@gaurav_rs 

I think that it's too much trouble for such fine-tuning in Maple.
There are much better approaches e.g. TikZ in LaTeX.

About the converse: it is theoretically possible to write a CAS in LaTeX (which is after all a programming language);
of course nobody is going to do this.

Sorry, I do not have this information. I have used it only occasionally, just instead of soving by hand.

BTW, it would be interesting a speed comparison versus Sage.

@marekszpak 

OK, but the algorithm is not very complicated and the code is clean, see e.g.

showstat(LinearAlgebra:-HermiteForm);

I'd suggest to test it before trying to program a better one.

@maple2015 

You must change the BC conditions. As stated the problem cannot be solved because w(x)=0 satisfies the ODE and the BC ; so, for psi you only have the condition psi'(K/2) = psi'(-K/2) = 0, which of course is not enough (the infimum of phi is obviously 0).

@maple2015 

The second objection remains. You cannot speak about "the" integration constant.
For example, the integral of 1/(x+1) can be written as ln(x+1) + C or  ln(2*x+2) + C  or ...
Also, what is this condition when w(x)=sin(sin(x))?

Now about the new formulation. In this case the problem seems to be without solution: taking y=0 the ODE for w(x) has order 2.

The formulation of the problem is not clear.
1. Which are the independent variable(s)? w = w(x) and psi=psi(y)? G seems to be constant (at least in the worksheet).

2. What do you mean by

(it is senseless, being defined up to an additive constant).

3. Explain your solving method mathematically in order to follow the Maple code.

@kuwait1 

Why should Maple give a warning? It's a legitimate integral but hard to compute.

Note that the argument of the hypergeometric function is not in the unit disk and it is considered by analytic continuation; also a branch point is present.

Even if we alter a bit the parameters of the hypergeometric function to [1/2, 1/2], [3/2] (in order to make an idea of its nature) we get something like:

Int(Int(arcsin((1/2)*sqrt(2)/(k-(1/5)*r))*(k-(1/5)*r)/(k+r)^4, k = 0 .. 10), r = 1 .. 2);

It's not simple but evalf works.

Did this function appeared from some computations or you simply took a more or less random complicated function with singularities?

@John Fredsted 

Actually the answer of solve is correct but incomplete: a=b-5,  but for b in 2..8 only!
A similar situation occurs for
solve(arcsin(sin(x))=x,x);
      x

@Markiyan Hirnyk 

Yes, it's a typical conversation with you.

I know what I said and it is true, see Fubini/Tonelli. No need for screenshots.

@Markiyan Hirnyk 

Ypu forgot theta = Pi/6.

@Markiyan Hirnyk 

f is OP's integrand of course.

@Markiyan Hirnyk 

A brief inspection confirms divergence:
int(f, k=1..10) assuming cos(p)>-1, cos(p)<-1/2, Z>0, Z<1/10;
         infinity

But the justification for extrema is easy: the feasible region is compact and the function is differentiable there.

@Kitonum