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How to evaluate a one-dimensional improper integra...

Asked by: sursumCorda 922
integration numeric shill
May 11 2023
1  2

The highly oscillatory integrand is: 

(sin(x + sqrt(x)) + x*BesselJ(0, x^2))/(1 + x): # int(%, x = 0 .. infinity, numeric); # Note that the upper limit of `x` is NOT finite.

Unfortunately, Maple still cannot return a result in a few minutes. For instance, 

> 

restart;
infolevel[`evalf/int`] := 1:
> 

smartplot((sin(x+sqrt(x))+x*BesselJ(0, x^2))/(1+x))

 
> 

timelimit(0.1e3, evalf(Int((sin(x+sqrt(x))+x*BesselJ(0, x^2))/(1+x), x = 0 .. infinity)))

Control: Entering NAGInt

trying d01amc (nag_1d_quad_inf)
Control: d01amc failed
evalf/int/control: NAG failed result = result
evalf/int/improper: integrating on interval 0 .. infinity
evalf/int/improper: applying transformation x = 1/x
evalf/int/improper: interval is 0 .. 1 for the integrand:

                        /     (1/2)\            /    2\
                     sin\x + x     / + x BesselJ\0, x /
                     ----------------------------------
                                   1 + x               

and interval is 0 .. 1 for the integrand:

                                                /   1 \
                                         BesselJ|0, --|
                        /       (1/2)\          |    2|
                        |1   /1\     |          \   x /
                     sin|- + |-|     | + --------------
                        \x   \x/     /         x       
                     ----------------------------------
                                 /    1\  2            
                                 |1 + -| x             
                                 \    x/               

Control: Entering NAGInt
Control: trying d01ajc (nag_1d_quad_gen)
Control: d01ajc failed
evalf/int/control: NAG failed result = result

evalf/int/control: singularity at left end-point
evalf/int/transform: series contains {x^(1/2), x^(3/2), x^(5/2), x^(7/2), x^(9/2), x^(11/2), x^(13/2), x^(15/2), x^(17/2), x^(19/2), x^(21/2), x^(23/2), x^(25/2)}

evalf/int/singleft: applying transformation x = x^2
evalf/int/singleft: interval is 0 .. 1. for the integrand:

                  /   / 2            \    2        /    4\\  
                2 \sin\x  + csgn(x) x/ + x  BesselJ\0, x // x
                ---------------------------------------------
                                        2                    
                                   1 + x                     

evalf/int/control: Applying simplify/ln, integrand is 2*(sin(x^2+x)+x^2*BesselJ(0,x^4))/(1+x^2)*x
evalf/int/CreateProc: Trying easyproc
evalf/int/CreateProc: Trying makeproc
evalf/int/ccquad: n = 2 integral estimate = .7758263476953
                    n = 6 integral estimate = .8097413335347
evalf/int/ccquad: n = 18 integral estimate = .8097470386638
                                  error = .8097470386638e-11
From ccquad, result = .8097470386638 integrand evals = 19 error = .8097470386638e-11
Control: Entering NAGInt
Control: trying d01ajc (nag_1d_quad_gen)
Control: d01ajc failed
evalf/int/control: NAG failed result = result

evalf/int/control: singularity at left end-point

evalf/int/transform: series contains {1/x^(3/2), x^(1/2), cos(1/x^2+1/4*Pi), sin((x+x^(1/2))/x^(3/2)), sin(1/4*(4+Pi*x^2)/x^2)}
evalf/int/singleft: applying transformation x = x^2
evalf/int/singleft: interval is 0 .. 1. for the integrand:

