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Context: Blazars are the most luminous and variable active galactic nuclei (AGNs). They are thus excellent probes of accretion and emission processes close to the central engine. Aims: We concentrate here on PKS 1510-089 (z = 0.36), a blazar belonging to the flat-spectrum radio quasar subclass, an extremely powerful gamma-ray source and one of the brightest in the Fermi-LAT catalog. We aim to study the complex variability of this blazar's bright multiwavelength spectrum, to identify the physical parameters responsible for the variations and the timescales of possible recurrence and quasi-periodicity at high energies. Methods: The blazar 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 the Nordic Optical Telescope (NOT), in addition to the constant Fermi monitoring. We also measured the optical polarization in several bands on 18 Jan. 2010 at the NOT.Using these and archival data we constructed historical light curves at gamma-to-radio wavelengths covering nearly 20 yr and applied tests of fractional and correlated variability. We assembled spectral energy distributions (SEDs) based on these data and compared them with those at two previous epochs, by applying a model based on synchrotron and inverse Compton radiation from blazars. Results: The modeling of the SEDs 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 distance of ~1000 Schwarzschild radii(~0.03 pc) from the central engine - a distance muchsmaller than the broad line region (BLR) radius. The emission at a few hundred GeV can be reproduced with inverse Compton scattering of highly relativistic electrons off far-infrared photons if these are located much farther than the BLR, that is, around 0.2 pc from the AGN, presumably in a dusty torus. We determine a luminosity of the thermal component due to the inner accretion disk of Ld ≃ 5.9 × 1045 erg s-1, a BLR luminosity of LBLR ≃ 5.3 × 1044 erg s-1, and a mass of the central black hole of MBH ≃ 3 × 108 M⊙.The fractional variability as a function of wavelength follows the trend expected if X- and gamma-rays are produced by the same electrons as radio and optical photons, respectively. Discrete correlation function (DCF) analysis between the long-term Steward observatory optical V-band and gamma-ray Fermi-LAT light curves yields a good correlation with no measurable delay. Marginal correlation where X-ray photons lag both optical and gamma-ray ones by time lags between 50 and 300 days is found with the DCF.Our time analysis of the RXTE PCA and Fermi-LAT light curves reveals no obvious (quasi-)periodicities, at least up to the maximum timescale (a few years) probed by the light curves, which are severely affected by red noise. [ABSTRACT FROM AUTHOR]