@Kitonum This is exactly what I needed. Thanks!

@rlopez Okay, it appears I was looking at the wrong package. This is exactly what I was looking for! Thanks!

@tomleslie Just what I was looking for. Thank you!

@tomleslie A path is a walk in which no vertices are repeated. So, with regard to your example, an undirected graph with three vertices would have a longest path of length 2: A-B-C, C-A-B, etc. For a cycle, only the initial and terminal vertices are allowed to be the same. Again, from your example, a couple cycles are C-A-B-C, A-C-B-A, all of length 3. Hope this clarifies my question.

@mmcdara Thank you. This does what I need it to do!

@acer Well, I've been playing around with the density option, but I am not getting it to work consistently. Maybe I'm not using it correctly. 

For a 10x10 matrix, I don't think speed will be much of an issue. Although, it would be nice to have a solution that is scalable for larger matrices.

@vv Yes, I find it a bit perplexing that Maple attempts the same methods for both, but is only able to evaluate one of them. The rewriting you've done is what is typically taught in a standard integration techniques unit, which facilitates a straightforward substitution. 

Moreover, I observe a similar behavior when attempt to evaluate comparable integrals involving powers of tangent and secant. Risch is being employed, I believe.

trig_integral.mw
 

Download trig_integral.mw

@vv 

I know how to get Maple to perform the integration, the question is why does the second integral require more? By hand, the same approach to integration works, particularly under the assumptions you stated. Other CASs, for example Mathematica, is able to handle both integrals easily.

@Carl Love The standard definition of continuity of a real-valued function, in a first semester calculus course is as follows. A function f:A->B, where A and B are subsets of R, is continous at a in A if:

(1) f(a) exists,

(2) lim f(x) as x->a exists, and

(3) (1) = (2)

Many books refer to this as a `continuity checklist`. This can then be modified using one-sided limits to reflect right- or left-continuity. 

@nm I completely agree with this. How do we make suggestions to the Maple engineers?

@Carl Love Ideally, I would like to restrict the domain and codomain to R. I didn't know is there was a system-wide way of achieving this. This isn't specific to just the sqrt function. It is true also for arcsine and arccosine. I have first-year college students exploring properties such as continuity using Maple. In these classes we use only the real number system, not the complex one. So, some of these results are initially confusing for the students. The follow-up discussion tends to be a good one, however. Thanks for your help!

@Carl Love Is it possible to restrict it to the real domain? 

@acer I like this approach a lot. Originally, I simply wanted the set of possible rational zeros. But the answers have given me more to think about. Much appreciated!

@Kitonum 

Thanks! Just a couple changes, in FullRandom, since you are allowing for a polynomial of arbitrary degree, the if statement should read if nops(q)=N+1. Also, in the definition of S1, it would have to be coeff(q,x,N) so as to grab the coefficient of the leading term.

This is much appreciated!

@tomleslie I am aware of the ability to export graphs from Maple to formats that are easily included in a LaTeX document. That is not what I'm looking for. I want to generate the LaTeX code that produces the desired graphs. Thanks, anyway!