                   /   /        /1\  \                    \
                   |   |1 + csgn|-| x|                    |
                   |   |        \x/  |  2          /   1 \|
                 2 |sin|-------------| x  + BesselJ|0, --||
                   |   |      2      |             |    4||
                   \   \     x       /             \   x //
                 ------------------------------------------
                                 3 /     2\                
                                x  \1 + x /                

evalf/int/control: Applying simplify/ln, integrand is 2*(sin((1+x)/x^2)*x^2+BesselJ(0,1/x^4))/x^3/(1+x^2)
evalf/int/control: Applying simplify/trig, integrand is (2*sin((1+x)/x^2)*x^2+2*BesselJ(0,1/x^4))/(x^3+x^5)
evalf/int/control: singularity at left end-point

evalf/int/transform: series contains {cos(1/x^4+1/4*Pi), sin((1+x)/x^2), sin(1/4*(4+Pi*x^4)/x^4)}
evalf/int/transform: no transform found
evalf/int/series: integrating on 0 .. .2493468135214 the series:

       2 (%2)     2 (%2) x   /2 (%2)      \  3   /  2 (%2)      \  5   
      --------- - -------- + |------- - %3| x  + |- ------- + %3| x    
        (1/2)       (1/2)    |  (1/2)     |      |    (1/2)     |      
      Pi      x   Pi         \Pi          /      \  Pi          /      
                                                                       
           /       2 (%2)      \  7   /     2 (%2)      \  9           
         + |- %3 + ------- - %4| x  + |%3 - ------- + %4| x            
           |         (1/2)     |      |       (1/2)     |              
           \       Pi          /      \     Pi          /              
                                                                       
                                                                       
           /2 (%2)           \  11   /          2 (%2) \  13           
         + |------- - %4 - %5| x   + |%4 + %5 - -------| x             
           |  (1/2)          |       |            (1/2)|               
           \Pi               /       \          Pi     /               
                                                                       
           /       2 (%2)      \  15   /     2 (%2)      \  17    / 19\
         + |- %5 + ------- + %6| x   + |%5 - ------- - %6| x   + O\x  /
           |         (1/2)     |       |       (1/2)     |             
           \       Pi          /       \     Pi          /             
                                                                       
                                                                       
                      /        4\                                      
             (1/2)    |4 + Pi x |                                      
      %1 := 2      sin|---------|                                      
                      |     4   |                                      
                      \  4 x    /                                      
                    /1 + x\   (1/2)                                    
      %2 := %1 + sin|-----| Pi                                         
                    |  2  |                                            
                    \ x   /                                            
             (1/2)    /1    1   \                                      
            2      cos|-- + - Pi|                                      
                      | 4   4   |                                      
                      \x        /                                      
      %3 := ---------------------                                      
                      (1/2)                                            
                  4 Pi                                                 
                        /        4\                                    
               (1/2)    |4 + Pi x |                                    
            9 2      sin|---------|                                    
                        |     4   |                                    
                        \  4 x    /                                    
      %4 := -----------------------                                    
                       (1/2)                                           
                  64 Pi                                                
                (1/2)    /1    1   \                                   
            53 2      cos|-- + - Pi|                                   
                         | 4   4   |                                   
                         \x        /                                   
      %5 := ------------------------                                   
                        (1/2)                                          
                  512 Pi                                               
                           /        4\                                 
                  (1/2)    |4 + Pi x |                                 
            1371 2      sin|---------|                                 
                           |     4   |                                 
                           \  4 x    /                                 
      %6 := --------------------------                                 
                          (1/2)                                        
                  16384 Pi                                             

evalf/int/CreateProc: Trying easyproc
evalf/int/CreateProc: Trying makeproc

evalf/int/CreateProc: Trying procmake
evalf/int/control: applying double-exponential method
evalhf mode unsuccessful -- retry in software floats
evalf/int/quadexp: applying double-exponential method

Error, (in tools/sign) time expired

  
> 

time() =

3999.500

 (1)

 =

%?

Download oscillatoryInt.mws

I tried to specify another method (as described in evalf/int), but it seems that this doesn't work. Any ideas?

Isolate variable...

