<p>Hm here is my separate try, note that there should be 4 slashes, not 2:
\begin{align*}
    -N-\frac{1}{N}\sum_{j=1}^N (2j-1) \left(\ln\left(\frac{1}{2}+\int_{\bar{x}}^{x_j}
\exp\left(-\frac{(t-\bar{x})^2}{2s^2}\right)\mathrm{dt}\right)\\
+\ln\left(\frac{1}{2}-\int_{\bar{x}}^{x_{N-j+1}}
\exp\left(-\frac{(t-\bar{x})^2}{2s^2}\right)\mathrm{dt}\right)\right)
\end{align*}</p>
