As part of a research problem I am working on, I have stumbled across a problem that I wish I could solve in closed-form. Unfortunately, my attempts haven't been successful so far.
The problem is the following: given $X \sim SN(\mu,\sigma,\lambda)$, i.e. $X$ is distributed as a skewed-normal distribution whose pdf is given by $ \frac{2}{\sigma} \phi\Big(\frac{x-\mu}{\sigma}\Big)\Phi\Big(\lambda\Big(\frac{x-\mu}{\sigma}\Big)\Big)$, with $\phi$ and $\Phi$ the standard Normal pdf and cdf, evaluate the following expectation: \begin{equation} \mathbb{E}[\ln \Phi(X)]\;. \end{equation}
I would highly appreciate any insights on how to approach such a problem, or do you think that this can only be achieved numerically?