Asked by: Andre Luis Alves Neves 15
isolate
May 10 2023
1  3

Hello,

How do I isolate the variable ng in the following expression:

rho := `ρg`*ng + `ρp`*np + `ρw`*nw;
B := np/Kp + nw/Kw + ng/Kg;

C := (B/rho)^(1/2);

As you can see, I am running the isolate command, but it is not including the variable C, or it returns the wrong result.

isolate(((np/Kp + nw/Kw + ng/Kg)/(`ρg`*ng + `ρp`*np + `ρw`*nw))^(1/2), ng);
                               ( Kp nw + Kw np) Kg
                   ng = -     -----------------

                                          Kw Kp       

 

isolate(C=((np/Kp + nw/Kw + ng/Kg)/(`ρg`*ng + `ρp`*np + `ρw`*nw))^(1/2), ng);
                             0 = 0

Thanks.                

How to solve some polynomial equation over the dom...

Asked by: sursumCorda 922
solve multivariate shill
May 09 2023
0  7

In certain tasks, I need to find all accurate positive (not just nonnegative) roots that exist of some multivariate polynomials like: 

nsd := 16*a*b*c*(9 + a^2 + b^2 + c^2)*(b*c + a*(b + c) + 3*(a + b + c)) - (3 + a + b + c)^2*(a*b + 3*c)*(3*b + a*c)*(3*a + b*c): # assume((a, b, c) >~ 0);

According to fsolve/details, for one general equation, the fsolve command only computes "a single real root", so it is inadequate to tackle this question. But if I use the solve command with floating-point values, the computation cannot finish in ten minutes instead! (Maybe a longer time will suffice, yet this is rather unacceptable.) 

> 

restart;
> 

"Digits+=Digits:"
> 

#assume((a, b, c) >~ 0);
nsd := 16*a*b*c*(a^2 + b^2 + c^2 + 9)*(a*(b + c) + 3*(a + b + c) + b*c) - (a + b + c + 3)^2*(a*b + 3*c)*(a*c + 3*b)*(b*c + 3*a):
> 

fsolve({`~`[`>`](a, b, c, ` $`, 0), nsd = 0}, fulldigits)

Error, (in fsolve) expecting an equation or set or list of equations, but received inequalities {16*a*b*c*(a^2+b^2+c^2+9)*(a*(b+c)+3*a+3*b+3*c+b*c)-(a+b+c+3)^2*(a*b+3*c)*(a*c+3*b)*(b*c+3*a) = 0, 0 < a, 0 < b, 0 < c}

  
> 

fsolve([`$`(nsd = 0, 3)], fulldigits, {`~`[`=`](a, b, c, ` $`, 0 .. infinity)}, avoid = {{a = 0}, {b = 0}, {c = 0}})eval(nsd, %)

{a = 3.0000000005817604971, b = 3.0000000004676996798, c = 3.0000000004610312655}

 (1)
> 

timelimit(0.6e3, RealDomain[solve]([`~`[`>`](a, b, c, ` $`, 0), nsd = 0.], allsolutions))

Error, (in gcd/gcdchrem1) time expired

  
> 

timelimit(0.6e3, `assuming`([solve(nsd = 0., useassumptions, allsolutions)], [`~`[`>`](a, b, c, ` $`, 0)]))

Error, (in modp1/DistDeg) time expired

  
> 

""(* However, there are (at least) five positive solutions to 'nsd = 0'. *)" map2(eval,nsd,[{a=1,b=1,c=1},{a=3,b=3,c=3},{a=3,b=3,c=9},{a=3,b=9,c=3},{a=9,b=3,c=3}])"

[0, 0, 0, 0, 0]

 (2)
> 

time()

11127.031

 (3)
> 

NULL


Download solve_numerically.mws

Is there any workaround to obtain those (finitely many) positive solutions completely?

What is wrong with my nested if statement?...

Asked by: Carlos36r 25
conditional rtfm if-then
May 07 2023
3  6

Please excuse my thickness if any.

In the following IF statement:

if 1 <> 2 then
    OuterThen;
    if 1 <> 2 then
        InnerThen;
    else
        InnerElse;
    end if;    
else
    OuterElse;
end if;

I expect the output:

OuterThen
InnerThen

but I only get:

OuterThen

Why?

Why is the performance of LinearAlgebra[MatrixPowe...

