Suppose I have a very large expression,
myEq[x1,x2,x3,x4] in four variables,
x4. (The general problem could of course use any number of variables.) Some (constant) terms do not depend on any variables. Some of the terms depend solely on
x1, others solely on
x2, ... some on both
x2, others on both
x3, .... and finally some on all four
x4. There are, then, $2^4 = 16$ classes of terms based on their minimum variables.
I'd like to perform a four-variable (symbolic) integration of myEq, so I'd like to simplify the process by separating out all terms that depend just on
x1 and perform that integration over
x1, and likewise for all the terms.
If the terms are all polynomial in the variables, then
Collect might be useful. As far as I can see, though, if I use
Collect[myEq, x1] (say) I still get terms that involve joint terms, involving, say
x2 as well as
x4. Moreover, I don't see how
Collect is helpful for my problem of non-polynomial terms.
In short, what is the simplest way to take a complicated non-polynomial expression (e.g., trigonometric expression) and break it into parts that depend solely upon each variable and each conjunction of variables?
myEq[x1,x2,x3,x4] = 5 + Cos[x1] + Sin[x1/2] + Tan[x2] + Cot[x1 x2] + Sin[x1 x2 x3] - Cos[x3] - Tan[x1/x3] + Sin[x2 + x3/x4] - Tan[x2 - x4] + Cot[x1 + x2 + x3 + x4] - Sin[x1 - x2 - x3/x4]
Break this into $2^4$ terms based on the minimum variables.
If there are $n$ variables, then break the expression into $2^n$ such terms.