  We describe the optimal horoball packings of asymptotic Koszul type Coxeter
simplex tilings of $5$-dimensional hyperbolic space where the symmetries of the
packings are generated by Coxeter groups. We find that the optimal horoball
packing density of $\delta_{opt}=0.59421\dots$ is realized in an entire
commensurability class of arithmetic Coxeter tilings. Eleven optimal
arrangements are achieved by placing horoballs at the asymptotic vertices of
the corresponding Coxeter simplices that give the fundamental domains. When
multiple horoball types are allowed, in the case of the arithmetic Coxeter
groups, the relative packing densities of the optimal horoball types are
rational submultiples of $\delta_{opt}$, corresponding to the Dirichlet-Voronoi
cell densities of the packing. The packings given in this paper are so far the
densest known in hyperbolic $5$-space.
