  The scattering of temporally shaped intense laser pulses off electrons is
discussed by means of manifestly covariant quantum electrodynamics. We employ a
framework based on Volkov states with a time dependent laser envelope in
light-cone coordinates within the Furry picture. An expression for the cross
section is constructed, which is independent of the considered pulse shape and
pulse length. A broad distribution of scatted photons with a rich pattern of
subpeaks like that obtained in Thomson scattering is found. These broad peaks
may overlap at sufficiently high laser intensity, rendering inappropriate the
notion of individual harmonics. The limit of monochromatic plane waves as well
as the classical limit of Thomson scattering are discussed. As a main result, a
scaling law is presented connecting the Thomson limit with the general result
for arbitrary kinematics. In the overlapping regions of the spectral density,
the classical and quantum calculations give different results, even in the
Thomson limit. Thus, a phase space region is identified where the differential
photon distribution is strongly modified by quantum effects.
