I'm trying to derive equation of motion in Higgs scalar-tensor theory with the Lagrangian given by
$$\mathcal{L}=[\frac{1}{16\pi}\alpha \phi^{\dagger}\phi R+ \frac{1}{2}\phi^{\dagger}_{;\mu}\phi^{;\mu} - V(\phi)]\sqrt{-g} + \mathcal{L_M}\sqrt{-g}$$
and equation of motion for gravity is given by
$$ R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\frac{8\pi}{\alpha\phi^\dagger\phi}V(\phi)g_{\mu\nu}=
\frac{8\pi}{\alpha\phi^\dagger\phi}T_{\mu\nu}-\frac{4\pi}{\alpha\phi^\dagger\phi}[\phi^\dagger_{;\mu}\phi_{;\nu} + \phi^\dagger_{;\nu}\phi_{;\mu}] + \frac{4\pi}{\alpha\phi^\dagger\phi}\phi^\dagger_{;\lambda}\phi^{;\lambda}g_{\mu\nu}-\frac{1}{\phi^\dagger\phi}[(\phi^\dagger\phi)_{;\mu;\nu}-(\phi^\dagger\phi)^{;\beta}_{\;\; ;\beta}g_{\mu\nu}]$$
I'm studying this paper in which the equation of motion is derived. I've got the LHS and the first term on the RHS but how do i calculate the last three terms on the RHS.
Also I'm following this paper which was quite helpful. It suggest using Euler Lagrange equation
$$ \frac{\partial{\mathcal{L}}}{\partial{g^{\mu\nu}}} - \partial_\kappa \frac{\partial{\mathcal{L}}}{\partial g^{\mu\nu}_{\;,\kappa}} + \partial_\kappa\partial_\lambda \frac{\partial{\mathcal{L}}}{\partial g^{\mu\nu}_{\; ,\kappa \lambda}} = 0$$
but still I could not get the last three terms on the RHS.
