It is not stated in ?solve,identity but, apparently, the list of variables can be omitted. Here:

solve(identity(func_num = func, x));
  {c = 3.660000000, a = 1.330000000, b = 2.440000000}

int(cos(m*x)*sin(n*x), x = 0 .. Pi) assuming m::integer,n::integer;
collect(%,n);
map(simplify@subs,[m+n=k1,m+n=k2],%) assuming k1::even,k2::odd;

                                             2 n
                                       [0, --------]
                                             2    2
                                           -m  + n

int(cos(m*x)*sin(n*x), x = 0 .. Pi) assuming m::integer,n::integer;
collect(%,n);
map(simplify@subs,[m+n=k1,m+n=k2],%) assuming k1::even,k2::odd;

                                             2 n
                                       [0, --------]
                                             2    2
                                           -m  + n

Other topologies and curved manifolds would come after R^n...

My point of view is more close to differential geometry. In Physics it is typical to exploit the symmetries to simplify the calculations: eg look whether a system changes under a rotation or a translation, ie the action of a  transformation group on a configuration space.

Certainly the issue here is not invariance or transformations on the solution or jet space of differential equations.

And most  probably not  algebraic geometry. This  is a field  of which  I have just a  vague idea.

To learn by experience which is the best coordinate system for simplifying the calculations is what students do. But for an algorithmic based CAS another approach is needed. In particular if profit is to be get from "nonstandard" coordinate systems.

In applications, many of the integrals that are done analytically can be done because  they can be factorized.

What do you mean by "1-dim Fourier problems or periodics"?

In regards to Jacques ``rectifying transformation'', it is within the subject of general coordinate transformations ie differential geometry. Indeed, general coordinate transformations is a big issue.

of this first example seems quite simple and probably, for linear changes of coordinates, it would be not much more complex and profitable to generalize a little bit to regions bounded by segments  as  straight lines  map into straight lines by these transformations.

Eg a quadrilateral figure whose sides are given by equations of the form a_i*x+b_i*y=c_i, i=1..4, such that four pairwise intersections (corners of the quadrilateral exist).  Your square is a particular case.

Let  X be the column matrix of the coordinates (x,y), U idem for coordinates (u,v), A the transformation matrix such that X=AU, B the matrix of the coefficients (a_i,b_i), C the column matrix of the coefficients c_i, such that BX=C.

Then, the equations for the sides of the transformed quadrilateral are BAU=C. This seems trivial.

In regards to conditions, this needs thinking a bit. A necesary condition for the region to be a quadrilateral is that the triples (a_i,b_i,c_i) are not multiple of each other. A similar conditions should apply replacing with the coefficients of the matrix BA. An additional condition is that no three lines intersect at a point.

Is this what you point to?

I see in this patent info some background.

http://en.wikipedia.org/wiki/Gaussian_integral

http://en.wikipedia.org/wiki/Gaussian_integral

For D>=3 you do not have isomorfism with C. So, in general, you need to think geometrically and you need to identify the coordinate system from knowledge of the symmetry properties. As far as these symmetry properties are computable (I beleive that they are in many cases of interest), there should be no problem to identify the coordinate system.  

Let us take this example. What needs to be done first is calculate the integral over R^2:

Int(Int(exp(-x^2-y^2),x=-infinity..infinity),y=-infinity..infinity);

So, a sketch of an algorithm would be:

1. Recognize that the integrand is invariant under SO(2).

2. Verify that the domain R^2 is invariant under SO(2).

3. Identify the polar coordinate system (r,theta) as the orthogonal coordinate system  "adapted" to SO(2): the curves of constant r are the orbits of SO(2).

4. Transform the integral to these polar coordinates, factorizing it as a product of an integral on r and and one on theta.

5. Calculate these 1D integrals and multiply them.

May be that someone with an algorithmic mind could transform this sketch into something useful.

that Robert checks my typos :)

that Robert checks my typos :)

without java it would not work at all.

without java it would not work at all.