Integration by parts on a manifold

Question

$M$ is a closed Riemannian manifold, I want to break the following integration on $M$ using integration by parts techniques which I am not so familiar with. \begin{align*} \int_M\langle d\beta,\omega\rangle|\phi|^2 dV \end{align*} Here $\phi$ is a section of a complex line bundle with a compatible metric on it, $\beta$ is a $3$-form and $\omega$ is a Harmonic four form on the manifold.