In accordance with this statement obtained by Чебышёв (1853), each of 

simplify(int(x^(1/2)*(x^2 + 1)^(-3/4), x), symbolic);
simplify(int((x^(1)*(1 - x^2))^(1/3), x), symnolic);
simplify(int(x^(-1)*(x^6 + 1)^(-1/6), x), symnolic);
simplify(int(x^(17/2)*(x^2 + 1)^(1/4), x), symnolic);

can be reduced to an integral of rational functions, which can be expressed in terms of elementary functions. But it appears that Maple 2023.0 is still unable to completely calculate them. For instance: 
 

Download Chebyshev_theorem_on_the_integration_of_binomial_differentials.mw

However, closed-form (and readable) solutions in elementary forms exist (cf. Regression reports for Computer Algebra Independent Integration Tests. Summer 2022 version (12000.org)); in fact, Mathematica returns: 

So, why can't Maple find these compact antiderivatives (expressed by elementary functions) directly here? In other words, is there a way to resolve them in Maple without applying some change of the variable to these indefinite integrals manually?

Trying to solve in Maple:

restart;
f := 15;
fk := 7;
zm := 350;
ym := 200;
eps := 1 - fk^2*exp(-((z - zm)/ym)^2)/f^2;
dz := diff(z(x, bn), bn);
db := diff(b(x, bn), bn);
eq1 := diff(z(x, bn), x) = cot(b(x, bn));
eq2 := diff(b(x, bn), x) = subs(z = z(x, bn), -1/(2*eps)*diff(eps, z));
eq3 := diff(dz, x) = -bd/sin(b(x, bn))^2;
eq4 := diff(bd, x) = subs(z = z(x, bn), dz/(2*eps)*(diff(eps, z)^2/eps - diff(eps, z $ 2)));
sys := eq1, eq2, eq3, eq4;
cond := z(0, bn) = 0, b(0, bn) = bn, zd(0, bn) = 0, bd(0, bn) = 1;
dsolve({cond, sys}, [z(x, bn), b(x, bn)], numeric);
Error, (in dsolve/numeric/process_input) dependent variables must be functions of a single unknown, the independent variable. Got [z(x, bn), b(x, bn)]

What does "Error, (in dsolve/numeric/process_input) dependent variables must be functions of a single unknown, the independent variable. Got [z(x, bn), b(x, bn)] " mean?

How can this system be solved?

So that computed results do not take up many lines in Euler transforms and in various tensor math, how can I present the sine and cosine functions in Maple with this compact form:

• sin(x(t)) = s(x) or sx
• cos(x(t)) = c(x) or cx

The input and computed output of symbolic calculations equations needs to be output in this compact notation. Derivatives with the compact notation are understood to be function of time.

Here is an example implimentation in Mathematica:

Matricies Rx and Ry are similarly defined and the dot product can be computed:

Here's example derivative:

Any help or pointers on this is appreciated. I'm new on Maple Primes and am not sure how to find out if this is already posted somewhere.

Thanks,

David

The issue arises from solving the following ODEs in Maple (where a is a suitable real parameter): 

ode__1 := a*(diff(y(x), x) + 1)^2 + (y(x) - x)^2*diff(y(x), x) = 0: # dsolve(ode__1);
ode__4 := a*(x*diff(y(x), x) + y(x))^2 - (y(x) + x)^2*diff(y(x), x) = 0: # dsolve(ode__4);

However, dsolve cannot give fully simplified solutions, so I have to compute these unevaluated integrals (i.e., expr₁) manually: (For the sake of completeness, I list some related ODEs below.) 
 

Download senseless_results_of_int.mw
 

Download senseless_results_of_int.mw

As you can see, the lengthy output of is nearly meaningless! (And if you want to simplify it, Maple will simply return: Error, (in simplify/recurse) indeterminate expression of the form 0/0.) So, how do I get the simplified results in Maple?
The integrals are: 

expr__1 := [Int(1/(z^2 + sqrt(z^4 + 4*a*z^2) + 4*a), z), Int(-1/(z^2 - sqrt(z^4 + 4*a*z^2) + 4*a), z)]: # (value(expr__1));
expr__4 := [Int((z^2 - 4*a*z + sqrt(-4*a*z^3 + z^4 - 8*a*z^2 + 4*z^3 - 4*a*z + 6*z^2 + 4*z + 1) + 2*z + 1)/(z*(-4*a*z + z^2 + 2*z + 1)), z), Int(-(z^2 - 4*a*z + 2*z + 1 - sqrt((-4*a*z + z^2 + 2*z + 1)*(z + 1)^2))/(z*(-4*a*z + z^2 + 2*z + 1)), z)]: # (value(expr__4)):

Note. By the way, Mma can solve the original ODEs directly and explicitly: 

In[1]:= DSolve[a*(y'[x]+1)^2+(y[x]-x)^2*y'[x]==0,y[x],x,IncludeSingularSolutions->Automatic]

                                   2                3                    2
                  a - x C[1] - C[1]             16 a  - 4 a x C[1] - C[1]
Out[1]= {{y[x] -> ------------------}, {y[x] -> --------------------------}}
                       x + C[1]                     4 a (4 a x + C[1])

In[2]:= DSolve[a*(x*y'[x]+y[x])^2-(y[x]+x)^2*y'[x]==0,y[x],x,IncludeSingularSolutions->Automatic]

                     2 a C[1]       2 a C[1]     2  2 a C[1]
                  a E         (-(a E        ) + a  E         + x)
Out[2]= {{y[x] -> -----------------------------------------------}, 
                                     2 a C[1]
                                  a E         - x
 
                2 a C[1]    2 a C[1]
               E         (-E         + 2 a x)
>    {y[x] -> --------------------------------}}
                    2 a C[1]              2
              2 a (E         - 2 a x + 2 a  x)

Unfortunately, Maple fails to do so.

