  The massive black holes in most faint active galactic nuclei (AGNs) and even
normal galaxies are still accreting gases, though their accretion rates are
very low. Radiatively inefficient accretion flows (RIAFs) are supposed in these
faint sources, which should radiate mostly in the hard X-ray band. We calculate
the contribution to the X-ray background from both the bright AGNs and the
RIAFs in faint AGNs/normal galaxies. Our calculations show that both the
observed intensity and spectral shape of the XRB with an energy peak at ~30$
keV can be well reproduced without including the emission of Compton-thick
AGNs, if the massive black holes in faint AGNs/normal galaxies are spinning
rapidly with a~0.9 and accreting at rates ~1.0-3.0\times 10^{-4}. It indicates
that less than ~5 per cent of local massive black hole mass density was
accreted during radiatively inefficient accretion phases, which is obviously
only an upper limit, because Compton-thick AGNs have not been considered. If
the same number of the Compton-thick AGNs with log N_H=24-25 as those with log
N_H=23-24 is considered, the fraction of local black hole mass density
accumulated during inefficient accretion phases should be lower than ~2 per
cent.
