In Maple V you could also set the _EnvExplicit environment variable before calling solve on this example.

restart;                             

_EnvExplicit:=true:

solve({x^2+y^2=3,x^2+2*y^2=3},{x,y});

                                1/2                 1/2
                   {y = 0, x = 3   }, {y = 0, x = -3   }

That's one alternative to using the allvalues command on the result (for this example) where the RootOf appeared, which has already been given in another Answer.

Since you're apparently already using a 3-component custom color for the shading of the surface, how about making that produce "black" for a controlled subset of the (x,y) points?

That is, with your HSV coloring expressions, you could set the V component to 1 or 0 in a piecewise fashion. The resulting black lines would be part of the actual surface, and not be hidden. And you could have as many or as few as you wish, regardless of the fineness of the grid option.

Here's a simple example,

3dcolorgrid.mw

You may choose to further restrict the thickness (width) of the lined black portions -- using derivative values -- if the surface becomes very steep.

The name e has no special meaning for Maple as 1D plaintext input. It's just another unassigned name.

As plaintext code you'd need to enter that with exp(x) instead of e^x, if you want to raise x to the base of the natural logarithm.

There document you attached seems to have a corrupted and unclosed Input at its end.

If I delete that problematic fragment and close off the outstanding Sections then I get to the following (which I'll try and attach both zipped and as is). It end with the subsection "Lektion 18 Tyngdepunkt af volumer".

I saved it last with Maple 2017.2 (since yours was last saved with Maple 2017.1).

1._sem_Maple_-_Kopie_ac.zip

1._sem_Maple_-_Kopie_ac.mw

restart;
de := diff(y(t),t,t)=-1:
ic := y(0)=1, D(y)(0)=0:

Events := [y(t)=0,
           diff(y(t),t)=piecewise(abs(diff(y(t),t))<1e-2,
                                  abs(diff(y(t),t)),
                                  abs(0.5*diff(y(t),t)))]:

dsol := dsolve({de, ic}, numeric,
               events=[Events], range=0..5):

plots[odeplot](dsol, thickness=3, color=red);

The simplification that you are expecting is not generally valid. It is not generally valid when Z<0 . That's why Maple is not combining those powers without special direction.

The symbolic option to the simplify command instructs it to ignore such branch cut distinctions. I prefer to provide the special directions in the form of assumptions, so that it's more clear to me just what circumstances are supposed to be allowed. (Of course I still have to be careful about respecting those assumptions.)

For example,

Download radsimp.mw

For your posted example you could do one of,

simplify(combine(req)) assuming positive;

simplify(combine(req),symbolic);

Download areaprob.mw

The phi obtained from GetProperty(EQUATION1, expression) is the unassigned global name phi, not numtheory[phi].

I don't know whether you want to rethink your definition of FTest.

MAPLE_PLEASE_HELP_aGAIN_ac.mw

How about considering it as an integral over y=-1..1, ie,

int( 2-y^2 - y, y=-1 .. 1);

                       10/3

The above is a bit easier than splitting the domain at x=1 and adding the two portions, while integrating over x=-1..2.

int(x-(-1), x=-1..1) + int(sqrt(2-x)-(-sqrt(2-x)), x=1..2);

                       10/3

I'm supposing that you have realized that the first step is to solve for the upper point where the curves intersect, since that gives you information about the bounds of integration.

solve(y = 2-y^2, {y});

                {y = 1}, {y = -2}

@mpre 

restart;

kernelopts(version);
   Maple 2016.2, X86 64 LINUX, Jan 13 2017, Build ID 1194701

ff:=(sqrt(1/beta)+1)*beta/(beta+sqrt(beta)):

normal(combine(expand(ff))) assuming beta>0;
                               1

evala(combine(ff)) assuming beta>0;
                               1

simplify(expand(ff), symbolic);
                               1

The expression assigned to ff is not equal to 1 for real beta<0.