(x^(3))^(1/3) doesn't simplify to x.  I am missing something.

The ODE is: 

eqn := y(x)*(2*x*diff(y(x), x) + y(x)*(diff(y(x), x)^2 - 1)) = -1: # How about another ODE 'lhs(eqn) = +1' ?

Maple can solve it, but I find that (to get all four solutions) I have to execute the dsolve command twice: 
 

Download dsolve_twice.mw

However, in MATLAB^®, the complete solutions can be found just in one go: 

>> dsolve('y*(2*x*Dy + y*(Dy^2 - 1)) = -1', 'x') % require the Symbolic Math Toolbox™
ans =
                         1
                        -1
 -(-(x - 1)*(x + 1))^(1/2)
  (-(x - 1)*(x + 1))^(1/2)
 (C1^2 + 2*x*C1 + 1)^(1/2)
-(C1^2 + 2*x*C1 + 1)^(1/2)

Does anyone know why?

Ideally, I would like to find all roots of this RootOf expression for a given interval.

I tried defining a function from the argument of the RootOf expression and using fsolve to find solutions, but could not get all of them.

What I managed to do skips the interval, is not really elegant and raises additional questions.
I would be grateful for any hints and improvements.

Download RootOf_a_periodic_function.mw

Is there any setting that controls the extent of a plot?

Left hand plot has defined extent of the plot, while the plot on the right hand side has not. When panning the graphics on the right side the plot is clipped.

Any idea how to make Maple to use the whole extent of the plot component as a boundary?

Download plotpoint2.mw

restart;
with(plots);
with(plottools);
with(DEtools);
N := S(t) + In(t) + C(t);
                    N := S(t) + In(t) + C(t)

eqn1 := diff*(S(t), t) = lambda - (lambda + sigma)*S(t) - (beta + Zeta)*S(t)*In(t) - beta[1]*S(t)*C(t), S(0) = ic1;
 eqn1 := diff (S(t), t) = lambda - (lambda + sigma) S(t)

    - (beta + Zeta) S(t) In(t) - beta[1] S(t) C(t), S(0) = ic1

eqn2 := diff*(In(t), t) = beta*S(t)*In(t) - (lambda + gamma)*In(t), In(0) = ic2;
 eqn2 := 

   diff (In(t), t) = beta S(t) In(t) - (lambda + gamma) In(t), 

   In(0) = ic2

eqn3 := diff*(C(t), t) = Zeta*In(t) + Zeta*In(t)^2 - (rho + lambda)*C(t) - Zeta*C(t)*In(t), C(0) = ic3;
                                                     2
     eqn3 := diff (C(t), t) = Zeta In(t) + Zeta In(t) 

        - (rho + lambda) C(t) - Zeta C(t) In(t), C(0) = ic3

lambda := 0.117852;
                       lambda := 0.117852

mu := 0.035378;
                         mu := 0.035378

beta := 0.11;
                          beta := 0.11

beta__1 := 0.05;
                        beta__1 := 0.05

g := 1;
rho := 0.1;
                           rho := 0.1

zeta := 0.02;
                          zeta := 0.02

sigma := 0.066;
                         sigma := 0.066

ic1 := 2390000;
ic2 := 753;
ic4 := 358500;
                         ic1 := 2390000

                           ic2 := 753

                         ic4 := 358500

dsol := dsolve([eqn1, eqn2, eqn3], numeric);
Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations
 

Hi all, how to get radical of expression?

Example: sqrt(a*(a+b+c))/(a*b+a*c+b*c) is 2, (a*b+a*c+b*c)*((1/3)*a+(1/3)*b+(1/3)*c)^(1/3) is 3

Thanks

I have this tedious looking function that I want to write in terms of the other expression but the command i usually use does not work here because the expressions are not polynomials. I am wondering if there is an alternative to doing this manually.
Temp.mw

As an example, the second display in the web site below shows the 42 possible triangulations of a cyclic heptagon polygon.

https://en.wikipedia.org/wiki/Polygon_triangulation

I have a document with quite a few symbols saved to my favorites palette. When I close the file and then reopoen it the Favorites Palette has not changed-the symbols are right where I want them. However, if I open the file with another computer the Favorites Palette is empty! What is happening?  (The document is stored in Dropbox and both computers are Macs running Maple 2023.)

Say I have a data matrix with one dependent variable and 50 independent variable

The first column is the dependent variable columns my first row has header names of variables say.

Is their way to code such that I can do a Linear regression stepwise such that even interactions terms can be into account and check for a best fit.

As only matlab can do it easily as i  see and it is paid costly software.

If pssible any help kind help. 

If possible some code can be written in maple kind help.

I have some large systems of linear equations.  The solutions are probability generating functions.  I can get solutions in a few minutes for systems of up to n= 200 eqns or so, but Maple just cycles indefinitely if I try to solve much larger systems.  I really only need to perform Gaussian Elimination, as I only need to solve for one of the n solutions.  The matrices are sparse, there are only 3 non-zero entries per row.  I tried to get help from the manuals but I get the impression that sparse solutions are only available for numeric computations.   Doesn't Maple allow for sparse symbolic solutions?  If so, how to do it?