You could show us what you have got so far. You can upload (and attach a link!) your worksheet using the green up-arrow in the Mapleprimes editor.

@Rouben Rostamian  I see now, thanks.

I'll file the examples below, and also yours using setoptions up front.

Fwiw, you may be able to get by with " " the single black space. You could also use `` the empty name, but since that isn't protected it would more safely be used as ':-``' though even that can get accidentally evaluated. The empty string case is more robust, terse, and ideally would work.

In the following (without any setoptions call in an initialization file) the first two look wrong.

restart;

plots:-display(plot(x^2, x=-1..1),
               labelfont=[TIMES,16], labels=["",""]):
indets(%,specfunc(anything,AXESLABELS));

   {AXESLABELS(x, "", FONT(TIMES, 16))}

plots:-display(plot(x^2, x=-1..1),
               labels=["",""], labelfont=[TIMES,16]):
indets(%,specfunc(anything,AXESLABELS));

   {AXESLABELS(x, "", FONT(TIMES, 16))}

plots:-display(plot(x^2, x=-1..1),
               labels=["",""]):
indets(%,specfunc(anything,AXESLABELS));

         {AXESLABELS("", "")}

plots:-display(plot(x^2, x=-1..1),
               labelfont=[TIMES,16], labels=[``,``]):
indets(%,specfunc(anything,AXESLABELS));

   {AXESLABELS(``, ``, FONT(TIMES, 16))}

plots:-display(plot(x^2, x=-1..1),
               labels=[``,``], labelfont=[TIMES,16]):
indets(%,specfunc(anything,AXESLABELS));

   {AXESLABELS(``, ``, FONT(TIMES, 16))}

plots:-display(plot(x^2, x=-1..1),
               labels=[``,``]):
indets(%,specfunc(anything,AXESLABELS));

         {AXESLABELS(``, ``)}

ps. These examples are supposed to work without need of overrideoption. That is intended to force clobbering of options which get set "locally" inside plotting substructures which designate curves, surfaces, etc. The AXESLABELS substructure is "global" in the sense that it gets uniquified within the whole plotting structure, and it exists outside the plotting substructures which designate curves, surfaces, etc. I suppose that you were demonstrating merely that it did not help here.

@mapleatha You're very welcome. I'm sorry that I didn't think of the discrepency until I saw your followups.

(I suspect that Kitonum realized what the problem was, and gave his working equivalent, before I did.)

@mapleatha In Maple 13 the plots:-animate command did not automatically (graciously) invoke plots:-display in your case that the animating procedure returned a sequence of plotting structures.

Try this,

restart;

F := proc(t)
plots:-display(
plottools[line]([-2,0], [cos(t)-2, sin(t)], color=blue,thickness=3),
plottools[line]([cos(t)-2, sin(t)], [t, sin(t)], color=blue,thickness=3),
plot(sin(x), x=0..t, view=[-3..7, -5..5],thickness=3));
end proc:

plots:-animate(F,
               [theta],theta=0..2*Pi,
               background=plot([cos(t)-2,sin(t),t=0..2*Pi],thickness=3),
               scaling=constrained,axes=none);

Note that the example on the Online Help webpage is designed for current Maple. Hence it does not contain the (now unnecessary, in Maple 2021) extra call to plots:-display. But if you look at the example in Maple 13.01 it actually contains this variant,

with(plots):
# followed further by the following...

F := proc(t)
     plots[display](
plottools[line]([-2,0], [cos(t)-2, sin(t)], color=blue),
plottools[line]([cos(t)-2, sin(t)], [t, sin(t)], color=blue),
plot(sin(x), x=0..t, view=[-3..7, -5..5])
     );
end proc:
animate(F,[theta], theta=0..2*Pi,
        background=plot([cos(t)-2, sin(t), t=0..2*Pi]),
        scaling=constrained, axes=none);

So it appears that you may have mistakenly copied the example from the Online Help instead of from your Maple 13's own Help page.

@mapleatha Please upload and attach the worksheet which reproduces the problem.

Your last Reply contained a typo. That may not be the cause  but it's a hint that it's not productive to guess what code you are actually executing.

@Rouben Rostamian  If I rerun the followup example in your attached Worksheet in my Maple 2021.1 for Linux with the same Build ID, then the label "x" does not appear.

