Combining the catalogue of galaxy morphologies in the COSMOS field and the sample of H$alpha$ emitters at redshifts $z=0.4$ and $z=0.84$ of the HiZELS survey, we selected $sim$ 220 star-forming bulgeless systems (Sersic index $n leq 1.5$) at both epochs. We present their star formation properties and we investigate their contribution to the star formation rate function (SFRF) and global star formation rate density (SFRD) at $z < 1$. For comparison, we also analyse H$alpha$ emitters with more structurally evolved morphologies that we split into two classes according to their Sersic index $n$: intermediate ($ 1.5 < n leq 3 $) and bulge-dominated ($n > 3$). At both redshifts the SFRF is dominated by the contribution of bulgeless galaxies and we show that they account for more than 60% of the cosmic SFRD at $z < 1$. The decrease of the SFRD with redshift is common to the three morphological types but it is stronger for bulge-dominated systems. Star-forming bulgeless systems are mostly located in regions of low to intermediate galaxy densities ($Sigma sim 1 - 4$ Mpc$^{-2}$) typical of field-like and filament-like environments and their specific star formation rates (sSFRs) do not appear to vary strongly with local galaxy density. Only few bulgeless galaxies in our sample have high (sSFR $>$ 10$^{-9}$ yr$^{-1}$) and these are mainly low-mass systems. Above $M_* sim 10^{10}$ M$_{odot}$ bulgeless are evolving at a normal rate (10$^{-9}$ yr$^{-1} <$ sSFR $<$10$^{-10}$ yr$^{-1}$) and in the absence of an external trigger (i.e. mergers/strong interactions) they might not be able to develop a central classical bulge.
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