What's the meaning of the $R(f(x),g(x))$ in $\int R(f(x),g(x))?$

Question

Usually when I'm reading about integration, there is a notation for integrals on some forms, for example:

$$\int R(\sin(x),\cos(x)) \;dx$$

Obviously I've deduced that this represents functions that are ratios expressed with $\sin x$ and $\cos x$. But It's not really clear what is really allowed, for example, It seems that $R(\sin(x),\cos(x)) $ represents functions such as:

$$f(x)=\frac{\sin x}{\cos x}$$

But does it represent functions such as the ones given below?

$$g(x)=\frac{2+\sin x}{\cos x}\quad h(x)=\frac{x^{45} +\sin x}{\cos x+1}\\i(x)=\frac{\cos x}{\sin x}\quad j(x)=\frac{\cos x+\sin x}{2\cos x +\cos x}$$

I've looked at some books, but the idea behind this notation is not very clear and I want to be sure about it's meaning. In the books I've looked, there doesn't seems to have a discussion about the meaning of this notation or perhaps I didn't find yet.

Answer 1

Unless you provide us with more context, it seems that $R$ is just some function with $R:\mathbb{R}^2 \to \mathbb{R}$. What about this suggests it's a ratio?

Answer 2

The notation is by no means "standard", but based on the context you provided it might mean:

"$R(x,y)$ is a rational function in its arguments $x,y$"

That is,

$$R(x,y) = \frac{p(x,y)}{q(x,y)}$$

where $p$ and $q$ are polynomials in $x$ and $y$.

In the examples you gave, the functions $g(\sin(x),\cos(x)),i(\sin(x),\cos(x))$, and $j(\sin(x),\cos(x))$ are rational functions of the arguments $\sin(x)$ and $\cos(x)$ but the function $h$ is not. (The term $x^{45}$ is not a polynomial expression in terms of $\sin$ and $\cos$.)

The Wiki page on Abelian functions uses the same notation.