= Solution
By the <Itô formula>,
$$
f(X_t)-f(X_0)=\int_0^tf'(X_s)\,dX_s+\frac12\int_0^tf''(X_s)\,d[X]_s.
$$
To compute the <Stratonovich integral> correction, apply the <Itô formula> also to $f'$, which is twice continuously differentiable because $f$ is $C^3$. The continuous <local martingale> part of $f'(X)$ is $\int f''(X_s)dL_s$, where $L$ is the continuous <local martingale> part of $X$. A <finite-variation process> has zero <quadratic covariation> with $X$, and the <quadratic variation of a stochastic integral> together with <polarization identity> gives
$$
[f'(X),X]_t=\int_0^tf''(X_s)\,d[X]_s.
$$
All these statements can be localized to compact ranges of $X$, so unbounded derivatives create no global integrability requirement. Substitute this identity into the <Stratonovich integral> to obtain the <Stratonovich chain rule>:
$$
\boxed{f(X_t)-f(X_0)=\int_0^tf'(X_s)\,\partial X_s.}
$$
This is the integrated meaning of the requested differential identity; there is no extra second-order term after the <Stratonovich integral> correction has been included.
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