These first two will not work directly in Maple 2015.2, but I'm going to mention them anyway since that version is getting quite out of date and some people might be interested in more current versions's behavior.

Download simplify_floatPi.mw

[edit] Maple 2015 suffered from an additional weakness of needing a kick to get the trig identity applied. Here are two ways to do that kick, for each of the two approaches floatPi=true alongside rationals, and floatPi=false.

The approach below using freeze & thaw is an automation of a process to replace the sin & cos arguments temporarily with a symbol, which allows the trig identity simplification to succeed here. That's analogous to automating the steps subs(0.12 + epsilon=t, expr) and (potentially necessary) final replacement of t, in Kitonum's second approach.

For the applyrule apporach, note that the latter step involves simplify(...,trig) which succeeds, while simplify(...) without the option does not succeed in Maple 2015.2.

Download simplify_Pyth_floatPi_2015.mw

Lastly, a note on why floatPi matters here. Consider the following effect.

Download floatPi_consequence.mw

So, when floatPi=true (default) the inequality gets significantly altered. And in consequence some mixed floating-point & symbolic expressions can just be too much for the simplifier to recognize simplifications under mismatches due to roundoff error.

I used the particular formulation  epsilon<Pi-0.12  in above assumptions only because you had that in your Question. If you accept the formulation as  epsilon+0.12<Pi  and you replace that with a symbol then floatPi may not be relevant. Programatically picking off all arguments to sin&cos calls and freezing them may also get you those replacements. Earlier I used subsindets above on sin&cos calls, but indets can also work, also helping in programmatically generating assumptions.

So, as a more general and programmatic revision of one of my earlier four methods for Maple 2015,

Download simplify_trig_float_2015.mw

In the case of multiple trig calls with differing operands you could handle contradictory generated assumptions, say by catching the relevent error thrown by assuming.

Download Not_evaluating_theta_ac.mw

I'm not really sure what you mean by "Polynomiograph".

Do you mean something like a Newton method iterative fractal image, which could be produced using the Newton command from the Fractals:-EscapeTime package?

Suppose you wanted to use you given list of colors to shade the escape-time values for some polynomial.

You could use the Statistics:-HeatMap command for this.

First, let's see that that command acting on a small Matrix.

restart;

m:= Matrix([[1, 2, 3], [4, 5, 6]]):

cm := ["aquamarine","blue","brown",
       "coral","cyan","black"]:

#Statistics:-HeatMap(m,color=cm,axes=none,size=[600,400]);

Statistics:-HeatMap(m,color=cm,
                    axes=none,method=polygons,size=[600,400]);

Now lets apply that command to the first layer results from the Newton command.

with(Fractals:-EscapeTime):

bl, ur := -6 - 6*I, 6 + 6*I:

f := t^3 - t^2 - 12;

                           3    2     
                          t  - t  - 12

temp := Newton(400, bl, ur, f, output=layer1):

cm := ["aquamarine","blue","brown",
       "coral","cyan","black"]:

Statistics:-HeatMap(temp,color=cm,axes=none,size=[400,400]):

That use of (undocumented) method=polygons is there as a workaround for a Maple 2021.1 bug in which the default background image (method=image) extends ouside the axes. That might not be appealing if you remove the axes=none option.

You might also look at the various ColorTools color Palettes, as there are some alternatives for your named colors.

It's also possible that you might want to color by the particular root to which each point might converge. If that's the case then you should explain clearly what you want and how you expect the colors to be used. I recall once implementing that in a few ways (densityplot+caseswitch+colorscheme, or IterativeMaps+caseswitch), but I'd have to try and dig them out.

For now, you might change the (forced) order that the determinate functions appear in the list passed as second argument to dsolve.

analytic_sol:=dsolve(sys, [Sy(t), Xc(t)]);

plot(eval([Sy(t), Xc(t)], analytic_sol), t=0..10);

Curiously, the alternative of using a set (rather than a list) also works. It seems to work even if the lexicographic set-ordering matches the list order that fails. (I rename, to illustrate.)

Download div_by_zero_alt.mw

If you have a problem with your particular example then it would be better to upload and attach your worksheet. That can also help with syntax issues.

Here I've squinted and tried to reproduce the input for the int call from your image, in plaintext Maple notation. I prefer to use a different name for the upper limit of integration. 

Download integral_ex1.mw

I'm not sure whether you are trying to simplify or improve the code you gave, or obtain code to produce a table of a variety of such curves.

