$O(9,9)$ symmetry of NS-NS couplings at order ${\alpha }^{\mathrm{\prime }3}$

Recently, imposing the $O(1,1)$ symmetry on the circle reduction of the classical effective action of string theory, we have found all NS-NS couplings of type II superstring theories at order ${\alpha }^{\mathrm{\prime }3}$. In this paper we use the cosmological reduction on the couplings and show that, up to field redefinitions and total derivative terms, they are invariant under the $O(9,9)$ transformations.