I need to integrate $\int u\, \partial_x \left[(\partial_x u)^2 + (\partial_y u)^2\right]\, \mathrm{d}x$.
In other words we need $w$ such that $\partial_x w = u\, \partial_x \left[(\partial_x u)^2 + (\partial_y u)^2\right]$. The only thing I know is that $\partial_{x}^2 u + \partial_y^2 u = \Delta u = 0$.
I have tried searching for $w$ but I never make progress. Any hints? The original form of the integral is $\int u\, \partial_x \left(\partial_x^2 u^2 + \partial_y^2 u^2\right)\, \mathrm{d}x $, and I reduced it to the form given above (with the known antiderivatives left out), so solving either is fine and automatically solves the other.
Any help is greatly appreciated!