Here, procedure L produces the curve for a particular set of parameter values, from t=0 to t=T some final value. You could call L with some choice of parameter values, to see that. Procedure LL makes a table of different calls to L, using names lists of parameter choices. And LL can be animated across T values. (There are several other ways you could code this, expecially w.r.t. you choice of parameters for a table. Adjust as you see fit.)

restart;
dlist:=[0,1*Pi/4,2*Pi/4,3*Pi/4,4*Pi/4]:
ablist:=[[1,1],[2,1],[3,1],[3,2],[4,3],[5,3],[6,5]]:
adj:=[0,Pi/2,0,Pi/4,Pi/2,Pi,-Pi/2]:
L := proc(A,B,a,b,d,T) local ee,t; uses plots;
  display(subsindets(
            plot([A*sin(a*t+d),B*sin(b*t),t=0..T],
                 'color'="white",'thickness'=1),
            'specfunc(:-anything,:-THICKNESS)',u->':-THICKNESS(0.9)'),
          pointplot([A*sin(a*T+d),B*sin(b*T)],'color'="cyan",
                    'symbol'=':-circle','symbolsize'=6),
          'axes'=':-none','scaling'=':-constrained',background=black);
end proc:
LL := proc(T) local i,j; uses plots, plottools;
  display(seq(seq(translate(L(1,1,ablist[j][],dlist[i]+adj[j],T),
                            (nops(dlist)-i+1)*3,(nops(abslist)-j+1)*3),
                  i=1..nops(dlist)),j=1..nops(ablist)),
          'size'=[nops(dlist),nops(ablist)]*125,'axes'=':-none',
          'scaling'=':-constrained','background'="black");
end proc:
## A single call to LL
#LL(2*Pi/3);
plots:-animate(LL,[T],T=0..2*Pi,'paraminfo'=false,'frames'=51,
               'background'="black",'size'=[nops(dlist),nops(ablist)]*125);

Lissajous_anim1.mw

The code above is slightly more verbose than necessary, so that a few possible modifications might be a little more clear. If you wanted to get fancier you could also color the curves according to certain parameter values, or add captions below each to show values.

As for the code you originally supplied, I think that merging a multitude of calls to plots:-animate makes the coding unnecessarily complicated. One difficulty it brings is that the various curves and points cannot be easily seen together (as a visual check of correctness) as a single frame for a single value of the animating parameter.

As an alternative approach, the procedure F below generates any single frame, taking the animating parameter as its sole argument. Thus a call to F generates a single frame, which makes for a much easier and quicker way to test that a given frame looks right.

Also, the various plot and pointplot calls could be merged. An easy edit, but a bit tidier.

Download Lissajous_ac.mw

restart;
with(plots,display): with(geom3d):
point(A, [1, 2, 3]):
point(B, [-2, 3, -1]):
point(C, [0,-3, 1]):
segment(S1, A, B):
segment(S2, A, C):
display(
  display(draw(A(symbol=solidsphere, symbolsize=15)), colour=green),
  display(draw(B(symbol=solidsphere, symbolsize=30)), colour=red),
  display(draw(C(symbol=solidsphere, symbolsize=20)), colour=magenta),
  draw([S1(thickness=4, color=red),
        S2(thickness=4, color=blue)]));

The leading minus sign gets generated depending on what Maple considers the sign of some expressions (or subexpressions) involved in the computation (here, by solve).

If you can find a lexicographic ordering for which the sign of (say...) the numerator is -1 then you might apply sort w.r.t that ordering and reconstruct.

For example,

Download sign_sort.mw

I have not included code that might search (loop) through candidate orderings (say, permutations) until the desired sign might be found.

Maple's design in this regard is for efficiency over beautification of the final display. Note that the application above of the sort command will transform the uniquified expression that was numer(ee). This reorders the terms of that polynomial. That is done to the unique representation stored in the kernel, for that expression. Uniquification of expressions is for effiiciency. Reordering expressions for the purpose of beautification of display is a performance killer in general since in general that might require repeated waste of time as the same expression could get sorted and resorted for situations that vie against each other. Even for mere printing effects, searching for beautification may get expensive in the multivariate case, and being inconsistent in that regard would be very poor.

note: One might also (have to) examine sign(ee), and sign(denom(ee)), depending on the example.

