@asa12 

You should explain mathematically what wou want to obtain. Without Maple code!

The Groebner bases are related to the ideal generated by a set of polynomials. If you remove polynomials from a basis, the ideal is changed. What theoretical facts are you intending to use?

simplify needs occasionally to use evalf (for localizing rootofs etc).
If Digits is raised to 15 (for ee, or even to 11 for u), the bug disappears!

@tejolson 

evalf(u) seems to be ok here. It's only roundoff error ("catastrophic cancellation"):

evalf([op(numer(u))]);

But the bug is from acer's version. I don't see any problem with your a,b,c, except that they are a bit difficult to simplify.

@acer 

Could you tell us the source of this example and where is the bug?
(I recall that in some old Maple texts, simplify without any option was not considered very safe.)

Edit. A simplified version:

u:=(4*(28412308*I*sqrt(3)+44068556*sqrt(3)-76328978-49211561*I))/((-3-4*I+2*sqrt(3)+(2*I)*sqrt(3))^6);

evalf(u);

evalf[100](u): evalf(%);

simplify(sqrt(u));
    0

Yes, because the integral over 0..infinity is computed directly. Actually a human would do the same.

@Axel Vogt

f:=ln(x)^n*x^n/factorial(n);

int(f, x=1..x) assuming x>0;

(actually type int seems to be unrelated to integer)

@sarra 

So, you have an algebra and e[i] is a basis.

I think that a better approach would be to represent an element of the algebra sum( a[i]*e[i], i=0..n) by the vector <a[0],...,a[n]>, a[i] being the scalars.

Now, e[i] &x e[j] is also such a vector, but it seems that you do not know its  representation for all i,j. For example, what is e[2] &x e[3]  in terms of e[0],...,e[n]? It will be complicated to work in an algebra which is only partially defined.

@Rouben Rostamian  

I think I understand what OP wants:

1) e0, e1, ... are symbolic and are seen as a basis in an algebra (e.g. the quaternions)

2) * is a binary operator. In Maple it could be implemented as neutral; let's denote it by &x
It is actually the multiplicative operation in the algebra.

3) The operator  &x is bilinear and satisfies some relations, e,g. e0 &x e0 = e2

4) Using the bilinearity and these relations, expressions such as 
(e0 + 4*e3 - 2*e7) &x (-e0 + 2*e3 - 3*e4)
can be expanded and simplified using e.g. applyrule by giving the relations

5) f is a linear functional on the algebra and must be declared as such.
It is uniquely determined by f(e0), f(e1),...

Now, all OP's expressions have to be expanded and simplified.

@Bendesarts 

But in this case:
- the first answer works as well and it's shorter
- "periodic" in the title is not justified.

@Markiyan Hirnyk 

RandomSolid generates solids with no integer points (probably always or almost always).
It is easy to write a proc based on my algorithm (which I did).

P.S. It would be much better if you stop being so hostile! You have absolutely no reason to do this.

@Markiyan Hirnyk 

Of course. But this could happen with any other algorithm - this is the nature of the problem!

On the other side, your example with RandomSolid will be solved at once!

Edit.
P.S. Nothing works in (too) higher dims.

Unfortunately there are these problems:

1. LinearMultivariateSystem seems to work properly only for relatively simple systems
2. isolve is very slow and gives "Warning, solutions may have been lost"

I would recommend the following algorithm:

a.) Given the polyhedral P, find a minimal cube a1..b1, ..., an..bn  containing P.
ak, bk can be found using simplex:

ak = eval(xk, simplex[minimize](xk, P));
bk = eval(xk, simplex[maximize](xk, P));

b.) Use  cartprod for the intervals ceil(ak) .. floor(bk)
and return the elements which are in P.

It was a joke, see the :-) sign.

What I don't understand is why convert,exp acts like this.

@acer 

@Bendesarts 

For f defined in [a,b) use

f_per:=x -> f(x - floor((x-a)/(b-a)) * (b-a));

Should simplify to 0 even for complex s,t   :-)

[evalc assumes s,t real]

@acer