Suppose that we have two independent random variables, $x$ and $n$. They are both normally distributed: $x \sim N(\mu_1,\sigma_1^2)$, $n \sim N(\mu_2,\sigma_2^2)$.
Let: $$y = x + ln(1 + e^{(n-x)})$$
If somehow we know the approximation of the mean and variance of $ln(1 + e^{(n-x)})$ is $\mu_3$ and $\sigma_3^2$, respectively. (This information is provided if you cannot solve this question analytically.)
The question is to calculate this conditional expectation: $$E(x|y=y_i)$$
The difficulty lies in that $ln(1 + e^{(n-x)})$ is not normally distributed, otherwise an approximate solution is easy to be got.
Thanks in advance