  We present the results of the INTEGRAL monitoring campaign on the accreting
low mass X-ray binary pulsar GX 1+4 performed during the Galactic plane scan of
the INTEGRAL Core Programme. The source was observed in different luminosity
states ranging from L(20-40 keV)= 1.7x10^{-10} erg/cm^2/s, to L(20-40
keV)=10.5x10^{-10} erg/cm^2/s for about 779 ks from March 2003 until October
2004. Our observations confirm the secular spin down of GX 1+4 with the
spin-period varying from 139.63 s to 141.56 s. In the highest luminosity state
a spin-up phase is observed. The phase-averaged spectrum of the source was
modelled either with an absorbed cut-off power law or with a Comptonization
model with significantly different parameters in the two brightest luminosity
states. No evidence for any absorption-like feature is observed in the
phase-averaged spectrum up to 110 keV. At highest luminosity, the source is
found to pulsate up to 130 keV. Phase-resolved spectroscopy reveals a
phase-dependent continuum and marginal evidence of an absorption feature at 34
keV in the descending part of the pulse. If interpreted as due to electron
resonant cyclotron scattering, the magnetic field in the emitting region would
be 2.9x10^{12}(1+z) G where z is the gravitational red shift of the emitting
region. We also observed a very low luminosity state typical of this source,
which lasted for about two days during which the source spectrum was modelled
by a simple power-law and a pulsed signal was still detectable in the 15-100
keV energy range.