That happens for me, whether I execute line by line with the Enter key, or use the menubar's !!! button.

@nm The given syntactic/structural query will not consider 1/exp(z) the same as exp(-z). Nor would it consider exp(z/2)^2 the same as exp(z).

Perhaps you might consider some attempt at canonicalization prior to making your type queries. Eg,
  expr:=A*exp(z/2)^2*sqrt(h)+a*exp(z)*sqrt(h)+exp(z)*sqrt(h)+1/(3*exp(z)*sqrt(h));
  new := combine(expr,exp);
etc. Also of interest might be combine(...,power) .

There may be other cases that require attention. For example, would you ever want that some subexpr^(3/2) treated similarly to the handling of "sqrt"s?

And what about other equivalent forms? How about the following forms?
   1/3*(3*a*exp(z)*h+3*h*exp(z)+exp(-z))/h^(1/2)
   1/3*(3*a*exp(z)^2*h+3*exp(z)^2*h+1)/exp(z)/h^(1/2)

If you would want only some specific kinds of canonicalization but never certain others then the onus would be on you to be very specific in detailing the complete requirements.

@Rouben Rostamian  In which version, platform, and input environment are you getting that?

You can put your followup examples here, if someone's Answer's code here doesn't handle the additional cases which you forgot or couldn't be bothered to mention here in the first place.

But don't submit further followup examples in separate Question threads. (You've been told this several times before.)

This is a very good question.

Unfortunately, I don't know a way to completely prevent the AXEXLABELS substructure from being present (without forcibly removing it, as you already showed).

You are right that it is a waste of space, around the plots. And a smaller plot (eg. 200x200 or so pixels for the plot widow...) clearly reveal the wasted space.

As the author of several applications which utilize animated plots within PlotComponents I will say that this is the kind of sloppy GUI handling of plots that really gets in the way of producing better quality results. Jitter in the axes frame, loss of valuable space, lack of symmetry in the visible rendering, and trouble with aligning axes of stacked plots, are just some of the difficulties that application authors encounter that really ought all to be avoidable.

Here is another worksheet that demonstrates the problem. (Uncomment the plots:-display calls for yet another view of the wasted space.) This demonstrates, unfortunately, that Kitonum's suggestion to simply omit the labels option is wholly inadequate and does not fix the underlying issue.

wasted_label_space.mw

I am going to submit a bug report, not so much gainst the GUI's plot drivers for rendering the wasted space as against the Library-side plot constructors for including AXESLABELS("","")  when not requested. (Sometimes the latter kind of silliness gets introduced as a band-aid for irritations with GUI "plot persistence". But I get fed up with dealing with the ensuing difficulties of patches for patches...)

@shankrel You're very welcome.

There are many ways to use Maple to solve this problem.

Since you may be relatively new to Maple, here is one (of several possible) alternatives.

Worksheet_2_ac.mw

@mmcdara Sorry, I didn't read carefully enough, and thought you wanted only the standalone piecewise calls, instead of wanting the sum with only some terms retained. (I think I was confused by the use of the terms "extract" and "operators", which I over hastily took to mean extracting piecewise calls instead of restricting the addends of the sum.)

An adjustment to what I gave earlier is simple.

select(hastype,expr,And(specfunc(piecewise),
                        satisfies(pw->not depends(pw,[x]))));

That produces the result for the given example, and I believe it reads reasonably naturally.

In another, extended example it also excludes any term with a piecewise conditional such as t<0 or x[1]<0 .

You could also use depends(pw,x) in the above, but I thought the list-brackets in depends(pw,[x]) give a little visual hint that additional dependent names can be check together.

If you will eventually need to handle more involved conditionals then this is the time to mention them.

For example (making it up...),     

x(t)<0 and t<0

or,

x(t)<0 and t^2>0

@nitins You're welcome.

As a future consideration you might consider either:
1) Uploading and attaching the actual worksheet (as Georgios requested, using the Mapleprimes editor's green up-arrow),
or,
2) Telling us where you obtained the worksheet, so we don't have to Google it. Worksheets come from many different places. (This site's editor allows for URL links to be inserted in postings.)

@Carl Love The given code does subs(i1=id, eq1) where eq1 is not yet assigned anything. So some formula for eq1 is indeed needed prior to that. It is also true that an original definition of id is missing.

All three original definitions of eq1, eq2, and id are needed, yet missing.