Expectation of maximum among N independent but not identical distributed guassian random variables

Question

Suppose here are n independent gaussian random variables, $X_1,X_2\dots X_n$, such that $X_i\sim \mathcal{N}(\mu,\sigma_i^2)$, let $X^*=max(X_1,X_2\dots X_n)$. If $\sigma_1=\sigma_2=\dots\sigma_n$, I can apply the extreme value theorem to get $E[X^*]$ for sufficient large n. My question is if $\sigma_1\neq\sigma_2\neq\sigma_3\dots\neq\sigma_n$(and $\mu_1=\mu_2\dots=\mu_n=\mu$), how to get the approximation of $E[X^*]$ for sufficient large n? Any help is appreciated!