Does the following integral admit a closed form answer:
$$\int_{-\infty}^\infty\mathrm d x \exp\left(-\frac{(x-\mu)^2}{2\nu}\right) \ln(1+e^x)$$
where $\nu>0$ and $\mu$ are finite real parameters. This is just the average value of $\ln(1+e^x)$, when $x$ is normally distributed with mean $\mu$ and variance $\nu$ (forgetting the normalization constant).
I tried with symbolic processing software (Mathematica and WolframAlpha), and both return the input expression unevaluated, suggesting that a closed form solution does not exist, or at least it is not obvious.