  We investigate the Bjorken polarized sum rule (BSR) in three approaches to
QCD with analytic (holomorphic) coupling: Analytic Perturbation Theory (APT),
Two-delta analytic QCD (2$\delta$anQCD), and Three-delta lattice-motivated
analytic QCD in the three-loop and four-loop MOM scheme (3l3$\delta$anQCD,
4l3$\delta$anQCD). These couplings do not have unphysical (Landau)
singularities, and have finite values when the transferred momentum goes to
zero, which allows us to explore the infrared regime. With the exception of
APT, these theories at high momenta practically coincide with the underlying
perturbative QCD (pQCD) in the same scheme. We apply them in order to verify
the Bjorken sum rule within the range of energies available in the data
collected by the experimental JLAB collaboration, i.e., $0.05 GeV^2 <Q^2< 3
GeV^2$ and compare the results with those obtained by using the perturbative
QCD coupling. The results of the new frameworks with respective couplings
(2$\delta$ and 3$\delta$) are in good agreement with the experimental data for
$0.5 GeV^2<Q^2<3 GeV^2$ already when only one higher-twist term is used. In the
low-$Q^2$ regime ($Q^2 \lesssim 1 GeV^2$) we use $\chi$PT-motivated expression
or an expression motivated by the light-front holography (LFH) QCD used earlier
in the literature.
