We analyze the star formation history (SFH) of galaxies as a function of present-day environment, galaxy stellar mass and morphology. The SFH is derived by means of a non-parametric spectrophotometric model applied to individual galaxies at z ~ 0.04- 0.1 in the WINGS clusters and the PM2GC field. The field reconstructed evolution of the star formation rate density (SFRD) follows the values observed at each redshift (Madau & Dickinson 2014), except at z > 2 where our estimate is ~ 1.7x higher than the high-z observed value. The slope of the SFRD decline with time gets progressively steeper going from low mass to high mass haloes. The decrease of the SFRD since z = 2 is due to 1) quenching - 50% of the SFRD in the field and 75% in clusters at z > 2 originated in galaxies that are passive today - and 2) the fact that the average SFR of today's star-forming galaxies has decreased with time. We quantify the contribution to the SFRD(z) of galaxies of today's different masses and morphologies. The current morphology correlates with the current star formation activity but is irrelevant for the past stellar history. The average SFH depends on galaxy mass, but galaxies of a given mass have different histories depending on their environment. We conclude that the variation of the SFRD(z) with environment is not driven by different distributions of galaxy masses and morphologies in clusters and field, and must be due to an accelerated formation in high mass haloes compared to low mass ones even for galaxies that will end up having the same galaxy mass today. ; Comment: 16 pages, 10 figures. Published on MNRAS