The classical equality problem is discussed in the class of means \(M_{\varphi, \mu}:I^{2}\to \mathbb{R}\) defined by \[ M_{\varphi, \mu}(x,y)=\varphi^{-1}\left(\int_{0}^{1}\varphi(tx+(1-t)y)d\mu(t)\right) \qquad (x,y \in I) \] where \(I\) is a nonempty open real interval, \(\varphi:I\to \mathbb{R}\) is a given continuous and strictly monotone function and \(\mu\) is a probability measure on the Borel subsets of \([0,1]\). This class includes the quasi-arithmetic as well as the Lagrangian means. Under at most fourth-order differentiability assumptions for the unknown functions \(\varphi, \psi:I\to \mathbb{R}\) the complete description of the solutions of the functional equation \(M_{\varphi, \mu}(x,y)=M_{\psi, \nu}(x,y)\) is obtained.