ie t(P), it becomes easier:

with(IntegrationTools):
eq:= diff(P(t),t) = .48*P(t) - .028*P(t)^2 - 2*P(t)^2 / (4 + P(t)^2);
dsolve({eq, P(0)=8}, P(t),implicit);
Combine(%);
select(has,lhs(%),Int);
tP:=subs(P(t)=P,%);

tP := Int(250/_a/(-480-120*_a^2+528*_a+7*_a^3)*(4+_a^2),_a = 8 .. P);

Then, apparently, this plot gives the solution in the region 0<t<15:

plot(tP,P=6..8);

See eg ?MapletsOverview and this blog.

Check ?LieAlgebras

Beyond the reason, to get a single 'cos' term you can do this:

diff(y(x),x$2)+4*y(x)=cos(2*x):
dsolve(%):
s:=collect(%,cos);

s := y(x) =(_C1+1/8)*cos(2*x)+sin(2*x)*_C2+1/4*sin(2*x)*x 

subs(select(has,rhs(s),cos)/cos(2*x)=_C3,s);

y(x) = _C3 cos(2 x) + sin(2 x) _C2 + 1/4 sin(2 x) x

For some expressions the output pasted between <maple> tags produces a correct image, and for some others it fails. I test the output in preview mode. In case of failure, I paste  the 2D ascii output produced by setting interface(prettyprint=1).

Or look for a workaround...

1/2*int(2*f, x = -1 .. 1.);

Probably attributes are a better option (I think that it might be better if the assume facility were rewritten in terms of them). Nevertheless, it seems possible to unassume without a priory knowledge of their names.

Properties are stored in the table `property/object`. The session starts with some "system" properties, put them in the set S0, and your 'assume' statements add entries to this table, generating a set S1, say. So, you can get those that were added and their indices:

restart:
op([1,2],`property/object`):
S0:=convert(%,set):
assume(x>0);assume(y<0);
op([1,2],`property/object`):
S1:=convert(%,set):
S1 minus S0;
B:=map(lhs,%);

{x = RealRange(Open(0),infinity), y = RealRange(-infinity,Open(0))}

B := {x, y}

Then you can "unassume" them by evaluating to a name their entries.

for u in B do
`property/object`[u]:=evaln(`property/object`[u]);
end do;

`property/object`[x] := '`property/object`[x]'
`property/object`[y] := '`property/object`[y]'

about(x);about(y);

Originally x, renamed x~:
  nothing known about this object

Originally y, renamed y~:
  nothing known about this object

just for those statements that need assumptions. If you can use it, you will have no need to "unassume" anything.

with(numapprox):
with(orthopoly):
J:=Int(sin(u^3)*tan(u),u=0..x);
f:=x->evalf(Int(sin(u^3)*tan(u),u=0..x)):

f(.1);
                                         -5
                          0.2004776617 10

p:=pade(f(x), x, [5,5]);

            5
           x
-------------------------
  /          2    23   4\
5 |1 - 5/21 x  - ---- x |
  \              1323   /

chp:=chebpade(f(x), x, [5,5]);

                                     3                 5
-0.0001075954404 x + 0.002076036171 x  + 0.2184405434 x
--------------------------------------------------------
                                2                 4
    1.148539583 - 0.3990232808 x  + 0.1359254864 x

evalf(eval(p,x=.1));
                                         -5
                          0.2004776764 10

evalf(eval(chp,x=.1));
                                          -5
                          -0.5678239432 10

plot([f(x),p,chp],x=0..0.2,color=[red,green,blue]);

Presumably you should read the papers listed in ?convert/ratpoly > References.

This article at MathWorld seems useful.

Tried in Maple 8:

with(GUESS):
L:=[seq(1/(n^2+1), n=1..10)]:
Guess(L,one);

    1
----------
         2
1 + _i[0]

also works:

evala(a);
                                   1/2
                               14 2

with(MultiSeries):
series(binomial(n,k),n=infinity,0) assuming n::posint,k::posint;

O((1/n)^(-k))

In Standard GUI, another typesetting trick:

isolate(subsindets(eq,integer,u->nprintf("#mi(\"%A\")",u)),x);

occurs here:

14/7;
2

You cannot avoid it.

gives exactly the same solution {x=63, y=-384, z=780} as Maple 11. On the other hand {x=39, y=-216, z=420} is not a solution. Think about this system of equations geometrically.