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We present exact expressions for certain integrated correlators of foursuperconformal primary operators in the stress tensor multiplet of$\mathcal{N}=4$ supersymmetric Yang--Mills (SYM) theory with classical gaugegroup, $G_N$ $= SO(2N)$, $SO(2N+1)$, $USp(2N)$. These integrated correlatorsare expressed as two-dimensional lattice sums by considering derivatives of thelocalised partition functions, generalising the expression obtained for $SU(N)$in our previous works. These expressions are manifestly covariant underGoddard-Nuyts-Olive duality. The integrated correlators can also be formallywritten as infinite sums of non-holomorphic Eisenstein series with integerindices and rational coefficients. Furthermore, the action of the hyperbolicLaplace operator with respect to the complex coupling $\tau=\theta/(2\pi) +4\pi i /g^2_{_{YM}}$ on any integrated correlator for gauge group $G_N$ relatesit to a linear combination of correlators with gauge groups $G_{N+1}$, $G_N$and $G_{N-1}$. These "Laplace-difference equation" determine the expressions ofintegrated correlators for all classical gauge groups for any value of $N$ interms of the correlator for the gauge group $SU(2)$. The perturbationexpansions of these integrated correlators for any finite value of $N$ agreewith properties obtained from perturbative Yang--Mills quantum field theory,together with various multi-instanton calculations which are also shown toagree with those determined by supersymmetric localisation. The coefficients ofterms in the large-$N$ expansion are sums of non-holomorphic Eisenstein serieswith half-integer indices, which extend recent results and make contact withlow order terms in the low energy expansion of type IIB superstring theory inan $AdS_5\times S^5/\mathbb{Z}_2$ background.

Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups