Perhaps something like this? (with integration from $-\infty$ to $\infty$ to arrive at a nicely symmetric answer): $$\int_{-\infty}^\infty f \ H (f') dx=\frac{1}{\pi}\text{P.V.}\,\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{f(x)f'(y)}{x-y}\,dxdy$$ $$\qquad=\frac{1}{\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty f(x)f'(y)\frac{d}{dx}\log|x-y|\,dxdy$$ $$\qquad=-\frac{1}{\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty f'(x)f'(y)\log|x-y|\,dxdy.$$ Alternatively, with both integrals from $0$ to $\infty$, $$\frac{1}{\pi}\text{P.V.}\,\int_{0}^\infty \int_{0}^\infty \frac{f(x)f'(y)}{x-y}\,dxdy$$ $$\qquad=\frac{1}{\pi}\int_{0}^\infty \int_{0}^\infty f(x)f'(y)\frac{d}{dx}\log|x-y|\,dxdy$$ $$\qquad=-\frac{1}{\pi}f(0)\int_{0}^\infty f'(y)\log y\,dy-\frac{1}{\pi}\int_{0}^\infty \int_{0}^\infty f'(x)f'(y)\log|x-y|\,dxdy.$$