It's not always easy or practical, but I usually try and avoid implicitplot3d since it has several drawbacks:

1) The surface looks very rough unless a very large number of points is used, and even then it still looks rougher than a usual plot3d surface.
2) With a large number of points manual rotation of the plot is highly problematic in the Standard GUI. Attempting a manual rotation usually means I'm going to have to wait for the GUI to pause for a minute or so.
3) With a large number of points I often have to wait a few minutes if I right-click on the plot, and sometimes the context-menu doesn't appear at all (so I can't export to .png using the GUI's drivers).
4) The plot structure doesn't actually contain data that delineates the surface, and so it is of little or no use in terms of exporting the plot data for use by a 3rd party application.

The following produces Pgrid as two conjoined surfaces produced by plot3d. It behaves reasonably for me under manual rotation and (discounting the seam at z=0) is reasonably smooth.
 

Download impl3d_m.mw

The results embedded by DocumentTools:-Tabulate will always appear separately from any regular output (including that produced by print), if executed within the same Document Block (paragraph) or Execution Group. All the regular output will always appear together. It cannot be done exactly like you are asking, with regular output split to be partly before and partly after the tabulation.

This is because an Execution Group has just one Output area, and just one embedded Task area, and they are separate.

You might consider a different approach. Could you make those `continue` text pieces be part of the tabulation? Or are you relying on the output appearing piece by piece over time, as more intermediate results get computed?

Side note: It is wrong to have those calls to with inside the procedure body. It won't work reliably as you'd expect. If you want it inside the procedure body then either use the long name like DocumentTools:-Tabulate or change the with calls to an appropriates uses call.

The subexpression in red, containing diff(f(eta),eta,eta,eta), appears more than once in your full expression (although in one other instance it is part of a product).  I'm guessing that you want to isolate that subexpression completely, to the LHS of an equation.

If instead you only want to isolate for the single instance (in red) of that third derivative then please clarify. In that case it's a minor adjustment to define temp:=isolate(expr,S) instead.

You haven't mentioned what criteria you have for simplification of the RHS of such an equation. Do you want the numerator of the RHS to be collected according to the remaining derivatives, or just to be "small"?

Here are some ideas. I haven't tried to utilize any assumptions (eg. to combine radicals, if it might help) because you've given us no such information.

Download something.mw

restart;

kernelopts(version);

