The INTEGRAL view of the pulsating hard X-ray sky: from accreting and transitional millisecond pulsars to rotation-powered pulsars and magnetars

In the last 25 years a new generation of X-ray satellites imparted a significant leap forward in our knowledge of X-ray pulsars. The discovery of accreting and transitional millisecond pulsars proved that disk accretion can spin up a neutron star to a very high rotation speed. The detection of MeV-GeV pulsed emission from a few hundreds of rotation-powered pulsars probed particle acceleration in the outer magnetosphere, or even beyond. Also, a population of two dozens of magnetars has emerged. INTEGRAL played a central role to achieve these results by providing instruments with high temporal resolution up to the hard X-ray/soft, γ-ray band and a large field of view imager with good angular resolution to spot hard X-ray transients. In this article we review the main contributions by INTEGRAL to our understanding of the pulsating hard X-ray sky, such as the discovery and characterization of several accreting and transitional millisecond pulsars, the generation of the first catalog of hard X-ray/soft γ-ray rotation-powered pulsars, the detection of polarization in the hard X-ray emission from the Crab pulsar, and the discovery of persistent hard X-ray emission from several magnetars.