A difference in sign:

Download int_cpv_ex1.mw

You have not provided your problematic module, so I'm going to guess.

This is not allowed in Maple 2026,

restart;
kernelopts(version);
   Maple 2016.2, X86 64 LINUX, Jan 13 2017, Build ID 1194701

M := module()
  local x;
  x := 5;
  export F := proc(x) sin(x); end proc:
end module:
Error, unexpected `export` declaration in `module` body

But it's ok in Maple 2020,

restart;
kernelopts(version);
   Maple 2020.2, X86 64 LINUX, Nov 11 2020, Build ID 1502365

M := module()
  local x;
  x := 5;
  export F := proc(x) sin(x); end proc:
end module:

You might consider,

restart;
kernelopts(version);
   Maple 2016.2, X86 64 LINUX, Jan 13 2017, Build ID 1194701

M := module()
  local x;
  export F := proc(x) sin(x); end proc:
  x := 5;
end module:

M := module()
  export F;
  local x;
  x := 5;
  F := proc(x) sin(x); end proc:
end module:

Perhaps the simplest is to store the data in a file.

If the data is in a Matrix, Array, or table then you could save it as a .m file. Or if it's a Matrix or Array you could export it to a file.

Then you could read (or import, if exported) that file back into your separate worksheet.

It's not quite clear (to me) from your Question whether you "table" might be an Matrix, Array, or table data-structure.

If you don't mind your worksheets being bundled together then you could also store both those and the data in a workbook. Or you could share a variable across them.

I really dislike making unnecessary assignments, since then the assigned symbolic names are no longer directly available for further symbolic manipulations (unless you also unassign, ie. wasting time and effort).

So, after what you had so far, you could do the following, if like me you prefer Qb11,etc remain unassigned. This does substitution instead of assignment.

eqs1 := Equate(lhs(eq1), rhs(eq1));

plot(eval(Qb11, eval(eqs1, [Q22 = 1, s = 1])), p = 0 .. 9);

plotting_elements_of_matrix_ac.mw

For your example, the rational factor of 1/3 gets distributed by a process within the Maple engine called automatic simplification.

That cannot be prevented through unevaluation, eg. it's not prevented by uneval quotes.

'1/3*(1-exp(-3))';

          1   1        
          - - - exp(-3)
          3   3        

So, about the best you can do here is turn some part(s) of the expression into a form that pretty-prints as you want it (even if it isn't the same value, internally).

Your example is rather short, and there are a few ways to accomplish such pretty-printing workarounds for it. Some of the approaches render slightly differently across versions. You didn't say which version you're using.

For example,

Download inert_autsimp.mw

You seem to expect that the condition x>=-4 implies only one explicit solution (for y). But that's incorrect. As the following plot illustrates, there are two y-solutions for each x>=-4.

plots:-implicitplot(x=subs(x=y,x^2+8*x), x=-18..5, y=-10..1,
                    rangeasview, size=[400,300]);

Perhaps you mean instead that y>=-4 ?  With that condition one may obtain a single, restricted solution. That can be accomplished either by utilizing assumptions or by supplying the extra condition in solve's first argument.

Download solve_restr.mw

Similarly, for y<=-4,

solve(x=y^2+8*y, y, useassumptions, parametric)
  assuming y::RealRange(Open(-infinity),-4);
           /                       [[                 (1/2)]]\
  piecewise\x < -16, [], -16 <= x, [[y = -4 - (x + 16)     ]]/

solve({x=y^2+8*y, y<=-4}, y, parametric);
           /                       [[                 (1/2)]]\
  piecewise\x < -16, [], -16 <= x, [[y = -4 - (x + 16)     ]]/

A Plot Component has a programmable property for the delay between frames.

You can programmatically construct, embed, and control the playing an properties of an animation inside a Plot Component. (see attachment)

You can even programmatically embed an animation which automatically starts playing, with your specified frame delay.

Ignoring some (usually small) rendering overhead, the frame-delay can be taken approximately as the reciprocal of the frame-rate -- accomodating also for the scaling conversion seconds vs milliseconds.

anim_auto_fps.mw

Somewhere I had a version of this which included play/stop/fastforward/etc Buttons below the Plot Component. I can't find it now -- it was years ago. I might have included a Dial for frame-rate/fame-delay.

Of course, this does not provide any way to alter the frame-rate for exporting to animated .GIF file. For that purpose all I know is the crude slowing down of an animation by duplication of certain/all frames.