  The X-ray spectra of X-ray binaries are dominated by emission of either soft
or hard X-rays which defines their soft and hard spectral states. Cygnus X-3 is
amongst the list of X-ray binaries that show quite complex behavior, with
various distinct spectral states. Because of its softness and intrinsic low
flux above typically 50 keV, very little is known about the hard X/soft
gamma-ray (100-1000 keV) emission in Cygnus X-3. Using the whole INTEGRAL data
base, we aim to explore the 3-1000 keV spectra of Cygnus X-3. This allows to
probe this region with the highest sensitivity ever, and search for the
potential signature of a high-energy non-thermal component as sometimes seen in
other sources. Our work is based on state classification carried out in
previous studies with data from the Rossi X-Ray Timing Explorer. We extend this
classification to the whole INTEGRAL data set and perform a long-term
state-resolved spectral analysis. Six stacked spectra were obtained using 16
years of data from JEM-X, ISGRI, and SPI. We extract stacked images in three
different energy bands, and detect the source up to 200 keV. In the hardest
states, our phenomenological approach reveals the presence of an component > 50
keV in addition to the component usually interpreted as thermal Comptonization.
We apply a more physical model of hybrid thermal/nonthermal corona to
characterize this component and compare our results with those of previous
studies. Our modeling indicates a more efficient acceleration of electrons in
states where major ejections are observed. We find a dependence of the photon
index of the power law as a function of the strong orbital modulation of the
source in the Flaring InterMediate (FIM) state. This dependence could be due to
a higher absorption when Cygnus X-3 is behind its companion. However, the
uncertainties on the density column prevent us from drawing conclusions.
