Since the series is alternating, I decided to pair consecutive terms to make it a series of positive terms:

Download AltSum.mw

I don't know why my output doesn't display the normal way. It looks normal in the worksheet. I tried creating-from-scratch and uploading the worksheet 3 times.

It can be done by applying a small command to the plot after the VennDiagram command:

V:= VennDiagram(...): #exact same command that you had before

(`&<`, `&>`, `&[`):= (curry, rcurry, ()-> index &> args):
fl:= s-> (thisproc(s):= try parse(s[..-2]) catch: s end try):

subsindets(
    V,
    And(string, "%" &under &[(-1), float &under fl),
    sprintf &< "%4.1f%%" @ fl
);

The Answer by @dharr is missing the partial-fractions aspect. The Answer by @nm doesn't reduce the results mod 11 after the partial fractions are obtained. The following code takes care of both of those issues.

Download ModParfrac.mw

Maple's series data structure does not allow for an order (O(...)) term when the low-order terms are already an exact representation of the function. You'd get the same thing from the series command:

series(x, x= 0)

If for some syntactic reason you need the series data structure and order term, you could do

series(sol + x^Order, x= 0)

The messages that you're getting from mint are neither warnings nor complaints, nor are they labeled as such; they are just information. When you see a message such as

• These names were used as global names but were not declared:  A

it is up to you to decide whether A has been used appropriately. The purpose of showing you those messages is that they'd also be shown if A had been a misspelling (of, for example, a). How else could mint be able to tell you about possibly misspelled global names?

Do not declare names as global inside a procedure or module unless you intend to allow the possibility that they be changed within that procedure or module. It is dangerous to do so because then you will not be warned if you unintentionally make assignments to those names. 

Maple has two types of command procedures: library and builtin. The Answer by @nm showed how to use the LibraryTools package to browse the library commands. Library commands are written in the Maple language. The builtin commands are not written in Maple; rather, they are implemented directly in the kernel (a.k.a. "engine"). They tend to be basic and very efficiently implemented commands. 

Many writers here, including some who are quite experienced, use the adjective "builtin" (or the more grammatically correct "built-in") to refer to any of the many 1000s of commands that come pre-installed with Maple, including those in the Library. It's understandable simply from an English-language perspective why they'd use "builtin" or "built-in'" that way; however, that usage is completely contrary to Maple documentation, where builtin refers strictly to the few hundred commands shown below.

Download AllBuiltins.mw

Your idea to use subsindets was the right idea! However, I don't think that MapleTA:-Builtin:-decimal(2, ...) does quite what you want; replace it with evalf[2]. Also, replace float with complex(float). If you want, you can operate on all of ans at once by making it [ans]. Summarizing all that, make the command

P1:= subsindets([ans], complex(float), evalf[2]);

The resulting expression should only be used to enhance human readability of the display. Use the original for any further computation.

All of your results can be explained by basic complex arithmetic. There is no significance to your use of ln(x); it could be replaced by a generic complex variable z and you'd get analogous results. Maple is not doing anything special or unusual here.

For any complex z and positive integer n, it is always true that (z^(1/n))^n = z; however, it's not always true that (z^n)^(1/n) = z; indeed, that's not even necessarily true for real z < 0. Try z = -1, n = 3. This explains why your f(3) and f(1/3) are different when you have no assumptions.

Here, I'm addressing your newer expression A, but the simplification process is essentially the same as for the expression in your original Question. Hence, this Answer is essentially the same as Kitonum's, only applied to your new expression.

A:= -797250051*((-(1284505600*a^2*sin(phi)^(7/2))/797250051 +
   (1284505600*a^2*sin(phi)^(3/2))/265750017)*cos(phi)^(5/2) - 
   (1284505600*a^2*sin(phi)^(3/2)*cos(phi)^(9/2))/265750017)*2^(1/4) *
   sqrt(cos(phi)^(5/2)/sin(2*phi)^(5/2)) /
   (2569011200*sin(phi)^(1/4)*cos(phi)^(5/2)*x^2)
:

I guessed that the variable ts that you had in one place was a typo meant to be phi.

Your large integers do not add any complexity or difficulty for simplify to this; they only make the expression harder to read.

You used the assumption phi>0. Your desired simplified form is not valid under that assumption. You can test with phi=4 to verify that. The assumption sin(phi) >= 0 is sufficient. When simplifying trig functions whose arguments have coefficients, such as sin(2*phi), it's necessary to use expand.

I recommend against using simplify(..., symbolic) unless absolutely necessary. Its simplifications are not always valid, as is clearly stated on its help page.

 AS:= simplify(expand(A)) assuming sin(phi) >= 0;
                                      2  2
                              sin(phi)  a 
                      AS := - ------------
                                     2    
                                  2 x     

Here is a breadth-first-search procedure for it. If one wanted to find all k-cycles containing u, breadth first is certainly faster than depth first. To find only one such cycle (if it exists), perhaps depth first would be faster. My procedure constructs only the (k-1)-paths (not walks) that start with u, then completes to k-cycles those that can be completed.

(* 
Returns all k-cycles in G containing the vertex vx.
Works for all graphs, including directed and those with self-loops.
Undirected cycles are represented twice in the output, once for
each direction of transversal.

