I think -- using the chain rule -- it's
$$ e^{\int_{t}^{t}\cdots d\tau} c(t)dt \cdot e^{\int_{s}^{t} \sigma(\tau)dW(\tau)+(r(\tau)-\frac{1}{2}\sigma(\tau)^{2})d\tau}\cdot \left(\sigma(t)dW(t)+(r(t)-\frac{1}{2}\sigma(t)^{2})dt\right) \\ =c(t)dt \cdot e^{\int_{s}^{t} \sigma(\tau)dW(\tau)+(r(\tau)-\frac{1}{2}\sigma(\tau)^{2})d\tau}\cdot \left(\sigma(t)dW(t)+(r(t)-\frac{1}{2}\sigma(t)^{2})dt\right) $$
Is this right?