I'm interested in the integral:
\begin{equation*} \int_0^{\infty} x\left(\Phi(x) - 1/2\right)^{n} \varphi(x) \varphi(xt)dx \end{equation*} Where $n \ge 1$ is an integer and $t \in [-1, 1]$ is a parameter. $\Phi$ and $\varphi$ are the gaussian CDF and PDFs respectively.
I was thinking that to start with I might expand the binomial term to get the integrals:
\begin{equation*} \int_0^{\infty} x \Phi(x)^{k} \varphi(x) \varphi(xt)dx \end{equation*}
I looked in Owen's tables, but this exact integral doesn't seem to be present. If it were I think it would be at index n111. Does anyone know if there's a close form solution to either the full integral or even just the monomial terms?