Symbolic integration is the application of computer software to solving problems in mathematics of find the integral of an expression, but finding an expression rather than a value.
Example:
is a symbolic result rather than a numerical value for the answer.
Finding the derivative of an expression is a straight forward process for which it is easy to determine an algorithm. The reverse question of finding the integral (of finding an expression whose derivative is the specified expression) is much more difficult. Many expressions that are relatively easy to state do not even have integrals that can be expressed in closed form.
A procedure called the Risch algorithm exists which is capable of determining if an integral exists and returning it if it does, for many classes of expressions. Such algorithms are still being expanded.
References
Symbolic Integration 1 (transcendental functions) by Manuel Bronstein, 1997 by Springer-Verlag, ISBN 3540605215
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