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We use the general Riemann approach to define the Stratonovich integral with respect to Brownian motion. Our new definition of Stratonovich integral encompass the classical Stratonovich integral and more importantly, satisfies the ideal Itô formula without the “tail” term, that is, $$f\left(W_t\right)=f\left(W_0\right)+{\int }_0^t{f}^{\text{'}}\left(W_s\right)\circ \mathrm{d}{W}_s.$$ Further, the condition on the integrands in this paper is weaker than the classical one.