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