     Maple 2017.2, X86 64 LINUX, Jul 19 2017, Build ID 1247392

eq:=-44.51913564*sinh(sqrt(x))*x^5*sin(sqrt(x))*cos(0.6232678986e-1*x)
+5.872275982*x^(11/2)*cosh(sqrt(x))*sin(sqrt(x))*cos(0.6232678986e-1*x)
+5.872295982*x^(11/2)*sinh(sqrt(x))*cos(sqrt(x))*cos(0.6232678986e-1*x)
-0.1e-5*x^6*sinh(sqrt(x))*sin(sqrt(x))*cos(0.6232678986e-1*x)
+11465.08352*x^6*cosh(sqrt(x))*cos(sqrt(x))*cos(0.6232678986e-1*x)
-0.10000e-4*x^(11/2)*sin(sqrt(x))*cos(sqrt(x))*cos(0.6232678986e-1*x)
+.1246535797*cosh(sqrt(x))*x^4*cos(sqrt(x))*cos(0.6232678986e-1*x)
+158.9969129*x^(9/2)*sinh(sqrt(x))*cos(sqrt(x))*cos(0.6232678986e-1*x)
-94.84329962*cosh(sqrt(x))*x^5*cos(sqrt(x))*sin(0.6232678986e-1*x)
-0.2e-2*x^7*sinh(sqrt(x))*sin(sqrt(x))*sin(0.6232678986e-1*x)
+0.4000e-2*x^(13/2)*cosh(sqrt(x))*sin(0.6232678986e-1*x)*sinh(sqrt(x))
+0.10000e-4*x^(11/2)*cosh(sqrt(x))*sinh(sqrt(x))*cos(0.6232678986e-1*x)
-158.9969129*cosh(sqrt(x))*x^(9/2)*sin(sqrt(x))*cos(0.6232678986e-1*x)
+38209.64552*sinh(sqrt(x))*x^6*sin(sqrt(x))*sin(0.6232678986e-1*x)
-3761.932636*x^(13/2)*cosh(sqrt(x))*sin(sqrt(x))*sin(0.6232678986e-1*x)
-3761.924636*x^(13/2)*sinh(sqrt(x))*cos(sqrt(x))*sin(0.6232678986e-1*x)
-0.4000e-2*x^(13/2)*sin(sqrt(x))*cos(sqrt(x))*sin(0.6232678986e-1*x)
-2.*10^(-7)*x^(13/2)*cosh(sqrt(x))*sin(sqrt(x))*cos(0.6232678986e-1*x)
-11465.08352*x^6*cos(0.6232678986e-1*x)+.1246535797*x^4*cos(0.6232678986e-1*x)
-94.84329962*x^5*sin(0.6232678986e-1*x)+0.1e-5*x^6*cosh(sqrt(x))^2*cos(0.6232678986e-1*x)
-0.1e-5*x^6*cos(sqrt(x))^2*cos(0.6232678986e-1*x)+0.3e-5*cosh(sqrt(x))^2*x^6*sin(0.6232678986e-1*x)
-0.2e-2*x^7*cos(sqrt(x))^2*sin(0.6232678986e-1*x)+0.2e-2*x^7*cosh(sqrt(x))^2*sin(0.6232678986e-1*x)
+2.*10^(-7)*x^(13/2)*sinh(sqrt(x))*cosh(sqrt(x))*cos(0.6232678986e-1*x)
+2.*10^(-7)*x^(13/2)*cos(sqrt(x))*sin(sqrt(x))*cos(0.6232678986e-1*x)
-2.*10^(-7)*x^(13/2)*sinh(sqrt(x))*cos(sqrt(x))*cos(0.6232678986e-1*x)
+1.159305284*10^5*cosh(sqrt(x))*x^(11/2)*sin(sqrt(x))*sin(0.6232678986e-1*x)
-1.159305284*10^5*x^(11/2)*sinh(sqrt(x))*cos(sqrt(x))*sin(0.6232678986e-1*x)
-8.359616334*10^6*x^7*cosh(sqrt(x))*cos(sqrt(x))*sin(0.6232678986e-1*x)
+8.359616334*10^6*x^7*sin(0.6232678986e-1*x) = 0:

#plot(lhs(eq),x=0..0.2,-1e-7..1e-7);

Digits:=10:
rt[1]:=0.0:
for i from 1 to 5 do
  rt[i+1]:=RootFinding:-NextZero(unapply(lhs(eq),x), rt[i]);
end do:
convert(rt,list);

     [0., 0.04191897329, 0.1445773188, 22.37101613, 50.40561001, 61.67210358]

Digits:=10:
op(sort(simplify({RootFinding:-Analytic(eq,x=0-0.1*I..100+0.1*I)},zero)));

     0., 0.04191897312, 0.1445773137, 22.37101614, 50.40561000, 61.67210360

Digits:=20:
op(sort(simplify({RootFinding:-Analytic(eq,x=0-0.1*I..100+0.1*I)},zero)));

     0., 0.041918973291358269200, 0.14457731887993756592, 
     22.371016133870070862, 50.405610019621053775, 61.672103588567668205

posroots.mw

You could use the explicit location of the .mpl file in the script file, eg,

/Library/Frameworks/Maple.framework/Versions/2017/bin/maple /Users/myname/Desktop/maple_test/test.mpl > output.txt

Note that output.txt may still appear in the directory from which you call test.command , but that may be what you want.