I've been trying to prove this equation: $$ \delta G_{\alpha\beta}=-\frac{1}{2}\left(\square\bar{h}_{\alpha\beta}+2R{}_{\gamma\alpha\delta\beta}\bar{h}^{\gamma\delta}\right)+\frac{1}{2}\left(\bar{h}_{\alpha;\gamma\beta}^{\gamma}+\bar{h}_{\beta;\gamma\alpha}^{\gamma}-g_{\alpha\beta}\bar{h}^{\rho\delta}{}_{;\rho\delta}\right), $$
for the first order Einstein tensor, where $\bar{h}_{\alpha\beta}:=h_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}\left[g_{\gamma\delta}h^{\gamma\delta}\right]$. I've tried it a dozen times and always get this expression:
$$ \delta G_{\alpha\beta}=-\frac{1}{2}\left(\square\bar{h}_{\alpha\beta}-2R{}_{\gamma\alpha\delta\beta}\bar{h}^{\gamma\delta}\right)+\frac{1}{2}\left(\bar{h}_{\alpha;\gamma\beta}^{\gamma}+\bar{h}_{\beta;\gamma\alpha}^{\gamma}-g_{\alpha\beta}\bar{h}^{\rho\delta}{}_{;\rho\delta}\right). $$
Can anyone prove the right expression from:
$$ \delta G{}_{\alpha\beta}=\frac{1}{2}\left(-h_{\gamma;\beta\alpha}^{\gamma}+h_{\beta;\alpha\gamma}^{\gamma}+h_{\alpha;\beta\gamma}^{\gamma}-h_{\alpha\beta}{}^{;\gamma}{}_{;\gamma}\right)-\frac{1}{4}g{}_{\alpha\beta}\left[g{}^{\gamma\delta}\left(-h_{\rho;\delta\gamma}^{\rho}+h_{\delta;\gamma\rho}^{\rho}+h_{\gamma;\delta\rho}^{\rho}-h_{\delta\gamma}{}^{;\rho}{}_{;\rho}\right)\right]. $$