I need to compute the expected entropy of a Bernoulli random variable whose probability of success is given by $\frac{1}{1+\exp(-x)}$ where $x$ is normally distributed. This leaves me stuck with solving the following integral
$$ \int_{-\infty}^{\infty} \frac{1}{1+\exp(-x)} \ln\left(\frac{1}{1+\exp(-x)}\right) \frac{1}{2\pi}\exp(-0.5 x^2) \, {\rm d}x .$$
Is there any hope for a closed form solution?