  We perform for the first time a joint analysis of the monopole and
quadrupoles for power spectrum, bispectrum and integrated trispectrum
(i-trispectrum) from the redshift space matter field in N-body simulations.
With a full Markov Chain Monte Carlo exploration of the posterior distribution,
we quantify the constraints on cosmological parameters for an object density of
$n_\mathrm{p}=5\times10^{-4} (h\,\mathrm{Mpc}^{-1})^{3}$, redshift $z=0.5$, and
a covariance corresponding to a survey volume of $V_\mathrm{survey}
=25\,(h^{-1}\mathrm{Gpc})^3$, a set up which is representative of forthcoming
galaxy redshift surveys. We demonstrate the complementarity of the bispectrum
and i-trispectrum in constraining key cosmological parameters. In particular,
compared to the state-of-the-art power spectrum (monopole plus quadrupole) and
bispectrum (monopole) analyses, we find 1D $68\%$ credible regions smaller by a
factor of $(72\%,78\%,72\%,47\%,46\%)$ for the parameters
$(f,\sigma_8,f_\mathrm{nl},\alpha_\parallel,\alpha_\perp)$ respectively. This
work motivates the additional effort necessary to include the redshift-space
anisotropic signal of higher-order statistics in the analysis and
interpretation of ongoing and future galaxy surveys.
