@mmcdara Yes, there is even a "chaotic" billiard.
For large rational m,n, the curve appears to be dense due to the physical thickness of the line.

plot([sin(127/11*t), sin(63/5*t), t=0..200*Pi], view=[(0.50 .. 0.51) $ 2]);

(but the plotting algorithm is not very accurate for such graphs).

Note that 
{ (sin(127/11*t), sin(63/5*t)) | t in R } =
{ (sin(127*5*t), sin(63*11*t)) | t in [0, 2*Pi] } 
so, it is a compact set (hence not dense).

@mmcdara  Actually the curve is dense (for t in R) iff m and n are incommensurable i.e. m/n irrational (e.g. m = sqrt(2), n = 1).

@Zeineb They have the same values, so one of the graphs will cover the other.
To see both, you may use the transparency, e,g.

plot([(x+1)^2, x^2+2*x+1], x=-2..1, 
color=[red,green], transparency=[0,0.5], thickness=[4,15]);

@AHSAN You should be aware that an antiderivative has very seldom a closed form. E.g.
int((4*x + 3*x^2*exp(x^3))/(a*x^2 + exp(x^3))^2, x)

is elementary only for a=2 (probably).

@AHSAN You have changed the expression!

V:=simplify(v); # v is your original expression

@Ugurgozutok If you know that g divides f, simply compute the remainder rem(f, g, x) (or rem(f, g, y)) and equate its coefficients with 0.

They are called Lissajous curves (see ?Lissajous and the wiki article) and are usually plotted via

plot([sin(4*t), sin(5*t), t=0..2*Pi]);

However, when x0 is a pole, the order corresponds to the number of terms:

series(1/x^7*exp(x),x=0,3);

        1/x^7 + 1/x^6 + 1/2/x^5 + O(1/x^4)  

@tomleslie Without this "main" condition the problem is trivial: a and x are arbitrary (x<>a) and
m/n = a^2*x^2/(x-a)^2+x^2 

@tomleslie You forgot about the main condition: four integer roots.

restart;
eval(a^2*x^2/(x-a)^2+x^2 = m/n, {a = 21, m = 15616744, n = 73}):
solve(%);

    22, 462, -221 - sqrt(58123), -221 + sqrt(58123)

(isolve is useless here)

@Maple_lover1 Now, for a=1, the two formulae agree for B = r*Pi/k * 2*n/(n-1) but w2 is set to 0 when the argument of sin is not in the interval (-Pi,Pi). Note however that in Maple w2 will be complex when sin(...) < 0 (for n>3).

@mmcdara In the last example, the warning is for g which will be (implicitly) local in the procedure f. X is global in g. Without declaring global X: (in f) , X would be local in g (because you have an assignment X:=X+1). 

@minhthien2016 If you are interested in programming, don't use the document mode.
Add ";" after restart (it should be in a separate execution group).

@tomleslie Thank you. I also had normal results with Usage, but I am puzzled (and the question is) about the ratio t2/t1 and the role of the GUI (with/without output). On a laptop with Maple 2018 I have t2/t1~1.

@Carl Love I was always unhappy with Maple terminology. In math classes, the educators struggle to explain that sin is a function and sin(x) is not; in Maple it's "upside down".