Option truefalse can be specified if one only wants to know whether
some k-cycle containing vx exists without displaying the cycles.
*)
kCycles:= proc(vx, k::posint, G::Graph, {truefalse::truefalse:= false})
option Author= `Carl Love <carl.j.love@gmail.com> 2025-Apr-22`;
local (V,E):= op(op~([3,4], G)), v, u, C, c;
    if not member(vx,V,v) then error "vertex %1 does not exist", vx fi;
    C:= [[v]];   
    to k-1 do
        C:= [seq](seq([c[],u], u= E[c[-1]] minus {c[]}), c= C)            
    until C=[];
    if truefalse then orseq(v in E[c[k]], c= C)
    else [for c in C do if v in E[c[k]] then V[c] fi od]
    fi
end proc
: 
#Examples:
G:= GraphTheory:-SpecialGraphs:-PetersenGraph():
kCycles(1, 5, G);
    [[1, 2, 3, 4, 5], [1, 2, 3, 7, 6], [1, 2, 9, 8, 5], 
      [1, 2, 9, 10, 6], [1, 5, 4, 3, 2], [1, 5, 4, 10, 6], 
      [1, 5, 8, 7, 6], [1, 5, 8, 9, 2], [1, 6, 7, 3, 2], 
      [1, 6, 7, 8, 5], [1, 6, 10, 4, 5], [1, 6, 10, 9, 2]]

G:= GraphTheory:-CompleteGraph(14):
C:= CodeTools:-Usage(kCycles(1,6,G)):
memory used=42.10MiB, alloc change=2.36MiB,
cpu time=687.00ms, real time=599.00ms, gc time=218.75ms

nops(C) = (14-1)!/(14-6)!;
                        154440 = 154440

CodeTools:-Usage(kCycles(1,6,G,truefalse));
memory used=23.25MiB, alloc change=-1.18MiB,
cpu time=406.00ms, real time=348.00ms, gc time=156.25ms

                              true

You asked:

• Also, is there a way to disable the use of remember tables permanently in Maple?

[Disclaimer: I don't claim that you'll achieve any benefit from using the code below. Rather, I provide this code so that you can easily, safely, and reversibly explore the possibility that you asked for above.]

I think that I have below something better than what you asked for. You can permanently, yet reversibly, remove options remember and cache from all Library procedures that have them. This will prevent the accumulation of new data in remember or cache tables when that accumulation is caused by the option remember or option cache mechanism. It doesn't affect direct assignments to the tables as in

f(3):= 7;

where f is a procedure. It affects only Library procedures; it doesn't prevent you from using the options in your own code. It doesn't remove the default initial remember or cache tables which commands have stored in the Library (which would be a very bad thing to do!). The only cost of using this procedure is a one-time use of 30 seconds of time, and permanent storage of 1.6M of files to hold the modified procedures. Once that's done, the new procedures are used by doing restart and updating libname. To return to the original procedures, just use restart.

Download RemoveRemOpts.mw

Yes, in almost all cases worksheets created with a newer version of Maple can be opened and edited and resaved with a version several years old. That part is usually still true even if the worksheet has new features. New programmatic features will not execute correctly and new GUI features will not display correctly, but that usually has no impact on reading, editing, and saving.

The Options dialog is in a weird place now: the bottom right corner of the File menu. Being on the right side of a menu is very weird. And why was it moved from the Tools menu? Tools seemed appropriate, but Options is certainly not a File operation.

I fretted for a long time during the beta testing over not being able to find the Options dialog, but I stumbled upon it last night.

I finally found a help page saying that Options was now on the File menu (without specifying the weird exact location). That page is
?worksheet,managing,preferences. Since that's the16th hit for ?options, it's not very helpful. If I capitalize it (?Options), then the relevant help becomes the 21st entry.

Your system has 2 variables,1 equation, and several inequalities, the number of which is irrelevant. Thus, you can only solve for 1 variable. It could be x or y, but you need to specify which. This will get you the solution (without removing the warning):

solve(Sol union area, {y})

If for some strange reason you actually want a solution containing RootOf, change {y} to {x}.

I believe that the warning comes because of your index= real[2]. The solver is able (at least in this case) to convert the RootOf equation to a polynomial before passing it to SemiAlgebraic, but I guess that it's unable to generate the appropriate inequality constraint that's implied by index= real[2]. Since you mean (I assume) the positive square root of y, that constraint is x >= 1. So, if you replace 

x = RootOf(_Z^2 - y, index= real[2]) + 1

with this equivalent

x = RootOf(_Z^2 - y) + 1, x >= 1

then the warning goes away.

There's no need to go through your entire code to modify all occurences of any command procedure (such as simplify). All you need to do is write a simple wrapper for simplify and name it simplify:

restart:
`simplify/orig`:= eval(simplify):
protect(`simplify/orig`):
unprotect(simplify):
simplify:= proc()
local r:= `simplify/orig`(args);
    forget(simplify, 'forgetpermanent', 'subfunctions', 'reintialize');
    r
end proc:
protect(simplify):

That is the "clobbering" approach that acer mentioned. Here is a more-elegant way:

Simplify:= module()
option package;
export 
    simplify:= proc()  
    local r:= :-simplify(args);
        forget(:-simplify, 'forgetpermanent', 'subfunctions', 'reintialize');
        r
    end proc
;
end module:
with(Simplify);

However, that module-based way will not change fhe way that your existing code works.

Using printlevel is one of the crudest possible debugging tools. Why not just use trace on all procedures whose names begin "simplify", like this?:

`&<`:= curry: `&>`:= rcurry:
((trace @ cat&<`` @ index&>(..-3))~ @ select)(
    type, index~(LibraryTools:-ShowContents(), 1), suffixed("simplify")
);

Your example expression for simplify --- 3*x^3/x + sin(x^2)/4 --- is a bad example because x^3/x is converted to x^2 by automatic simplification which has nothing to do with the simplify command. Thus, the command only sees 3*x^2 + sin(x^2)/4, which is already fully simplified,