Integrated correlators in $\mathcal{N}=4$ SYM beyond localisation

We study integrated correlators of four superconformal primaries $\mathcal{O}_{p}$ with arbitrary charges $p$ in $\mathcal{N}{=}4$ super Yang-Mills theory (SYM). The $\langle \mathcal{O}_{2} \mathcal{O}_{2} \mathcal{O}_{p} \mathcal{O}_{p} \rangle$ integrated correlators can be computed by supersymmetric localisation, whereas correlators with more general charges are currently not accessible from this method and in general contain complicated multi-zeta values. Nevertheless we observe that if one sums over the contributions from all different channels in a given correlator, then all the multi-zeta values (and products of zeta's) cancel leaving only $\zeta(2\ell{+}1)$ at $\ell$-loops. We then propose an exact expression of such integrated correlators in the planar limit, valid for arbitrary 't Hooft coupling. The expression matches with the known exact localisation-based results for specific charges, as well as with all existing perturbative and strong-coupling results in the literature for more general charges. As an application, our result is used to determine certain $7$-loop Feynman integral periods and fix previously unknown coefficients in the correlators at strong coupling.