Integral relation for periodic functions and their derivatives

Let $w:\mathbb{R}\to\mathbb{R}$ be a $2\pi$-periodic function of class $C^2(\mathbb{R})$. Integral relation. The function $w$ satisfies

$$\int_0^{2\pi}\frac{w(\theta)}{\sqrt{1+w^2(\theta)}}\,d\theta=\int_0^{2\pi}\left(w(\theta)+w”(\theta)\right)\cdot\frac{\sqrt{1+w^2(\theta)}}{1+w^2(\theta)+w'^2(\theta)}\,d\theta.$$

The relation is presented as a generalization of an identity obtained for rosette widths, but the surrounding discussion notes that it is not characteristic of widths: some functions that are not widths still satisfy it, while the stated regularity assumption is necessary.