1. Open the attachment above in 2023.

2. do

3. Try to execute the whole document

@sursumCorda 

I am mouth open for two reasons:

• First, I stopped my “elliptic investigations” at the point where I felt the need to transform moduli but could not find any reliable reference. I overlooked the connection formulas…
• Second, I did not know that eval can be used this way.

What is missing in your Maple trick is a formula connecting E(z,k) with I*E(z,k). I checked Maple’s function advisor for it. Perhaps it is not general enough to be listed there but it can probably be found in DLMF (If ones know where to look…).

You use two times 1/… in f which is not required. I would remove it.

Consider making your reply an answer to nm. It is at least a complex trick using assumptions and formulas not(?) available in Maple. It is the best way I have seen so far how to simplify with Maple such complex output.
 

Excellent!

PS.: The assumptions in your simplification match my complex3D plot above

Same for me. I try to upload an image

@nm 
I do not know MMA. Can I assume that FullSimplify is fully implemented in the complex domain?

Maple cannot simplify the intermediate result (simplify,symbolic can)

sqrt(sin(x))*csgn(sin(Pi/4 + x/2))

without the assumption

(0 <= x, x <= Pi/2)

Can FullSimplify do this? If it can and if this simplification is possible without assumptions, it would be desirable that Maple can do it as well.

Interesting to note that the derivation performed with Maple returns the substitution you mentionned but with a complex sign term.

sqrt(1 - 2*cos(Pi/4 + x/2)^2)*csgn(sin(Pi/4 + x/2))

Something does not fit together. Either the substitution cannot be applied in the complex domain (and hence the short result does not apply for the complex domain) or Maple could be improved when dealing with expressions without assumptions.

It looks like that MMA does more than Maple. To your original question about Maple tricks to shorten it would be interesting to see whether MMA can simplify Maples complex result. My guess would be no because of nonexisting(?) theorems and identities (see my reply to @sursumCorda).

Update:

Plotting the difference between Maple's differentiation and the original integrand in the complex plane reveals the domain where the expression are equal. Beyond that, the magnitude of the expressions is the same but the argument is different

plots:-complexplot3d(sqrt(sin(z))*csgn(sin(Pi/4 + z/2)) - sqrt(sin(z)), z = -2*Pi - (2*Pi)*I .. 2*Pi + 2*I*Pi, orientation = [35, 25, 0], title = sqrt(sin(z))*csgn(sin(Pi/4 + z/2)) - sqrt(sin(z)))

Taking the derivative of Maples output results in the integrand of the original integral

int(sqrt(sin(x)), x);
simplify(diff(%, x));
           1          /            (1/2)                (1/2) 
 - ------------------ |(sin(x) + 1)      (-2 sin(x) + 2)      
                (1/2) \                                       
   cos(x) sin(x)                                              

            (1/2) /           /            (1/2)  1  (1/2)\
   (-sin(x))      |2 EllipticE|(sin(x) + 1)     , - 2     |
                  \           \                   2       /

               /            (1/2)  1  (1/2)\\\
    - EllipticF|(sin(x) + 1)     , - 2     |||
               \                   2       ///


                                (1/2)
                          sin(x)     

Now the same with MMA output

MmaTranslator:-FromMma(`-2*EllipticE[1/2*(Pi/2-x),2]`);
simplify(diff(%,x));
combine(%,trig)assuming x>0,x<Pi/2 ;

                         /   /1      1  \         \
             -2 EllipticE|sin|- Pi - - x|, sqrt(2)|
                         \   \4      2  /         /

                              (1/2)                      
      /                     2\                           
      |         /1      1  \ |          /   /1      1  \\
      |1 - 2 cos|- Pi + - x| |      csgn|sin|- Pi + - x||
      \         \4      2  / /          \   \4      2  //

                                (1/2)
                          sin(x)     

The required restriction to a part of the real domain is an indication that the MMA response may be too short for the complex domain.

I have not found a way to use the assumptions on the original integral to shorten the output.

