@one man 
My point was (is) that a "true" spiral on a cylinder does not have self-intersections.
You can convince yourself:
with(plots):with(plottools):
display(cylinder([0,0,0], 1, 6, style=surface, color=yellow),
spacecurve([cos(s/20*cos(s)),sin(s/20*cos(s)),3+s/20*sin(s),s=0..8*Pi],
color=red,thickness=5),scaling=constrained,axes=none);

Animated version:

@one man 

In Mathematics the definitions are clear. If there are non-equivalent definitions for a notion, we must say which one is used.
For example, here is something that is not spiral in the sense of "monotone curvature of constant sign".
(You should say what definition is satisfied by your spiral if your animation refers to a spiral). 

r:= t + 4*sin(t/2):
plot( [r*cos(t), r*sin(t), t=0..8*Pi]);

@one man 

Are you sure that you have a "true" spiral here? According to the definition it should have a monotone curvature of constant sign.

@acer 

It seems that codegen[optimize] is not always efficient. For

E3:=(x^2-x+2-y)^25+x+(1+y)^25:

it does not work.
[  (1+y)^25 annihilates the collect "trick" ].

@acer 

It is interesting to see how/when simplify/size is able to shrink an expression (polynomial here):

restart;
n:=75:
E1 := add(i*x[i],i=1..n)*add(i^2*x[i],i=1..n)+add(x[i]^2,i=1..n):
E2:=(x^2-x+2)^30+x+1:
E3:=(x^2-x+2-y)^25+x+1+y:
E:=[E1,E2,E3]:
                          
for e0 in E do
e1:=expand(e0):
e2:=simplify(e1,size):
print(length~([e0,e1,e2]));
od:

                      [3009, 84700, 44505]
                        [34, 1287, 1287]
                       [42, 15924, 6938]

@TomM 

Such a phenomenon is normal when working with floats.
If you convert the foats to exact rationals (i.e. 0.333 becomes 333/1000), then the coefficient of x^9 happens to be exactly 0
(btw, why didn't you rename your vars for us?, a search & replace is so simple ...).

Using floats, the coeff of x^9 may of course be a (usually) small nonzero (due to cancelations, catastrophic or not).
But in practice when a float such as -53417.4875669262 appears as input, it usually means that the actual quantity could have been -53417.48756692623... or -53417.48756692618... and in such a case the coefficient of x^9 could be of course <> 0. It is very simple!

@Kitonum 

An unpleasant fact is that the plot3d routine is not able to draw this cylinder properly
when z is given explicitely.

expr:=(x-y)^2+(y-z)^2+(z-x)^2 = 3:
z1,z2:=solve(expr,z):
plot3d([z1,z2],x=-3..3,y=-3..3,grid=[100,100], scaling=constrained);

plot3d([z1,z2],x=-3..3,y=x-sqrt(2)..x+sqrt(2),grid=[100,100], scaling=constrained);

@Bachatero 

Unfortunately this is theoretically impossible due to Richardson's_theorem.

Note that even now Maple is not useless here:

E:=-n*mu+ln(product(mu^x[i]/factorial(x[i]), i = 1 .. n)):
E4:=eval(E,n=4):
extrema( E4,{},mu,'sol'):sol;


It remains to extrapolate ...

For other curves, use Path(...) instead of Line(...). Type
?PathInt
for examples.

The integrals of P dx + Q dy ...   are line integrals. See
?LineInt

@1valdis 

As I understand (not being a specialist or working in string theory) the link refers to a projection of a higher dimensional manifold into R^3.
Before asking such a question you should provide the wanted projection, i.e. the parametric surface, preferable in Maple notation.

@acer 

Yes, I forgot about it. So, no need to return unevaluated by hand as I did.

@Bachatero 

Maple is not intended (prepared?) for such manipulations.
Just consider:

restart;
fn:=sum(x[i]^2,i=1..n);
f5:=eval(fn,n=5);

diff(fn, x[1]) assuming n::posint;
                               0
diff(f5, x[1]);
                             2 x[1]

[You can't use Maple e.g. in Functional Analysis. Here one needs pen+paper+brain for the moment].

@WilburC 

If f is a procedure (such as f:=sin) then

evalf(Int(f, x=0..1))

is incorrect because actually f has nothing to do with x.
(even if Maple could have been less strict and accept it).

Executing
evalf(Int(f, 0..1))
Maple will call f only with numeric arguments
which is not the case for
evalf(Int(f(x), x=0..1))
where f(x) is evaluated first, x being a symbol, not a numeric.
Your initial g called with a nonnumeric argument will produce an error, that's why it must be "unevaluated".
My previous version returns unevaluated when the argument is nonnumeric, so it needs not being unevaluated..

@WilburC 

Probably you have an older version of Maple.
Search for "premature evaluation" to understand the role of the quotes here.

Here is a version of g which does not need quotes.

g:=proc(a)
local y;
if type(a,numeric) then fsolve(a*y^2-sin(y),y=2)
else 'g'(a) end if;
end proc:

evalf( Int(g(x),x=1..2));