  Using the assumption that the independent gauge invariant couplings on the
world-volume of the non-perturbative objects in the string theory are
independent of the background, we find the four and the six gauge field
strength and/or the second fundamental form couplings on the world volume of a
D$_p$-brane in the superstring theory at order $\alpha'^2$ in the normalization
that $F$ is dimensionless. We have found them by considering the particular
background which has one circle and by imposing the corresponding T-duality
constraint on the independent couplings. In particular, we find that there are
12+146 independent gauge invariant couplings at this order, and the T-duality
constraint can fix 150 of them. We show that these couplings are fully
consistent with the partial results in the literature. This comparison also
fixes the remaining 8 couplings.
