Is there a relationship between AGN and star formation in IR-bright
  AGNs?
  We report the relationship between the luminosities of active galactic nuclei
(AGNs) and the rates of star formation (SF) for a sample of 323 far-infrared
(FIR)-detected AGNs. This sample has a redshift range of 0.2 $< z <$ 2.5, and
spans three orders of magnitude in luminosity, ${\rm L_{X} \sim
10^{42-45}}$erg$s^{-1}$. We find that in AGN hosts, the total IR luminosity
(8-1000$\mu$m) has a significant AGN contribution (average$\sim$20%), and we
suggest using the FIR luminosity (30-1000 $\mu$m) as a more reliable star
formation rate (SFR) estimator. We also conclude that monochromatic
luminosities at 60 and 100\,$\mu$\,m are also good SFR indicators with
negligible AGN contributions, and are less sensitive than integrated infrared
luminosities to the shape of the AGN SED, which is uncertain at
$\lambda>$100\micron. Significant bivariate $L_{\rm X}$-$L_{\rm IR}$
correlations are found, which remain significant in the combined sample when
using residual partial correlation analysis to account for the inherent
redshift dependence. No redshift or mass dependence is found for the ratio
between SFR and black hole accretion rate (BHAR), which has a mean and scatter
of log (SFR/BHAR) $=3.1 \pm$ 0.5, agreeing with the local mass ratio between
supermassive black hole and host galaxies. The large scatter in this ratio and
the strong AGN-SF correlation found in these IR-bright AGNs are consistent with
the scenario of an AGN-SF dependence on a common gas supply, regardless of the
evolutionary model.
