Is there a finite set $P$ of nonelementary functions $f_n$ such that the derivative of any function $f$ from that set is not elementary, but expressible with functions from the same set $P$ plus elementary functions?
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closed as unclear what you're asking by Misha, Ricardo Andrade, Stefan Kohl, Andrey Rekalo, Chris Godsil Mar 17 '14 at 11:49Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. 


Let $g$ be an elementary function whose indefinite integral $f$ is nonelementary, and set $P = \{\cos(t)f(t), \sin(t)f(t)\}$. 


And also many other solution sets to classes of differential equations. You can also replace $x$ by any elementary function in these classes. 

