Write an expression as a full derivative

Question

Is there a command that tells Mathematica to try to write the terms (if possible) of an expression as total derivates? Example: I want to tell mathematica to write

-Sin[t]+g[t] 

as

D[Cos[t],t]+g[t] 

instead. Is this possible?

Mathematically, this is basically (up to some constants) equivalent of Mathematica testing if the terms are analytically integrable (i.e. if they have closed form expressions).

Answer 1

Here I define the expression and integrate it term by term using Distribute. Then I differentiate each term but with a Defer wrapping it, so the derivatives aren't actually carried out. FInally, I replace the terms that weren't integrated by the original function:

expression = -Sin[t] + g[t]

(* ==> g[t] - Sin[t] *)

integrals = Distribute[Integrate[expression, t]]

(* ==> Cos[t] + \[Integral]g[t] \[DifferentialD]t *)

original = 
 Map[Defer[D[#, t]] &, integrals] /. 
  Defer[D[Integrate[f_, __], _]] :> f

$\partial_t \text{Cos}[t] + \text{g}[t]$

The reason I used Defer is that it allows you to copy the output and evaluate it by pasting it in a new cell. You could alternatively replace Defer with HoldForm if you want a more "stable" output. Then you have to explicitly use ReleaseHold if you want to evaluate the derivatives in the output expression.