  The power spectrum has long been the workhorse summary statistics for
large-scale structure cosmological analyses. However, gravitational non-linear
evolution moves precious cosmological information from the two-point statistics
(such as the power spectrum) to higher-order correlations. Moreover,
information about the primordial non-Gaussian signal lies also in higher-order
correlations. Without tapping into these, that information remains hidden.
While the three-point function (or the bispectrum), even if not extensively,
has been studied and applied to data, there has been only limited discussion
about the four point/trispectrum. This is because the high-dimensionality of
the statistics (in real space a skew-quadrilateral has 6 degrees of freedom),
and the high number of skew-quadrilaterals, make the trispectrum numerically
and algorithmically very challenging. Here we address this challenge by
introducing the i-trispectrum, an integrated trispectrum that only depends on
four $k$-modes moduli. We model and measure the matter i-trispectrum from a set
of 5000 \textsc{Quijote} N-body simulations both in real and redshift space,
finding good agreement between simulations outputs and model up to mildly
non-linear scales. Using the power spectrum, bispectrum and i-trispectrum joint
data-vector covariance matrix estimated from the simulations, we begin to
quantify the added-value provided by the i-trispectrum. In particular, we
forecast the i-trispectrum improvements on constraints on the local primordial
non-Gaussianity amplitude parameters $f_\mathrm{nl}$ and $g_\mathrm{nl}$. For
example, using the full joint data-vector, we forecast $f_\mathrm{nl}$
constraints up to two times ($\sim32\%$) smaller in real (redshift) space than
those obtained without i-trispectrum.
