@Kitonum It's sad that Maple cannot simplify:
simplify(tan(x+k*Pi))    assuming   k::integer;
simplify(sin(x+2*k*Pi)) assuming   k::integer;
 

So, you have two m x n matrices A and B.
You need a (column) permutation matrix P and a (row) permutation matrix Q  such that Q.A.P = B, if such P, Q exist.
(actually, it seems that you are interested only in P, and for Q the existence is enough).
Is this correct? Do you need all the possibilities for P?
 

@JAMET Then, do not assign X,Y:

[X = (a*m^2 + 2*m^2*p + 2*p)/(2*m^2), Y = a/(2*m)]:
eliminate(%, m);

(a parabola)

Actually, in modern mathematics these symbols are considered redundant.

You've got four answers and no reaction. It's not a polite attitude!

@Earl The method is mentioned in the help page for EulerLagrange (for the case of a single function). They are not important, but they could simplify the computations sometimes.

In Windows both examples work fine.

@mmcdara The Riemann sphere is defined via the stereographic projection, see https://encyclopediaofmath.org/wiki/Riemann_sphere

It does not introduce singularities. Unfortunately it will not help  the OP in better understanding complex functions.

Int(cos(x)/(x^2+1),x=-infinity..+infinity) = 2*Pi*I*residue(exp(I*x)/(x^2+1), x=I);

        

@janhardo Edwards's book contains several obsolete notations including some used originally by Riemann himself. It is not a good idea to try to learn Complex Analysis from such a source (even if the book is a valuable monograph).   

You are a beginner in Complex Analysis. So, you should start with standard exercises. In this one, the path must be exactly defined; probably here we actually have a limit of a complex integral (e.g. tha radius of the arc around the origin tends to 0). 
The branches could be a problem; maybe it is not the Maple principal branch.
Not to mention z; the integral could diverge for some z, depending on how the path is defined.

dx / x is just for a convenient way of writing the integral: ∫ f(x)/x dx =  ∫ f(x) dx/x.

BTW, the notation Int( ..., x = +oo .. +oo)  is very exotic. Where did you find it?

@John2020 The methods do not work when the parameters depend on t; actually in such a case the problem does not make much sense.

@John2020 see above

@John2020 The method in my answer can be adapted here too.

restart;
r := x*(diff(theta(t), t))^2+y*(diff(varphi(t), t))^2+z*(diff(theta(t), t$2))+w*diff(varphi(t), t$2)+p*m:
g := (4*(f+T))*(diff(theta(t), t))^2+u*(diff(varphi(t), t))^2+(f+9)*(diff(theta(t), t$2))+4*s*diff(varphi(t), t$2)+p*4*cos(varphi(t)):

eval(r-g, [theta=(t->1+t+t^3), varphi=(t->1+t+t^7)]): # or similar
[coeffs(convert(series(%, t), polynom), t)]:
solve(%, [x,y,z,w,p]):
solve(%[1], [x,y,z,w,p]);
#      [[x = 4 T + 4 f, y = u, z = f + 9, w = 4 s, p = 0]]
simplify(eval(r-g, %[1]));   #check
#                               0

Note that you cannot take m as parameter; it must be p (it appears in both r and g); for a fixed p there is no solution.

@ecterrab  

eq:=sin(x)+cos(x) = a*sin(d*x^2+c*x+b):
solve( identity(eq, x), {a,b,c,d} );  # works
PDEtools:-Solve(eq, {a,b,c,d}, independentof=x); # must be interrupted