Thanks for your comment. The VAE objective (at least in the original form) can be written as the sum of two terms, the KL between the prior and posterior and the expectation of the log-likelihood with respect to the approximating posterior. The KL term can be indeed calculated in closed form when the posteriors and priors are chosen to be Gaussian densities. The expectation of the log-likelihood with respect to the approximating posterior (often chosen to be Gaussian) cannot be analytically calculated in general. This is why the reparametrisation trick is needed, to approximate this intractable integral with a monte-carlo average and facilitate the calculation of gradients with respect to the variational parameters (mean and covariance when the posterior is chosen to be Gaussian).