  We develop N=4 d=4 bi-harmonic superspace and use it to derive a novel form
for the low-energy effective action in N=4 super Yang-Mills theory. We solve
the N=4 supergauge constraints in this superspace in terms of analytic
superfields. Using these superfields, we construct a simple functional that
respects N=4 supersymmetry and scale invariance. In components, it reproduces
all on-shell terms in the four-derivative part of the N=4 SYM effective action;
in particular, the F^4/X^4 and Wess-Zumino terms. The latter comes out in a
novel SO(3) x SO(3)-invariant form.
