  We calculate analytically the two-loop triangle integrals entering the
$\mathcal{O}(\alpha\alpha_s)$ corrections to the $HZV$ vertex with
$V=Z^*,\gamma^*$ using the method of differential equations. Our result
provides a prototype to study the analytic properties of multi-loop multi-scale
Feynman integrals, and also allows fast numeric evaluation for phenomenological
studies. We apply our results to the leptonic decay of the Higgs boson and to
$ZH$ production at electron-positron colliders. Besides the top quark loop, we
include also the bottom quark loop contributions, whose evaluation takes a lot
of time using purely numeric methods, but is very efficient with our analytic
results.
