It is known since the work of Dyachenko and Zakharov (Phys Lett A 190:144–148 (1994), [25]) that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non-trivial resonant manifold. In this paper we study this partial integrability from a different point of view. We construct, directly in the physical space, a sequence of energy functionals \(\mathfrak E_j(t)\) which are explicit in the Riemann mapping variable and involve material derivatives of order j of the solutions for the 2d water wave equation, so that \(\frac{d}{dt} \mathfrak E_j(t)\) is quintic or higher order. As a consequence we present a longtime existence result which states that if some scaling invariant norm, and a norm involving one spatial derivative above the scaling of the initial data are of size no more than \(\varepsilon \) , then the lifespan of the solution for the 2d water wave equation is at least of order \(O(\varepsilon ^{-3})\) , and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size \(\varepsilon \) , then the lifespan of the solution is at least of order \(O(\varepsilon ^{-5/2})\) .