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Using an $e^+e^-$ annihilation data sample corresponding to an integratedluminosity of $2.93\,\rm fb^{-1}$ collected at a center-of-mass energy of 3.773GeV with the BESIII detector, the doubly Cabibbo-suppressed decay $D^+\toK^+\pi^+\pi^-\pi^0$ is studied with a semileptonic tag method. After removingthe decays containing narrow intermediate resonances, $D^+\to K^+\eta$, $D^+\toK^+\omega$, and $D^+\to K^+\phi$, the branching fraction for the decay $D^+\toK^+\pi^+\pi^-\pi^0$ is determined to be $(1.03 \pm 0.12_{\rm stat} \pm0.06_{\rm syst})\times 10^{-3}$. The ratio of the branching fraction for$D^+\to K^+\pi^+\pi^-\pi^0$ to its Cabibbo-favored counterpart $D^+\toK^-\pi^+\pi^+\pi^0$ is measured to be $ (1.65\pm0.21)\%$, corresponding to$(5.73\pm0.73)\tan^4\theta_C$, where $\theta_C$ is the Cabibbo mixing angle.These results are consistent with our previous measurement with hadronic tagsbut are significantly larger than other doubly Cabibbo-suppressed decays in thecharm sector.

Measurement of the doubly Cabibbo-suppressed decay $D^+\to K^+\pi^+\pi^-\pi^0$ with semileptonic tags