I ran across an interesting footnote in Wolfram Koepf’s book Computer Algebra.
Gosper’s algorithm [1] was probably the first algorithm which would not have been found without computer algebra. Gosper writes in his paper: “Without the support of MACSYMA and its developer, I could not have collected the experiences necessary to provoke the conjectures that lead to the algorithm.”
When I first glanced at the footnote I misread it, coming away with the impression that Gosper said he could not have proved that his algorithm was correct without computer algebra. But he said he would not have formulated his algorithm without his experience using MACSYMA.
If a sum of hypergeometric terms is itself hypergeometric, Gosper’s algorithm gives a way to find the sum. Gosper’s algorithm, discovered using computer algebra, is itself an important computer algebra algorithm.
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[1] Koepf cites Gosper as R. W. Gosper, Jr. Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA, 75, 1978, 40–42.