Asked by: sursumCorda 922
performance linearalgebra shill
May 07 2023
0  9

In practice, I need calculate the (principal) squareroot of some suitable large matrix exactly (so the desired result should not involve floating-point numbers, otherwise the decomposition will be of no theoretical value as a certificate …). But I find it difficult to do so in Maple. (What about the Efficient Computations - Maple Help (maplesoft.com)?)
Below are some matrices: (They are not contrived academic examples.) 

M__4 := <1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0|0,3,0,-6,0,3,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,-3,0,0,0,0,0,0,0,0,6,0,0,0,0,-3,0|-2,0,7,0,-8,0,3,0,0,0,0,0,0,0,0,1,0,-7,0,2,0,0,0,0,0,0,4,0,3,0,0,0,0,-3,0,0|0,-6,0,13,0,-8,0,1,0,0,0,0,0,0,0,0,6,0,-4,0,4,0,0,0,0,0,0,-1,0,-11,0,0,0,0,6,0|1,0,-8,0,13,0,-6,0,0,0,0,0,0,0,0,4,0,8,0,-1,0,0,0,0,0,0,-11,0,-6,0,0,0,0,6,0,0|0,3,0,-8,0,7,0,-2,0,0,0,0,0,0,0,0,-3,0,-1,0,1,0,0,0,0,0,0,2,0,4,0,0,0,0,-3,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,0,0,1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,4,0,0,0,-2,0,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,-2,0,-1,0,0,0,0,2,0,2,0,0,-1|0,0,0,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,-6,0,-6,0,0,0,0,0,0,6,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,-2,0,-6,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,0,0,-3,0,3,0,0,0,0,0,0,6,0,3,0,-6,0,0,0,0,-3,0,3,0,0,0|-2,0,1,0,4,0,-3,0,0,0,0,0,0,0,0,7,0,-1,0,2,0,0,0,0,0,0,-8,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|2,0,-7,0,8,0,-3,0,0,0,0,0,0,0,0,-1,0,7,0,-2,0,0,0,0,0,0,-4,0,-3,0,0,0,0,3,0,0|0,3,0,-4,0,-1,0,2,0,0,0,0,0,0,0,0,-3,0,7,0,-7,0,0,0,0,0,0,-2,0,8,0,0,0,0,-3,0|-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0|0,-3,0,4,0,1,0,-2,0,0,0,0,0,0,0,0,3,0,-7,0,7,0,0,0,0,0,0,2,0,-8,0,0,0,0,3,0|0,0,0,0,0,0,0,0,-6,0,-1,0,-6,0,6,0,0,0,0,0,0,13,0,8,0,-11,0,0,0,0,-8,0,4,0,0,1|0,0,0,0,0,0,0,0,0,-6,0,-6,0,6,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,-2,0,-3,0,3,0,0,0,0,0,0,8,0,7,0,-4,0,0,0,0,-7,0,-1,0,0,2|0,0,0,0,0,0,0,0,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,-6,0,12,0,0,0,0,0,0,-6,0,0,0,0|0,0,0,0,0,0,0,0,6,0,-1,0,6,0,-6,0,0,0,0,0,0,-11,0,-4,0,13,0,0,0,0,4,0,-8,0,0,1|1,0,4,0,-11,0,6,0,0,0,0,0,0,0,0,-8,0,-4,0,-1,0,0,0,0,0,0,13,0,6,0,0,0,0,-6,0,0|0,0,0,-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,6,0,-11,0,4,0,1,0,0,0,0,0,0,0,0,-6,0,8,0,-8,0,0,0,0,0,0,-1,0,13,0,0,0,0,-6,0|0,0,0,0,0,0,0,0,3,0,2,0,3,0,-3,0,0,0,0,0,0,-8,0,-7,0,4,0,0,0,0,7,0,1,0,0,-2|0,0,0,0,0,0,0,0,0,-2,0,6,0,-2,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,4,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,2,0,-3,0,3,0,0,0,0,0,0,4,0,-1,0,-8,0,0,0,0,1,0,7,0,0,-2|0,0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,-6,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,0,0,0,0,-2,0,-2,0,0,1>:
M__3 := 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Unexpectedly, LinearAlgebra[MatrixPower] does not work well. For instance: 
 

