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.

翻译：我们提出穷举符号积分（Exhaustive Symbolic Integration，ESI）方法，该方法能在指定算子基下枚举复杂度不超过$k$的所有符号函数，并确定其中哪些函数在该类内存在闭形式原函数。据此可计算"可积性分数"$ρ(k)$（即导数仍属于同一类函数的比例）。我们在五种算子基（包括有理函数、幂函数、指数函数、对数函数和三角函数的组合）上完成了该计算。研究发现，$ρ(k)$在高复杂度区域呈下降趋势，且算子基的影响显著——特别地，引入对数函数会使$ρ(k)$提升约3倍，并在$k=6$处产生或加剧一个明显峰值。我们还将ESI作为新型积分算法加以应用，识别出三个在SymPy、Mathematica、RUBI、FriCAS、Maxima和Giac所有已测策略下均无法求解的积分。当多个方法可找到原函数时，ESI通常能返回最简形式。这些结果表明，符号可积性图景主要由算子选择决定，而穷举枚举能系统性地发现包括新型可积形式在内的可积函数——这些形式往往被计算机代数系统所遗漏。