If this is perhaps a win in terms of correctness for Maple it would still be nice to evaluated the integral to something shorter if assumptions can be made.

@sursumCorda 

I have spend guite a while on the question why Maple is unable to simplify such expressions without finding an answer. Here is where I stopped:

Technically, to reduce such output, addition theorems or identities would be required that allow elliptic integrals of the first and second kind to be combined (with one of the them having a coefficient). I did not find any suitable theoremes or identities neither in Maple nor in DLMF (I did not look any further).

An essential technical step to simplify elliptic expressions would be to include the coefficient of an elliptic integral in the argument (i.e. transforming the elliptic integral with a coefficient into an elliptic integral without coefficient). I could not find identities that would allow for this in general. It might be possible in cases where the coeffient, the argument and the module depend on each other in a certain way.

So my guess is that there are no simple ways to simplify algorithmically and smart substitutions as here are not applied before evaluation of an integral. I assume that Mathematica provides lookup tables for such integrals and Maple does not do the same in this case.

Integration is still an art.

@AHSAN 

If I understand correctly you are interesed in solutions of the kind x=x(lambda) and not in the inverse, which is trivial

subs(h = x^2 + 1, solve(Expression = 0, [lambda])[][])

In case you are interested in x=x(lambda) the problem gets worse since we are now dealing with a polynom of 10th degree.

Taking random values for the "constants" will not change the situation. Maple will allways return a RootOf expression because it cannot provide a general solution for a polynom of 5th degree. Try for example

allvalues(subs(m = 1, Br = 0.2, N = 5, beta = 45, k = 2, lambda = 6, solve(Expression = 0, h)))

Maple returns the five indexed solutions that can only be found numerically (see ?RootOf).

If you can find a combination of values for the "constants" that make the coefficient of h^5 zero

(-210*N + (50*Br + 140)*m)*k + 210*N + (-40*Br - 70)*m = 0

you will get geenral solutions.

I think the hardware should be powerful enough.

Your observations are GUI related. Maples GUI is implemented on Java and it communicates with a Maple server that does the computations. The Java GUI uses resources from the operating system which is Linux in your case. Do you have Java installed on your machine? Perhaps Maples Java interferes  with an existing installation (this is just a guess).

If no Linux&Windows user shares his experience, I would do a comparison with Windows to see if the experience is the same.

@AHSAN 

There is no general solution for polynoms of fifth order accoridng to Abel-Ruffini Theoreme. For this reason I ttought that you wanted "somehow" identify and factor out one root and solve the remaining polynom of fourth-order. Without any assumptions on the indeterminants there is little hope that you will get a symbolic solution.

Solve sometimes can find solutions with assumptions that are not necessarily numeric. Q could be larger than m for example. Looking at the complex coefficients, I don't think that assumptions of this kind will change the situation.

Do you mean by separating a value: Factoring out a root of the ploynom?

Are all the other indeterminats {Br, N, Q, beta, k, lambda, m} real valued?

Can you give values for their ranges that could be used as assumptions for solve?

The operator is listed in Chapter 5 of the programming guide, but it is not explained. It is possible that it is mentioned and explained somewhere in a Maple file, but those files are unfortunately not searchable.

@mmcdara 
maybe acers answer here

@acer 

this way

DAE is key in this tricky case.
 

Thanks again

@acer

This page is really hidden. Not dsolve,numeric but dsolve,numeric,DAE

Many thanks!

By the way: Reading your answer, I have just tried two so called “AIs” on this by asking: “In Maple dsolve: what does the option differential do”. Although the help pages are very precisely written and were used for training, AI did not get the semantics at all. Since the word differential is so extensively used in the help pages an option differential (although textually in close proximity) as an own sematic entity is washed out in the training process.
Only one AI returned the right answer when the question was written in a way that pointed to the answer: "In Maple dsolve numeric for DAEs: what does the option differential stands for"

Not a yet a big help...

@MagnusMJ 

sometimes pays off!

Maybe you find the ini file in this hidden folder (you -> your account)

C:\Users\you\AppData\Roaming\Maple\2023