> 

restart;
> 

interface(version)

`Standard Worksheet Interface, Maple 2023.0, Windows 10, March 6 2023 Build ID 1689885`

 (1)
> 

M__4 := <1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0|0,3,0,-6,0,3,0,0,0,0,0,0,0,0,0,0,-3,0,3,0,-3,0,0,0,0,0,0,0,0,6,0,0,0,0,-3,0|-2,0,7,0,-8,0,3,0,0,0,0,0,0,0,0,1,0,-7,0,2,0,0,0,0,0,0,4,0,3,0,0,0,0,-3,0,0|0,-6,0,13,0,-8,0,1,0,0,0,0,0,0,0,0,6,0,-4,0,4,0,0,0,0,0,0,-1,0,-11,0,0,0,0,6,0|1,0,-8,0,13,0,-6,0,0,0,0,0,0,0,0,4,0,8,0,-1,0,0,0,0,0,0,-11,0,-6,0,0,0,0,6,0,0|0,3,0,-8,0,7,0,-2,0,0,0,0,0,0,0,0,-3,0,-1,0,1,0,0,0,0,0,0,2,0,4,0,0,0,0,-3,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,0,0,1,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,4,0,0,0,-2,0,0,0,0,0,0,0,0,-6,0,6,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,-2,0,-1,0,0,0,0,2,0,2,0,0,-1|0,0,0,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,-6,0,-6,0,0,0,0,0,0,6,0,0,0,0|0,0,0,0,0,0,0,0,3,0,0,0,3,0,-3,0,0,0,0,0,0,-6,0,-3,0,6,0,0,0,0,3,0,-3,0,0,0|0,0,0,0,0,0,0,0,0,-2,0,-6,0,4,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,-2,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,0,0,-3,0,3,0,0,0,0,0,0,6,0,3,0,-6,0,0,0,0,-3,0,3,0,0,0|-2,0,1,0,4,0,-3,0,0,0,0,0,0,0,0,7,0,-1,0,2,0,0,0,0,0,0,-8,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|2,0,-7,0,8,0,-3,0,0,0,0,0,0,0,0,-1,0,7,0,-2,0,0,0,0,0,0,-4,0,-3,0,0,0,0,3,0,0|0,3,0,-4,0,-1,0,2,0,0,0,0,0,0,0,0,-3,0,7,0,-7,0,0,0,0,0,0,-2,0,8,0,0,0,0,-3,0|-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,2,0,-2,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0|0,-3,0,4,0,1,0,-2,0,0,0,0,0,0,0,0,3,0,-7,0,7,0,0,0,0,0,0,2,0,-8,0,0,0,0,3,0|0,0,0,0,0,0,0,0,-6,0,-1,0,-6,0,6,0,0,0,0,0,0,13,0,8,0,-11,0,0,0,0,-8,0,4,0,0,1|0,0,0,0,0,0,0,0,0,-6,0,-6,0,6,0,0,0,0,0,0,0,0,12,0,-6,0,0,0,0,0,0,0,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,-2,0,-3,0,3,0,0,0,0,0,0,8,0,7,0,-4,0,0,0,0,-7,0,-1,0,0,2|0,0,0,0,0,0,0,0,0,6,0,-6,0,0,0,0,0,0,0,0,0,0,-6,0,12,0,0,0,0,0,0,-6,0,0,0,0|0,0,0,0,0,0,0,0,6,0,-1,0,6,0,-6,0,0,0,0,0,0,-11,0,-4,0,13,0,0,0,0,4,0,-8,0,0,1|1,0,4,0,-11,0,6,0,0,0,0,0,0,0,0,-8,0,-4,0,-1,0,0,0,0,0,0,13,0,6,0,0,0,0,-6,0,0|0,0,0,-1,0,2,0,-1,0,0,0,0,0,0,0,0,0,0,-2,0,2,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0|0,0,3,0,-6,0,3,0,0,0,0,0,0,0,0,-3,0,-3,0,0,0,0,0,0,0,0,6,0,3,0,0,0,0,-3,0,0|0,6,0,-11,0,4,0,1,0,0,0,0,0,0,0,0,-6,0,8,0,-8,0,0,0,0,0,0,-1,0,13,0,0,0,0,-6,0|0,0,0,0,0,0,0,0,3,0,2,0,3,0,-3,0,0,0,0,0,0,-8,0,-7,0,4,0,0,0,0,7,0,1,0,0,-2|0,0,0,0,0,0,0,0,0,-2,0,6,0,-2,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,4,0,0,0,0|0,0,0,0,0,0,0,0,-3,0,2,0,-3,0,3,0,0,0,0,0,0,4,0,-1,0,-8,0,0,0,0,1,0,7,0,0,-2|0,0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,-6,0,-3,0,0,0,0,3,0,0|0,-3,0,6,0,-3,0,0,0,0,0,0,0,0,0,0,3,0,-3,0,3,0,0,0,0,0,0,0,0,-6,0,0,0,0,3,0|0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,2,0,1,0,0,0,0,-2,0,-2,0,0,1>:
> 

