Integration by part on a surface

Question

Let $f$ be a smooth positive function from $\mathbb{R}^3$ to $\mathbb{R}$. Can we "simplify" the integral $$ \int_{\mathbb{S}^2} \dfrac{\Delta f(x)}{f(x)} d \sigma(x), $$ where $\Delta f$ is the three-dimensional Laplacian of $f$. For instance, what nice conditions on $f$ ensure that this integral have a sign?