What does it mean the notation $\int{R\left( \cos{x}, \sin{x} \right)\mathrm{d}x} $ Sometimes I find this notation and I get confused:
 $$\int{R\left( \cos{x}, \sin{x} \right)\mathrm{d}x} $$
Does it mean a rational function or taking rational operations between $\cos{x}$ and $\sin{x}$ ?
Can you explain please?
Update: I think you did not understand the question well,
Here is an example (maybe it is a lemma or a theorem):

All the integrals of the form $\int{R\left( \cos{x}, \sin{x}
 \right)\mathrm{d}x} $ can be evaluated using the substitution
  $u=\tan{\dfrac{x}{2}} $.

I think that $R$ here does not stand for a rational function but for taking rational operations(addition, subtraction, multiplication, division) between $\cos{x} $ and $\sin{x}$
Update : I did not noticed that $R$ is a rational function of two variables and that means exactly that we are taking rational operations.
 A: Here $R$ is a function of two variables $s$ and $t$. For instance, if $$R(s,t) = \frac{s}{1+t}$$ then $$R(\cos x , \sin x) = \frac{\cos x}{1 + \sin x}.$$
A: In books with tables of integrals etc. (like Gradshteyn's book) $R(\cos x, \cos y)$ typically means a rational function of $(\cos x, \sin x)$. So take a rational function $R(a,b)=f^{1}(a,b)/f^{2}(a,b)$ and  plug in $a=\cos x, b=\sin x$. Here, $f^{i}(a,b), i=1,2$ are polynomials in the two variables $(a,b)$, i.e.,
$$f^{i}(a,b)=c^i_{0,0}+c^i_{1,0}a+c^i_{0,1}b+c^i_{2,0}a^2+c^i_{0,2}b^2+c^i_{1,1}ab+\dots$$
with coefficients $c^{i}_{k,l}$.
A: A polynomial in two variables is an expression
$$
P(x, y) = \sum_{i, j=0}^{\infty} a_{ij} x^{i} y^{j}
$$
with only finitely many non-zero coefficients $a_{ij}$. A rational function in two variables is a quotient of polynomials in two variables.
In your question, $R$ denotes an arbitrary rational function of two variables. (It's arguably reasonable to describe "evaluating a rational function of two variables $x$ and $y$" as "taking rational operations between $x$ and $y$", but the number of comments here and elsewhere suggests doing so is a recipe for ambiguity.)
We can be certain of this interpretation on grounds of mathematical culture: Thanks to the double-angle formulas for the circular functions and the chain rule, the substitution $u = \tan(x/2)$, or $x = 2\arctan u$, gives
$$
\cos x = \frac{1 - u^{2}}{1 + u^{2}},\qquad
\sin x = \frac{2u}{1 + u^{2}},\qquad
dx =  \frac{2\, du}{1 + u^{2}}.
$$
Consequently,
$$
\int R(\cos x, \sin x)\, dx
  = \int R\left(\frac{1 - u^{2}}{1 + u^{2}}, \frac{2u}{1 + u^{2}}\right) \frac{2\, du}{1 + u^{2}},
$$
a rational function in $u$.
The significance is, every rational function in one variable has an elementary primitive (antiderivative).