M__3 := 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,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1/2,0,-1/2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1/2,0,-1/2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1/2,0,0,0,0,0,0,-1/2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1/2,0,0,0,0,0,0,-1/2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1/2,0,0,0,0,-1/2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1/2,0,0,0,0,-1/2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0;0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0>:
> 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(M__4, 1/2)))

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 10 x 10 matrix

  

IntegerCharacteristicPolynomial: Used total of  2  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 10 x 10 matrix

  

IntegerCharacteristicPolynomial: Used total of  2  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 10 x 10 matrix

  

IntegerCharacteristicPolynomial: Used total of  2  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 6 x 6 matrix

  

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

  

Error, (in simplify/sqrt/fraction) time expired

  
> 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(M__3, 1/2)))

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 13 x 13 matrix

  

IntegerCharacteristicPolynomial: Used total of  4  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

  

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

  

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

  

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

  

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

  

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 4 x 4 matrix

  

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

  

memory used=2.84GiB, alloc change=8.00MiB, cpu time=3.05m, real time=3.01m, gc time=5.84s

  

_rtable[36893490642867319380]

 (2)
> 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(M__2, 1/2)))

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 6 x 6 matrix

  

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

  

IntegerCharacteristicPolynomial: Computing characteristic polynomial for a 6 x 6 matrix

  

IntegerCharacteristicPolynomial: Used total of  1  prime(s)

  

memory used=2.40GiB, alloc change=0 bytes, cpu time=2.85m, real time=2.82m, gc time=4.08s

  

_rtable[36893490642428505372]

 (3)
> 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(M__1, 1/2)))

CharacteristicPolynomial: working on determinant of minor 2

  

CharacteristicPolynomial: working on determinant of minor 3

  

CharacteristicPolynomial: working on determinant of minor 4

  

CharacteristicPolynomial: working on determinant of minor 5

  

CharacteristicPolynomial: working on determinant of minor 6

  

CharacteristicPolynomial: working on determinant of minor 2

  

CharacteristicPolynomial: working on determinant of minor 3

  

CharacteristicPolynomial: working on determinant of minor 4

  

CharacteristicPolynomial: working on determinant of minor 5

  

CharacteristicPolynomial: working on determinant of minor 6

  

CharacteristicPolynomial: working on determinant of minor 2

  

CharacteristicPolynomial: working on determinant of minor 3

  

CharacteristicPolynomial: working on determinant of minor 4

  

CharacteristicPolynomial: working on determinant of minor 5

  

CharacteristicPolynomial: working on determinant of minor 6

  

CharacteristicPolynomial: working on determinant of minor 2

  

CharacteristicPolynomial: working on determinant of minor 3

  

CharacteristicPolynomial: working on determinant of minor 4

  

CharacteristicPolynomial: working on determinant of minor 5

  

