I tried to answer your question, but I'm confused by the plethora of d-like symbols in your expressions. Do they all represent derivatives/differentials? In particular, what's the lowercase delta?

@jmalik7 Don't confuse FAIL with false. FAIL means that is was neither able to rigorously prove the universally quantified statement (the quantification being with respect to the assumptions on symbolic variables) nor was it able to find a counterexample. Since finding counterexamples (when they exist) is generally easier than proving a universal, FAIL results are more likely to be in reality true rather than false. 

Acer said, "you may receive a result of FAIL from is, which doesn't help." Yes, it doesn't help if you require rigor. But if just knowing what's likely is sufficient, I'd accept FAIL as true. Likewise, a FAIL from coulditbe (the existential quantifier version of is) is likely to be false in reality.

@Ronan The number of solutions is given by a well-known function of the modulus called the totient, Euler's totient, or Euler's phi. You should read it about it in Wikipedia. It may be the most important function in number theory. It is a simple function of the prime factorization of the modulus. Since 10000 = 2^4*5^4, its totient is (2-1)*2^(4-1)*(5-1)*5^(4-1) = 4000. Via Maple, it's NumberTheory:-Totient(10000).

This only works because 2391 and 10000 are relatively prime. If the coefficient and the modulus aren't relatively prime, the situation is more complicated.

@dharr IIRC, more than 15 years ago, the sole difference was that the maximum setable value of Digits was 100. That difference has long since been dropped.

@dharr AFAIK, there are no differences (other than the price) between the student edition Maple and the much-more-expensive editions.

I don't understand why VV's msolve solution has received three upvotes while mine has received none. My solution is well over 1000 times faster and just as easy to use and to understand.

@vv I guess that I should've said

• I don't think that it actually matters whether we consider x to be an independent variable or a constant.

If x is considered a dependent variable, as you suggest, then, yes, I agree that that would make a difference. 

@vv For an example of the former,

shake(sec(x));

   INTERVAL(-infinity .. -1,1 .. infinity)

So, shake implicity assumes that variables are real, which seems reasonable to me.

I haven't yet a meaningful example for INTERVAL(x, a..b), but it does seem very easy to generate nonsense with shake, evalr, and INTERVAL. Also, don't trust any prettyprinted output of these command; always examine with lprint. 
 

@Kitonum It is easy to avoid the need to enumerate or index the solutions:

seq(eval(f, s), s= Sol)

@dcatalanoferraioli Thank you. While you were typing the above Reply, I added two paragraphs to my Answer plus significant details about the composed operator. So, be sure to read those.

I am often asked about references that can help one learn Maple, but I haven't read any formal third-party references. All my knowledge comes from, roughly,

• 90% practice, experimentation (always pushing the boundaries of what the syntax will allow), reading the endogenous help files (including the Programming Guide, which is a book-length document available in-program through ?ProgrammingGuide), reading the code of Maple itself, and writing my findings here on MaplePrimes (which forces me to clearly organize my learnings in my mind);
• 9% reading other experts on here MaplePrimes, and earlier forums (both of which used to contain a much higher percentage of higher-quality material);
• 1% private discussions with programmers who work for Maplesoft.

I merged your two recent Questions. Please do not post multiple copies of essentially the same question. It's great to add additional details, but please do it by editing the original Question rather than by posting a new one. If you need assistance with editing, please ask. (Asking Questions that are only about MaplePrimes and not about Maple is fine.)

@mmcdara What you say is true, but I strongly advise the OP against wanting to do that, as I find it pedagogically unsound. In other words, there isn't much practical that can be learned by doing that or even by doing the computation in exact arithmetic at all.

@mmcdara You made at least two errors as far as I can tell:

• You didn't multiply the transformed integrand by the absolute value of the determinant of the jacobian of the transformation.
• You've exchanged the usual roles of sin(phi) and cos(phi) in your transformation. I suppose that this wouldn't matter if the integration range was also changed to phi= -Pi/2..Pi/2 rather than phi= 0..Pi.

When you finally do get the menu, in what language are its entries written? I wonder if language conversion has something to do with the slowness.

@sand15 Would you please define "bow" as you've used this word above?