  We apply the Picard and Magnus expansions to both the semiclassical and the
quantum Rabi model, with a switchable matter-field coupling. The case of the
quantum Rabi model ia a paradigmatic example of finite-time quantum
electrodynamics (QED), and in this case we build an intuitive diagrammatic
representation of the Picard series. In particular, we show that regular
oscillations in the mean number of photons, ascribed to the dynamical Casimir
effect (DCE) for the the generation of photons and to the anti-DCE for their
destruction, take place at twice the resonator frequency $\omega$. Such
oscillations, which are a clear dynamical "smoking gun" of the ultrastrong
coupling regime, can be predicted by first-order Picard expansion. We also show
that the Magnus expansion can be used, through concatenation, as an efficient
numerical integrator for both the semiclassical and the quantum Rabi model. In
the first case, we find distinctive features in the Fourier spectrum of motion,
with a single peak at the Rabi frequency $\Omega$ and doublets at frequencies
$2n\omega\pm\Omega$, with $n$ positive integer. We explain these doublets,
which are a feature beyond the rotating wave approximation (RWA), on the basis
of the Picard series.
