The present work is concerned with Gaussian integrals on simply-connected non-positively-curved Riemannian symmetric spaces. It is motivated by the aim of explicitly finding the high-rank limit of these integrals for each of the eleven families of classical Riemannian symmetric spaces. To begin, it deals with the easier complex case (where the isometry group admits a complex Lie group structure). To go beyond this case, it introduces a variational characterisation of the high-rank limit, as the minimum of a certain energy functional over the space of probability distributions on the real line. Using this new variational formulation, it is possible to recover the high-rank limit in closed form, from the expression originally found in the complex case. This two-step approach is illustrated through the examples of two kinds of symmetric spaces : symmetric cones and classical symmetric domains.