People say that Maple is leading when it comes to solving symbolical solutions to differential equations. On the other hand, the Mathematica team has more developers afaik, so they probably have some overall advantage.

u := a+b*x+a/x+b*x^7+c;

collect(u,{a,b,c});

Maple has no specifically for this purpose designed command. You have to guess which parts of your function could limit the (real) domain IR and restrict accordingly.

Here, for the root function to be >0, solve the inequality.

solve(16-x^2>=0,x);

One way is to use seq instead:
seq(i^2,i=1..10);

" think you have in mind "transform" by "trafo". Isn't so?"

yes. Since the transformation into the s-domain was successful (resulting equation could be solved algebraically) , I hoped the rest would be easy, I was wrong.

1.)

eval((x^2+y^2)^2=x^2-y^2,[x=cos(t)/(1+sin(t)^2),y=sin(t)*cos(t)/(1+sin(t)^2)]):simplify(%);
Is it this what you mean?

2.)

There is a convert,cos command, but it works silly, it just adds  pi/2 to the argument of sin. Better keep in mind that cos^2x+sin^2x=1

3.)

combine(cos(3*x)^3*cos(5*x));

restart:
eval((x^2+y^2)^2=x^2-y^2,[x=r*cos(t),y=r*sin(t)]);
simplify(%);solve(%,r);
plots[polarplot](sqrt(2*(cos(t))^2-1),t=-Pi..Pi);

I don't think it is possible to explicitly solve this equation for omega, for arbitrary parameters.

solve(alpha*x - konjugate(alpha)*x=1,x);

:-)

For some control, click insert->page break

wrong:
xcir(t):=0.3 + 0.15*sin(t):
better:
xcir:=t->0.3 + 0.15*sin(t):

inttrans[fourier](velocity, t, omega);

When writing text, you can toggle to math input by pressing F5.

f:=s->a+b*s+c*s^2;
solve([f(0)=0,D(f)(0)=0,(D)(f)(5)=1]);