The number of functions and variables is free. For deciding the existence of only, the explicit occurrence of the variables is arbitrary. In order to actually calculate explicitly, must have the property that all terms in must either contain an unknown function of or must be formally integrable w.r.t.
Iteration stops if no term with any -derivative of any is left.
If in the remaining un-integrated terms any itself occurs, then is not expressible with and its derivatives only. In case no occurs then any remaining terms can contain only explicitly. Whether they can be integrated depends on their formal integrability. The conditions that is exact are the following. The leading derivative must occur linearly before each partial integration step.