Are there rules for omitted bracket use? In older work I often came across a somewhat inconsistent use of brackets. Specifically, I am readings the following paper: Self-inductance of air-core circular coils with rectangular cross section (https://doi.org/10.1109/TMAG.1987.1065777). 
Usually any function surrounds its arguments by brackets. This is sometimes omitted for the comfort of the writer.
Are there strict rules for omitting brackets? 
More specifically my question regards the equations (13), (14) and (15) in the mentioned paper.

How would you read the third logarithm in equation (13)? 
I assume it is supposed to mean
$$ \cdots + \frac{2}{3} (1+\alpha^3)\beta^2 \ln(2) +  \cdots.$$ This would suggest that if no brackets are used only the first symbol after the function, in this case the number 2 after the logarithm, is the argument of that function. 
In equation (15) we obviously have $\frac{\phi}{2}$ as the argument of the sine.

This would suggest that the entire product after the natural logarithm in equation (13) is the argument of that logarithm.
This confusion also arises in equation (14) where it is not completely clear to me if only the angle $\phi$ is the argument of the cosine or the entire curly brace multiplied with the angle $\phi$.

To sum up: I assume the next following symbol after a function is its argument. But as illustrated this is also not completely rigorous when you have a fraction like $\frac{\phi}{2}$ because this is nothing else than a multiplication.
What do you think? Maybe someone can clear this confusion.
 A: 
Are there strict rules for omitting brackets?

There aren't any strict rules, but the aim should be that of clarity and unambiguity. Unfortunately, what can be notationally crystal clear for one can also be totally unclear for someone else.
For short fractions in arguments (like the case of the $\sin$ function in (14)) it's common to omit brackets, and so don't take that as hint that the argument of the logarithm is the full expression.
However, if the difference is important for you—in the sense that you need to use those equations—I really suggest you to re-derive them, because in papers, in reporting long expressions, it's way too easy to make mistakes and I wouldn't blindly trust those equations.
A: As far as I know, there are no strict rules here. I imagine in the instances that you give the correct interpretations are
$\displaystyle \frac {2 \ln(2) (1+\alpha^3)\beta^2}{3}
\\ \displaystyle \int \cos (\phi) \left[ \dots \sin^2 \left(\frac \phi 2 \right) \dots \right] d\phi
\\ \displaystyle \int \cos (\phi) \left\{ \dots \right\} d\phi$
but it would be clearer if the author had been more conscientious about including brackets.
