  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.
