If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be expressed in elementary functions plus generalized polygamma function?

Some examples where having integral in elementary functions results in having indefinite sum in elementary functions plus generalized polygamma:

http://en.wikipedia.org/wiki/User:MathFacts/Alternative

Under generalized polygamma I mean the generalization of Polygamma following this paper: http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf It differs from the polygamma used in Mathematica by being *balanced*.

This generalization allows to express Zeta function and the Bernoulli polynomials in terms of generalyzed polygamma function: http://en.wikipedia.org/wiki/Generalized_polygamma_function

Well probably this question should be confined to the continuous non-periodic functions.