Analytic expressions for the background evolution of massive neutrinos
  and dark matter particles
  We provide exact analytic expressions for the density, pressure, average
number density and pseudo-pressure for massive neutrinos and generic dark
matter particles, both fermions and bosons. We then focus on massive neutrinos
and we compare our analytic expressions with the numerical implementation in
the CLASS Boltzmann code. We find that our modifications including the exact
analytic expressions are in agreement to better than $10^{-4}\%$ with the
default CLASS implementation in the estimation of the CMB power spectrum; our
modifications do not have an impact on the performance of the code. We also
provide several specific limits of our expressions at the relativistic regime,
but also at late times for the neutrino equation of state.
