Blazars are the most luminous and variable AGNs, and thus excellent probes of accretion and emission processes close to the central engine. We focus on PKS 1510-089 ($z=0.36$), one of the brightest gamma-ray sources in the Fermi LAT catalog, to study its complex multi-wavelength variability. PKS 1510-089 was observed twice in hard X-rays with the IBIS instrument onboard INTEGRAL during the flares of Jan 2009 and Jan 2010, and simultaneously with Swift and NOT, in addition to the constant Fermi monitoring. The optical polarization was measured in several bands on 18 Jan 2010 at the NOT. Using our and archival data we constructed historical light curves at gamma-to-radio wavelengths covering nearly 20 years and applied variability tests. We assembled SEDs in 2009 and 2010 and compared them with those at two previous epochs and with a model based on synchrotron and inverse Compton (IC) radiation. The SED modeling suggests that the physical quantities that undergo the largest variations are the total power injected into the emitting region and the random Lorentz factor of the electron distribution cooling break, that are higher in the higher gamma-ray states. This suggests a correlation of the injected power with enhanced activity of the acceleration mechanism. The cooling likely takes place at a much smaller distance ($sim$0.03 pc) than the BLR radius. The emission at a few hundred GeV can be reproduced with IC scattering of highly relativistic electrons off FIR photons at $sim$0.2 pc, presumably in a dusty torus. DCF analysis between the long-term optical and gamma-ray light curves yields a good correlation with no measurable delay. Our time analysis of the RXTE PCA and Fermi LAT light curves reveals no obvious (quasi-)periodicities, up to the maximum time scale (a few years) probed by the light curves, which are severely affected by red noise.
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