Exhaustive Symbolic Integration: Integration by Differentiation and the Landscape of Symbolic Integrability

Abstract: We introduce Exhaustive Symbolic Integration (ESI), a method that enumerates all symbolic functions up to a given complexity $k$ within a specified operator basis and determines which admit closed-form antiderivatives within the same class. This allows us to compute the "integrability fraction" $ρ(k)$ (the fraction of functions whose derivatives lie within the same class), which we do for five operator bases including combinations of rational functions, powers, exponentials, logarithms and trigonometric functions. We find that $ρ(k)$ declines at high complexity and that the operator basis has a dramatic effect -- in particular, adding the logarithm boosts $ρ(k)$ by a factor of $\sim$3 and produces or exacerbates a clear peak at $k=6$. We also deploy ESI as a novel integration algorithm, identifying three integrals that resist SymPy, Mathematica, RUBI, FriCAS, Maxima and Giac under all tested strategies. When an antiderivative can be found by multiple methods, ESI often returns the simplest form. These results reveal that the landscape of symbolic integrability is shaped primarily by the choice of operators, and that exhaustive enumeration can systematically discover integrable forms -- including novel ones -- that elude computer albegra systems.