  In this work, we extend our previous study of the bulk viscosity of hot and
dense $npe$ matter induced by the Urca processes in the neutrino trapped regime
to $npe\mu$ matter by adding the muonic Urca processes as well as the purely
leptonic electroweak processes involving electron-muon transition. The nuclear
matter is modeled in a relativistic density functional approach with two
different parametrizations which predict neutrino dominated matter (DDME2
model) and antineutrino dominated matter (NL3 model) at temperatures for which
neutrinos/antineutrinos are trapped. In the case of neutrino-dominated matter,
the main equilibration mechanism is lepton capture, whereas in the case of
antineutrino-dominated matter this is due to neutron decay. We find that the
equilibration rates of Urca processes are higher than that of the pure leptonic
processes, which implies that the Urca-process-driven bulk viscosity can be
computed with the leptonic reactions assumed to be frozen. We find that the
bulk viscosity decreases with temperature as $\zeta\sim T^{-2}$ at moderate
temperatures. At high temperatures this scaling breaks down by sharp drops of
the bulk viscosity close to the temperature where the proton fraction is
density-independent and the matter becomes scale-invariant. This occurs also
when the matter undergoes a transition from the antineutrino-dominated regime
to the neutrino-dominated regime where the bulk viscosity attains a local
maximum. We also estimate the bulk viscous dissipation timescales and find that
these are in the range $\gtrsim$ 1 s for temperatures above the neutrino
trapping temperature. These timescales would be relevant only for long-lived
objects formed in binary neutron star mergers and hot proto-neutron stars
formed in core-collapse supernovas.
