There is no complete implementation of the Risch algorithm. None of the implementations fully implement the algebraic case. Besides, the Risch algorithm doesn't

There is no complete implementation of the Risch algorithm. None of the implementations fully implement the algebraic case. Besides, the Risch algorithm doesn't solve all symbolic integration problems in practice: * Even when it succeeds, it doesn't necessarily give you the simplest possible form of the solution. * It is concerned with elementary functions, but real-world problem often involve non-elementary functions (like Bessel and hypergeometric functions). You can extend the Risch algorithm to work with such functions, but this is complicated. * It doesn't allow computing parametric answers for common parametric families of integrals (this needs to be done using heuristics and lookup tables). * It doesn't give you an algorithm for definite integration, except in simple cases. Even with proper definite integrals, you need to be extremely careful with branch cuts when applying the fundamental theorem of calculus. For improper definite integrals, systems like Mathematica generally try to go via convolutions of Meijer G-functions, using huge lookup tables and simplification heuristics. * Even more fundamentally, the Risch algorithm (and symbolic integration more generally) puts heavy demands on the underlying symbolic computation engine. In particular, it depends on the ability to decide whether an expression is equivalent to zero, which in fact is an undecidable problem, and this strictly speaking makes the Risch "algorithm" a non-algorithm (though it is a proper algorithm when restricted to a ground field where zero testing is effective, such as Q).