few definite integrals related to special functions within a look-up table. However this particular method, involving differentiation of special functions with respect to its parameters, variable transformation, pattern matching and other manipulations, was pioneered by developers of the Maple system and then later emulated by Mathematica, Axiom, MuPAD and other systems. Recent advances The main problem in the classical approach of symbolic integration is that, if a function is represented in closed form, then, in general, its antiderivative has not a similar representation. In other words, the class of functions that can be represented in closed form is not closed under antiderivation. Holonomic functions are a large class of functions, which is closed under antiderivation and allows algorithmic implementation in computers of integration and many other operations of calculus. More precisely, a holonomic function is a solution of a homogeneous linear differential equation with polynomial coefficients. Holonomic functions are closed under addition and multiplication, derivation, and antiderivation. They include algebraic functions, exponential function, logarithm, sine, cosine, inverse trigonometric functions, inverse hyperbolic functions. They include also most common special functions such as Airy function, error function, Bessel functions and all hypergeometric functions. A fundamental property of holonomic functions is that the coefficients of their Taylor series at any point satisfy a linear recurrence relation with polynomial coefficients, and that this recurrence relation may be computed from the differential equation defining the function. Conversely given such a recurrence relation between the coefficients of a power series, this power series defines a holonomic function whose differential equation may be computed algorithmically. This recurrence relation allows a fast computation of the Taylor series, and thus of the value of the function at any point, with an arbitrary small certified error. This makes algorithmic most operations of calculus, when restricted to holonomic functions, represented by their differential equation and initial conditions. This includes the computation of antiderivatives and definite integrals (this amounts to evaluating the antiderivative at the endpoints of the interval of integration). This includes also the computation of the asymptotic behavior of the function at infinity, and thus the definite integrals on unbounded intervals. All these operations are implemented in the algolib library for Maple. See also the Dynamic Dictionary of Mathematical functions. Example For example: is a symbolic result for an indefinite integral (here C is a constant of integration), is a symbolic result for a definite integral, and is a numerical