Summary: 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 studying 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 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-vecispectrumtor, we forecast \(f_{\mathrm{nl}}\) constraints up to two times (\(\sim 32\)%) smaller in real (redshift) space than those obtained without i-tr.