An off-shell D-brane action at order ${\alpha }^{\mathrm{\prime }2}$ in flat spacetime

We use compatibility of the second fundamental form corrections to DBI action at order ${\alpha }^{\mathrm{\prime }2}$ which includes trace of the second fundamental form, with T-duality and with the linear S-duality as guiding principles to find an off-shell D-brane action at order ${\alpha }^{\mathrm{\prime }2}$ in type II superstring theories in flat spcetime.