Decide whether a function has an elementary indefinite integral without determining it! 
Risch, who developed the algorithm in 1968, called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral; and also, if it does, determining it.

Is there a way to only decide whether a function has an elementary function as an indefinite integral without determining it?
 A: For a given integrand, Liouville's theorem and Risch algorithm and its generalizations give equations and conditions that must be fulfilled if the integrand has an elementary antiderivative. Often, the fulfillment or non-fulfillment of these conditions is not known until the solution sets of the equations are known. If the solution sets are known, you already have the antiderivative. The decision problem is therefore usually just as expensive as the problem of determination.
But you can look for classes of functions that have elementary antiderivatives: e.g. the elementary standard functions and polynomials.
And you can look for classes of functions that don't have elementary antiderivatives:
List of Functions Without Antiderivatives
Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012
Yadav, D. K.: Six Conjectures in Integral Calculus. 2016
Yadav, D. K.: Six Conjectures on Indefinite Nonintegrable Functions or Nonelementary Functions. 2016
