We present near-infrared ( NIR ) spectroscopic observations of 28 X-ray and mid-infrared selected sources at a median redshift of z \sim 0.8 in the Extended Groth Strip ( EGS ) . To date this is the largest compilation of NIR spectra of active galactic nuclei ( AGN ) at this redshift . The data were obtained using the multi-object spectroscopic mode of the Long-slit Intermediate Resolution Infrared Spectrograph ( LIRIS ) at the 4.2 m William Herschel Telescope ( WHT ) . These galaxies are representative of a larger sample studied in a previous work , consisting of over a hundred X-ray selected sources with mid-infrared counterparts , which were classified either as AGN-dominated or host galaxy-dominated , depending on the shape of their spectral energy distributions ( SEDs ) . Here we present new NIR spectra of 13 and 15 sources of each class respectively . We detect the H \alpha line at \geq 1.5 \sigma above the continuum for the majority of the galaxies . Using attenuation-corrected H \alpha luminosities and observed Spitzer/MIPS 24 µm fluxes , and after subtracting an AGN component estimated using an AGN empirical correlation and multifrequency SED fits , we obtain average star formation rates ( SFRs ) of 7 \pm 7 and 20 \pm 50 M _ { \sun } ~ { } yr ^ { -1 } respectively ( median SFRs = 7 and 5 M _ { \sun } ~ { } yr ^ { -1 } ) . These values are lower than the SFRs reported in the literature for different samples of non-active star-forming galaxies of similar stellar masses and redshifts ( M _ { * } \sim 10 ^ { 11 } M _ { \sun } and z \sim 1 ) . In spite of the small size of the sample studied here , as well as the uncertainty affecting the AGN-corrected SFRs , we speculate with the possibility of AGN quenching the star formation in galaxies at z \sim 0.8 . Alternatively , we might be seeing a delay between the offset of the star formation and AGN activity , as observed in the local universe .