  The [CII] 158 micron line is one of the strongest IR emission lines, which
has been shown to trace the star-formation rate (SFR) of galaxies in the nearby
Universe and up to $z \sim 2$. Whether this is also the case at higher redshift
and in the early Universe remains debated. The ALPINE survey, which targeted
118 star-forming galaxies at $4.4 < z< 5.9$, provides a new opportunity to
examine this question with the first statistical dataset. Using the ALPINE data
and earlier measurements from the literature we examine the relation between
the [CII] luminosity and the SFR over the entire redshift range from $z \sim
4-8$. ALPINE galaxies, which are both detected in [CII] and dust continuum,
show a good agreement with the local L([CII])-SFR relation. Galaxies undetected
in the continuum with ALMA are found to be over-luminous in [CII], when the UV
SFR is used. After accounting for dust-obscured star formation, by an amount
SFR(IR)$\approx$SFR(UV) on average, which results from two different stacking
methods and SED fitting, the ALPINE galaxies show an L([CII])-SFR relation
comparable to the local one. When [CII] non-detections are taken into account,
the slope may be marginally steeper at high-z, although this is still somewhat
uncertain. When compared in a homogeneous manner, the $z>6 $ [CII] measurements
(detections and upper limits) do not behave very differently from the $z \sim
4-6$ data. We find a weak dependence of L([CII])/SFR on the Lyman-alpha
equivalent width. Finally, we find that the ratio L([CII])/LIR $\sim (1-3)
\times 10^{-3}$ for the ALPINE sources, comparable to that of "normal" galaxies
at lower redshift. Our analysis, which includes the largest sample ($\sim 150$
galaxies) of [CII] measurements at $z>4$ available so far, suggests no or
little evolution of the L([CII])-SFR relation over the last 13 Gyr of cosmic
time.
