Let F be a rational function of x over a field K of characteristic 0. The RationalCanonicalForm[i](F,x) calling sequence constructs the ith differential rational canonical forms for F, .

If the RationalCanonicalForm command is called without an index, the first differential rational canonical form is constructed.

The output is a sequence of 2 elements , called RationalCanonicalForm(F), where  are rational functions over K such that

.

If the third optional argument, which is the name 'polyform', is given, the output is a sequence of 4 elements , where  are polynomials over K,  monic such that , .

The use of RationalCanonicalForm[1] is for testing similarity of two given hyperexponential functions. For RationalCanonicalForm[2], the polynomials  are also pairwise relatively prime. RationalCanonicalForm[2] is used in a reduction algorithm for hyperexponential functions.

Geddes, Keith; Le, Ha; and Li, Ziming. "Differential rational canonical forms and a reduction algorithm for hyperexponential functions." Proceedings of ISSAC 2004. ACM Press, (2004): 183-190.