CharacteristicPolynomial: working on determinant of minor 6

  

CharacteristicPolynomial: working on determinant of minor 2

  

CharacteristicPolynomial: working on determinant of minor 3

  

CharacteristicPolynomial: working on determinant of minor 4

  

CharacteristicPolynomial: working on determinant of minor 5

  

CharacteristicPolynomial: working on determinant of minor 6

  

Error, (in LinearAlgebra:-MatrixFunction) time expired

  
> 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalhf[hfloat](M__4), 1/2)))

Eigenvalues: calling external function

  

Eigenvalues: CLAPACK hw_dgeevx_

  

memory used=468.66MiB, alloc change=0 bytes, cpu time=18.48s, real time=19.71s, gc time=1.09s

  

_rtable[36893490642747474396]

 (4)
> 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalhf[hfloat](M__3), 1/2)))

Eigenvalues: calling external function

  

Eigenvalues: CLAPACK hw_dgeevx_

  

Multiply: copying first Matrix to enable external call

  

Multiply: calling external function

  

Multiply: NAG hw_f06yaf

  

unknown: NAG hw_f06yaf

  

unknown: NAG hw_f06yaf

  

unknown: NAG hw_f06yaf

  

memory used=13.43GiB, alloc change=0 bytes, cpu time=8.66m, real time=8.46m, gc time=26.41s

  

_rtable[36893490642773280876]

 (5)
> 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalhf[hfloat](M__2), 1/2)))

Eigenvalues: calling external function

  

Eigenvalues: CLAPACK hw_dgeevx_

  

Multiply: copying first Matrix to enable external call

  

Multiply: calling external function

  

Multiply: NAG hw_f06yaf

  

unknown: NAG hw_f06yaf

  

memory used=13.45GiB, alloc change=0 bytes, cpu time=8.39m, real time=8.23m, gc time=26.12s

  

_rtable[36893490642902286332]

 (6)
> 

timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalhf[hfloat](M__1), 1/2)))

Eigenvalues: calling external function

  

Eigenvalues: CLAPACK hw_dgeevx_

  

Multiply: copying first Matrix to enable external call

  

Multiply: calling external function

  

Multiply: NAG hw_f06yaf

  

unknown: NAG hw_f06yaf

  

Error, (in LinearAlgebra:-MatrixFunction) time expired

  
> 

"Digits+=5:"timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalf(M__4), 1/2)))

Eigenvalues: calling external function

  

Eigenvalues: copying first Matrix, to enable external call

  

Eigenvalues: CLAPACK hw_dgeevx_

  

memory used=418.17MiB, alloc change=0 bytes, cpu time=16.33s, real time=16.16s, gc time=796.88ms

  

_rtable[36893490642540182332]

 (7)
> 

"Digits+=5:"timelimit(0.1e4, CodeTools:-Usage(LinearAlgebra:-MatrixPower(evalf(M__1), 1/2)))

Eigenvalues: calling external function

  

Eigenvalues: copying first Matrix, to enable external call

  

Eigenvalues: CLAPACK sw_dgeevx_

  

Multiply: copying first Matrix to enable external call

  

Multiply: copying second Matrix to enable external call

  

Multiply: calling external function

  

Multiply: NAG sw_f06yaf

  

unknown: copying second Matrix to enable external call

  

unknown: calling external function

  

unknown: NAG sw_f06yaf

  

Error, (in LinearAlgebra:-MatrixFunction) time expired

  

time() = 5731.593NULL


 

Download performance_of_`sqrtm`.mw

The last six numerical experiments all fail to compute the square root. (Accordingly, here it is impossible to convert each numeric element to one of the "simplest" algebraic numbers that approximates it well.) As you can see, if I execute them directly (i.e., without converting each entry to the nearest floating-point value), the elapsed time required to run the procedure is still unacceptable! (Note that they can be evaluated symbolically in fact.) Is this a bug of Maple? And how do I get the desired results in Maple efficiently?

Unable to handle exact expressions that contain th...

