} \gamma_1 = \operatorname{E}\Big[\big(\tfrac{X-\mu}{\sigma}\big)^{\!3}\, \Big] = \frac{\mu_3}{\sigma^3} = \frac{\operatorname{E}\big[(X-\mu)^3\big]}{\ \ \ ( \operatorname{E}\big[ (X-\mu)^2 \big] )^{3/2}} \end{align}
Now my question is, what is the Skewness of a mixture gaussian? How can I derive it? I searched for it, but I could not find anything. A mathematical derivation would be helpful, also I have estimated both densities and I have the estimates for $\mu$ and $\sigma$. I want a formula in what I can insert those values to get the skewness of the mixed density.