We continue to study the frequency-dependent complex bulk viscosities of one-dimensional Bose and Fermi gases with contact interactions, which, according to recent work, exhibit weakly strong duality. Here we show the contribution of the Drude peak, which typically diverges at zero frequency, to the transport coefficients of one-dimensional quantum integrable systems. In particular, their Drude weights are evaluated based on his Kubo formula for the high temperature limit at any bond and the weak and strong limit at any temperature. A systematic expansion with respect to small parameters is available here. In all three limits, Drude’s peaks are found at higher orders compared to the finite normal part.