Do you really want to differentiate with respect to a natural number? Maybe you want fracdiff.

@Rouben Rostamian  

Yes, the existing numerical method is not stable, but a solution exists (even C^oo).
Must we change a sign and solve another problem, just because it has a stable numerical method implemented?
Maybe the method will work with some adjustments: in our example, adding 'abstol'=0.1  ==> OK for t=0..3.
Or, maybe a change of variables will help.

@Rouben Rostamian  

I don't agree entirely with your argument. The backward heat equation could work quite nicely sometimes.

For example,

sol:=pdsolve(diff(u(x,t),t) = -diff(u(x,t),x,x), {u(x,0)=sin(x), u(0,t)=0, u(Pi,t)=0}, numeric):
sol:-plot3d(u(x,t), x=0..Pi, t=0..1);

It fully agrees with the exact solution u(x,t)=sin(x)*exp(t).

@sgils1 

Your problem is actually an integral (so, instead of dsolve, it's possible to simply use int).

Unfortunately, Maple does not find a continuous antiderivative in the symbolic case i.e. it is valid only locally.
(This is not a bug, it is documented; for example int(1/(2+cos(x)), x) is discontinuous and it is exactly the integral obtained by hand using the substitution tan(x/2)=t; note that for this example there is a simple continuous antiderivative, but in general a CAS does not try to find it).

For example:

Here, (3) and (4) are continuous, but (2) is discontinuous.
The interesting fact is that (3) is also valid for a<0, but Maple does not care.

For your simple example, there are workarounds (looking for discontinuities and adding constants if needed; or use assuming).

@torabi 

Not quite useful. I was hoping to see an example where both input and output are Maple expressions.
I see instead a problem which must be formalized and has lots of unessential parameters such as lambda, mu, J2 etc which seem to be constants.
I don't know what the circled triple integral means. What is V? Why that change of variables? It is not clear what quantities are constans. I don't understand the assumption (4) ; Aijk are unique? Are Jijk etc given? etc, etc.
In conclusion, I cannot help here.

@torabi 

It would be useful to see:
1. A short example (just the example, no subs)
2. A description of what you want to obtain
3. The expected result.

If 1-2-3 are clear enough, then it will be probably possible to find a general method for doing this.

@tomleslie 

Actually the integral is oo (diverges).

@MacDTU 

Your f(t) := int(1/x, x = 1 .. t) does not define the desired function (Maple procedure).
[Technically it defines an "empty" procedure having an entry in the remember table].
(This is a standard mistake for a beginner.)

The correct definition (in 1D math mode) is

f := t -> int(1/x, x = 1 .. t) ;

When I saw the problem I used the substitution z = 1 - x and I found the first term of the series by hand
z ~ LambertW(4*n^2)/n.

Note the the series for z fails too.

@Gourav 

After correction, you do not have any more singularities for omega>0, but omega=0 still remains and makes the inner integral divergent.

Note that the integral cannot be computed symbolically. In general we need numerical values for parameters to compute numerically (not a very simple task for an improper double integral).
 

@student_md 

if ... then ...   is a conditional statement, `if` is a condional function (alias to ifelse), see  ?ifelse.

Do you think that the integral is convergent? I don't.

@Carl Love 

There are many such problems in number theory and most of them are unsolved.
E.g. it is conjectured that n! + 1 is not a square number for n>7.

UMG:=proc(n::posint)
local N := Normal(0,1),
      S := Statistics:-Sample(N,[n,n]) + I*Statistics:-Sample(N,[n,n]),
      j;
Matrix(LinearAlgebra:-GramSchmidt([seq(S[..,j],j=1..n)],'normalized'))
end:

@Joe Riel 

Yes, you are right.