Symbolic integration

Symbolic integration is the process of finding a closed-form antiderivative of a given function, an explicit formula F such that its derivative equals the original function, rather than only a numeric estimate at particular points. It sits in contrast to numerical integration, which approximates a definite integral's value without producing a general formula. While differentiation is mechanical and can be automated in a straightforward way, integration is fundamentally harder, since many simple-looking functions have no antiderivative expressible in elementary closed form. The Risch algorithm is the central procedure in this area: given an elementary function, it decides whether that function's integral is itself elementary, and produces it when it is. The Risch algorithm was first implemented for transcendental functions in the Reduce computer algebra system, with James H. Davenport later handling the algebraic case and Manuel Bronstein completing a general implementation in the Axiom system. The algorithm addresses indefinite integrals only; for the definite integrals common in physics and engineering, computer algebra systems typically fall back on heuristic pattern matching and special functions such as the incomplete gamma function rather than a fully algorithmic method.