"s" is my variable, not yours :-)

 1/2*Int(Int(
   (1-cos(k*Pi*x)*cos(n*Pi*y))/(sin(1/2*Pi*x)^2+sin(1/2*Pi*y)^2),
   x = 0 .. 1),y = 0 .. 1);
 Change(%,{x=arccos(r)/Pi, y=arccos(s)/Pi}):
 subs(cos(k*arccos(r)) = T(k,r), %):
 subs(cos(n*arccos(s)) = T(n,s), %):
 combine(%): simplify(%): 
 ``=combine(%, Int);
 rhs(%): GetIntegrand(%): GetIntegrand(%): 
 #rationalize(%): simplify(%, size):
 h:=unapply(%, r,s);

                  1    1
                 /    /
                |    |   1 - cos(k Pi x) cos(n Pi y)
           1/2  |    |   --------------------------- dx dy
                |    |         Pi x 2       Pi y 2
               /    /      sin(----)  + sin(----)
                 0    0         2            2


           1    1
          /    /
         |    |              T(k, r) T(n, s) - 1
      =  |    |   ------------------------------------------ dr ds
         |    |     2    2     1/2    2     1/2
        /    /    Pi  (-s  + 1)    (-r  + 1)    (-2 + r + s)
          -1   -1


                                T(k, r) T(n, s) - 1
      h := (r, s) -> ------------------------------------------
                       2    2     1/2    2     1/2
                     Pi  (-s  + 1)    (-r  + 1)    (-2 + r + s)

 h(r,s):
 subs(k=3,n=3, %):
 
 eval(%, T=orthopoly[T]):
 Int(%, r=-1..1):
 value(%) assuming -1 < s, s <1: #simplify(%, size);
 
 Int(%, s=-1..1):
 value(%);
 
                                 46
                                -----
                                15 Pi

mial@mostafajani 

Then I either would use map(convert, theMatrix, string) or similar and then use the StringTool command like Kintonum.

Or probably I would try to export as text or RTF and use a text processor (like Word or an editor), which is fast, easy and painless - though I almost never use the ugly 2-dim Maple notations.

and I would add "simplify(%, size)" to vv's sheet, though it would make it to start with a '1'.

@Preben Alsholm 

I think it is a common error of Students with Math to call f(t) a function.
And then writing f(t)(t) in trying to correctly use it. Actually I learned
at Algebra to write f: x |--> something (plus domains) to avoid sloppyness.

@tazatel you want i * (...), not i(...)

If you see an error message like that you may feed some test numbers for N to see whether the expression evaluates to a number - else it would not make sense - see your error above

That Maple interface is not very friendly ...

Looking at it I have doubts that the system (with many parameters) can be solved by Maple. May be one should try inverse trigonometrics (with assumptions on real ranges)

PS: I always hate that kind of typing the input in the messy interface ... is walpha a variable or is it w*alpha ?

@ecterrab Thumbs up!

To rephrase (?): a classical coding style 0 <= a and a <= 1 should do, since using " <= " automatically tells Maple to use Reals.

@Carl Love Thank you, I agree.

Here is another way. It is quite natural to write the task as a cosinus
transform by reducing the input of cos to a variable (c.f. the approach
given by Robert Israel). Then it is a "half Fourier transform":

Int(f(x)*exp(omega*x*I),x = 0 .. infinity), f = D(LambertW), omega=1.

I remembered that years ago I 'played' with an approach given by Ooura
(one of the tough guys for the double exponential integration) and his
code and paper T. Ooura, A double exponential formula for the Fourier
transforms, Publ. RIMS, Kyoto Univ., 41, (2005).

http://www.kurims.kyoto-u.ac.jp/~ooura/papers/DE-FT-gen.pdf

For limitations one has to read the paper, for omega off ~ 1 one would
have to work and there are problems if f is oscillating as well.

I once put that to Maple and expanded it for complex valued f. Find it
appended. It is not very fast (I wanted to learn), and hopefully I have
cleaned up my old code correctly (may be 'some' sign problems).

The other stuff for highly oscillating integrals (Iserless or Olver)
seems to have gone after switching to a newer PC :-(

DEFT_complex.mws
DEFT_complex.pdf

@vv Yes and that might be due to Dirk Laurie (he wrote the chapter and does note cite anybody for it, which he does otherwise)

But what about the suggested modification as a test (of course Maple can not do it out of the box)?

Edited: I received a private mail that this solution is reminiscent of posts by usenet poster "CW" (search "Google" groups within sci.math.symbolic)

My first thought was: have they implemented all the 10 tasks or have they implemented a method?

Can you give a modified task on which one can test that?

Hm - are you sure about convergence?

Fine. But please note that there are different results given by MMA and
Maple, that's why I asked:

M_MMA:= " <what Markiyan posted as result of MMA on 08 Nov 2015>  "

Then 

convert(M_MMA, FromMma):
subs(k1=1, k2=0, R=2.3, %); evalf(%);

    -.473398684009699*I

While I said (above) it is -.693587157054838*I using Maple's symbolic
answer and a direct numerical computation of the integral.

Yes :-(

Anyway: has somebody done some (more) numerical checks?