  The collapsar model of long gamma-ray bursts (GRBs) indicates that they may
trace the star formation history. So long GRBs may be a useful tool of
measuring the high-redshift star formation rate (SFR). The collapsar model
explains GRB formation via the collapse of a rapidly rotating massive star with
$M> 30M_\odot$ into a black hole, which may imply a decrease of SFR at high
redshift. However, we find that the \emph{Swift} GRBs during 2005-2012 are
biased tracing the SFR, including a factor about $(1+z)^{0.5}$, which is in
agreement with recent results. After taking this factor, the SFR derived from
GRBs does not show steep drop up to $z\sim 9.4$. We consider the GRBs produced
by rapidly rotating metal-poor stars with low masses to explain the
high-redshift GRB rate excess. The chemically homogeneous evolution scenario
(CHES) of rapidly rotating stars with mass larger than $12M_\odot$ is
recognized as a promising path towards collapsars in connection with long GRBs.
Our results indicate that the stars in the mass range $12M_\odot<M<30M_\odot$
for low enough metallicity $Z\leq 0.004$ with the GRB efficiency factor
$10^{-5}$ can fit the derived SFR with good accuracy. Combining these two
factors, we find that the conversion efficiency from massive stars to GRBs is
enhanced by a factor of 10, which may be able to explain the excess of the
high-redshift GRB rate. We also investigate the cosmic reionization history
using the derived SFR. The GRB-inferred SFR would be sufficient to maintain
cosmic reionization over $6<z<10$ and reproduce the observed optical depth of
Thomson scattering to the cosmic microwave background.
