Consider a Gaussian variable $x$ with mean $\mu$ and variance $\sigma^2$, can I calculate or approximate $\mathbb{E}\!\left[x\, \phi(x)\right]$, such that $\phi$ is the sigmoid function given by $\phi(x)=\frac{1}{1+\exp{(-x)}}$ ?
Note that the cumulative density function of the logistic distribution is simply the sigmoid function, taking it into account we can approximate $\mathbb{E}\!\left[\phi(x)\right]\approx \phi\left(\frac{\mu}{\sqrt{1+\frac{3}{\pi^2}\sigma^2}}\right)$.