The Rational(f, k) command computes a closed form of the indefinite sum of  with respect to .

Rational functions are summed using Abramov's algorithm (see the References section). For the input rational function , the algorithm computes two rational functions  and  such that  and the denominator of  has minimal degree with respect to .  The non-rational part, , is then expressed in terms of the digamma and polygamma functions.

If the option failpoints=true (or just failpoints for short) is specified, then the command returns a pair , where

 is the closed form of the indefinite sum of  w.r.t. ,

 is a list containing the integer poles of , and

 is a list containing the poles of  and  that are not poles of .

See SumTools[IndefiniteSum][Indefinite] for more detailed help.

Abramov, S.A. "Indefinite sums of rational functions." Proceedings ISSAC'95, pp. 303-308. 1995.