Let $f(x) = x \left(1 - \Phi(x)\right)^{n-1} e^{-x^2/2}$, where $\Phi(x) = \int\limits_{-\infty}^xe^{-t^2/2}\mathrm dt$. I am interested in evaluating
$$ \int\limits_{\mathbb{R}} f(x) \mathrm dx = \int\limits_{\mathbb{R}} x\left(1-\int\limits_{-\infty}^xe^{-t^2/2}\mathrm dt\right)^{n-1} e^{-x^2/2}\mathrm dx.$$
I tried some numerical methods, using Taylor expansion, but I just couldn't evaluate it. Any advice will be welcome.