The Harish-Chandra-Itzykson-Zuber integral, also called spherical integral is defined as the expectation of exp(Tr(AUBU*)) for A and B two self adjoint matrices and U Haar-distributed on the unitary/orthogonal/symplectic group. It was initially introduced by Harish-Chandra to study Lie groups and it also has an interpretation in terms of Schur functions. Since then, it has had many kinds of applications, from physics to statistical learning. In this talk we will look at the asymptotics of these integrals when one of the matrices remains of small rank. We will also see how to use these asymptotics to prove large deviation principles for the largest eigenvalues for random matrix models that satisfy a sub-Gaussian bound. This talk is mainly based on a collaboration with Justin Ko.