I don't know what "tolerance = tolerance" means. If you use "res := eval(<vars>, fsolve(eval({e||(1..8)}, [y1[n]=iny1, y2[n]=iny2]), {vars})):" instead, your code will work well.

Not an answer. But if you use "[x]" (instead of "x" or "{x}"), the output does agree with each other: 

with(RealDomain):
sol1 := solve(x - 1 > 0, [x]);
sol2 := solve(1/x^2 > 0, [x]);
sol3 := solve(1/(x^2 - 1) > 0, [x]);
sol4 := solve(x^2 + 1 > 0, [x]);
sol5 := solve(x^2 + 1 > 0, x); = 
                       sol1 := [[1 < x]]

                       sol2 := [[x <> 0]]

                  sol3 := [[x < -1], [1 < x]]

                       sol4 := [[x = x]]

                           sol5 := []

@acer Thank you. Though this might produce useful solutions (especially in physics and engineering), I think that for the original problem, this does not guarantee the completeness of results.

@rcorless Thanks. I doubt that this solver will be stuck or hanged as well, but if complete exact solutions can be obtained at long last, so much the better.

@tomleslie Thank you. But what about [a = 3, b = 9, c = 3]? It seems that the "AllSolutions" opinion creates a false impression. Besides, I think that it should be "a::positive, b::positive, c::positive".
By the way, it appears that these are not enough to show that a "sixth" solution does not exist. (Right or not?)

@Carl Love Thanks for your constructive suggestions. This question is just a extension of one former question: MatrixPower doesn't give the right answer - MaplePrimes. When I try to search for the LinearAlgebra[MatrixPower] command at this forum, I just get few questions about the LinearAlgebra[MatrixPower] command, but these questions are not related to what I post here. Since no one concerned oneself with this topic these years, does this mean that this question make no sense? I'm not sure, but I still hope that this new question will benefit other (or further) users (which is also the major aim). (I'd like to cite this reply: It is *always* the user's responsibility to ensure correctness and fidelity.)  As for "Why do you ask "Is this a bug?" so often?", it is simply a hypophora. You may be of the opinion that this is only a feature rather than a bug, but the Wikipedia article claims that: A software bug is an error, flaw or fault in the design, development, or operation of computer software that causes it to produce an incorrect or unexpected result, or to behave in unintended ways. In the above example, the output is more or less unexpected (at least for me), so in my view, isn't this a bug? As for "Why do you constantly mention Mathematica?", someone may believe that comparisons are odious. Nonetheless, contrasting some computer algebra system (including Maple) with other computer algebra systems (including Mma) is natural and common. Many authors compare them (independently) in their mathematical papers (instead of certain marketing literature) as well. But why Maple and Mathematica? Actually, as List of computer algebra systems - Wikipedia says, the most "general" (not generic) CASs are still Maple and Mathematica. Since none of them acquires a monopoly, an attendant contrast is fairly inevitable. By the way, I am not unsatisfied with Maple; I am just unfamiliar with Maple. (So some misunderstandings could have been avoidable?) Yet this question is not directed against any third-party CAS.

@vv Thanks. The original problem is: 

evalf(Int(x^(x^x), x = 1 .. 6));

Unfortunately, Maple cannot evaluate it (Why?).
So I have to change the variable (x^x -> y, so diff(x(y), y) = 1/(y*(1 + LambertW(ln(y))))) manually, but Maple still cannot compute: 

evalf(Int(expr[2]/(y*(1 + LambertW(ln(y)))), y = 1 .. 6^6));
evalf(Int(expr[3]/(y*(1 + LambertW(ln(y)))), y = 1 .. 6^6));
evalf(Int(expr[4]/(y*(1 + LambertW(ln(y)))), y = 1 .. 6^6));

This is really strange. (And if one doesn't know expr₁ in advance, Maple can never simplify any of expr₂, expr₃, and expr₄ to expr₁ ….)

@acer Thanks. But I don't know why a singular simplify command is not enough here. Maybe there is still plenty of scope for improvement in Maple.

@vv You are right; such transformations are mentioned within the first link (Чебышёв, Čebyšëv, or Chebychev) in this post. But in my view, shouldn't this be done internally in Maple (just as Mma did)? (Another similar example is: integration - Does there exist a complete implementation of the Risch algorithm? - MathOverflow. Fortunately, Maple 2023 has fixed that "bug".)

@acer Thanks.
Besides, I find another way to avoid the lengthy results: 

[1/(4*a + z^2 + (z^4 + 4*a*z^2)^(1/2)), -1/(z^2 - (z^4 + 4*a*z^2)^(1/2) + 4*a)]:
int~(evala(%), z); # rather than just `int`
 = 
     [                   (1/2)                       (1/2)]
     [    / 2 / 2      \\             / 2 / 2      \\     ]
     [    \z  \z  + 4 a//             \z  \z  + 4 a//     ]
     [z - --------------------    z + --------------------]
     [             z                           z          ]
     [------------------------, - ------------------------]
     [          4 a                         4 a           ]

It seems that the putting the radical in the numerator is better than putting the radical in the denominator. (But in my opinion, this must be a bug instead of a feature.)

@ecterrab Thanks for your detailed descriptions! I shall read them later.
Recently, I read a note: https://www.12000.org/my_notes/CAS_ode_tests/reports/maple_2022_2_mma_13_2/indexchapter1.htm#x2-80001.3 (CAS Independent Differential Equations Tests - Maple 2022.2 and Mathematica 13.2). I find that certain ODEs that cannot be solved by Maple 2022.2 (yet can be solved by Mathematica 13.2.1) have been solved by Maple 2023.0, but many of them remain unsolved for the latest Maple. Will they be solved in a future release?

@vv In my opinion, Maple can "evalf" S, but cannot "evalf" s or v (at least in acceptable time); isn't this a bug?

@Thomas Richard Thanks. But since dilog is just a special case of the polylog, why can't Maple handle the expression directly? (And as @vv says, it appears that there are some unexpected limitations in Maple.)

@lcz I find an example: 

@Christian Wolinski Thanks. But in my opinion, numerical computations may suffer from floating-point errors (theoretically or mathematically). Can Maple carry out those evaluations exactly and symbolically?