I found this paper here. In equations (5) - (6), the paper says that suppose $(X_1,X_2)$ are jointly normal with means and variances $(\mu_1,\mu_2),(\sigma_1^2,\sigma_2^2)$, define $X = \max\{X_1,X_2\}$. Then the moment generating function for $X$ is defined in (5)-(6).
I'm trying derive that result, but I'm totally stuck. Can I get a hint? I know how to derive the mgf for normal random variables, but not the maximum of two non-iid normals.