Asked by: sursumCorda 922
simplify simplification lambertw
May 06 2023
0  6

Let y＞1. It can be proved (maybe by hand) that the following four expressions are mathematically equivalent: 

assume(y > 1); # Assumption!
expr := [0, 0, 0, 0]: # Preallocation.
expr[1] := exp(y*LambertW(ln(y))):
expr[2] := (ln(y)/LambertW(ln(y)))^y:
expr[3] := eval(x^(x^x), x = exp(LambertW(ln(y)))):
expr[4] := eval(x^(x^x), x = ln(y)/LambertW(ln(y))):

But unfortunately, when I tried to simplify expri - exprj (symbolically), I just got: 

seq(seq(ifelse(j <> i, [i, j, verify(expr[j], expr[i], equal)], NULL), j = 1 .. numelems(expr)), i = 1 .. numelems(expr)); # is(expr[j] = expr[i]) does not work as well.
 = 
   [1, 2, FAIL], [1, 3, FAIL], [1, 4, FAIL], [2, 1, FAIL], 

     [2, 3, FAIL], [2, 4, FAIL], [3, 1, FAIL], [3, 2, FAIL], 

     [3, 4, true], [4, 1, FAIL], [4, 2, FAIL], [4, 3, true]

In other words, Maple can only determine that expr[3] = expr[4].
One may check that, for example, 

MmaTranslator:-Mma:-Chop([seq](seq(evalhf(subs(y = log10(rand()), expr[i] - expr[j])), j = 1 .. numelems(expr)), i = 1 .. numelems(expr)), 2^(-26));
 = 
        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

However, the desired approach is simplifying them symbolically. Is there a way to do so in Maple?

Kudos for Maple 2023!

by: rcorless 530 Maple 2023
polynomial integration rootfinding
May 05 2023
2  3

Just installing Maple 2023 on my office machine (a mac); installed it on my travel computer (a Surface Pro running Windows) yesterday.

Configured Jupyter notebooks to use the 2023 Maple Kernel and it all went smoothly.  I was *delighted* to notice that plotting Lambert W in Jupyter with the command

plot( [W(x), W(-1,x)], x=-1..4, view=[-1..4, -3.5..1.5], colour=[red,blue], scaling=constrained, labels=[x,W(x)] );

produced a *better* plot near the branch point.  This is hard to do automatically!  It turns out this is a side effect of the better/faster/more memory efficient adaptive plotting software, which I gather from "What's New" was written for efficiency not for quality.  But the quality is better, too!  Nice!

I am working my. way through the "What's New" and I'm really pleased to learn about the new univariate polynomial rootfinder, *not least because it cites the paper describing the algorithm*.  Lots of other goodies too; the new methods of integration look like serious improvements.  Well done. (One thing there: "parallel Risch" is a term of art, and may lead people to believe that Maple is doing something with parallel computing there.  I don't think so.  Could a reference be supplied?)

The new colour schemes and plotting features in 3d and contour plotting look fabulous.

Direct Python language support from a code edit region is not at all what I expected to see---I wonder if it will work in a Jupyter notebook?  I'm going to have to try it...

I'm quite impressed.  The folks at Maplesoft have been working very hard indeed.  Congratulations on a fine release!

 

Why does the figure generated from the tex code ha...

Asked by: lcz 994
export plotting latex
May 04 2023
0  5

I followed the code on the website https://de.maplesoft.com/support/help/maple/view.aspx?path=updates/Maple18/GraphTheory to convert a graph to LaTeX code. However, after compiling with pdflatex, I found that some edges of the graph are jagged.

restart:    
with(GraphTheory):
with(SpecialGraphs):
S:=SoccerBallGraph():
Latex(S,FileTools:-JoinPath([currentdir(), "soccer.tex"]),300,300,true)

soccer.pdf

I suspect it's because of the converted LaTeX code.

PS: The PDF conversion issue from last time still remains unsolved in Maple 2023; see

https://www.mapleprimes.com/questions/236142-How-To-Remove-The-Mosaic-Of-Vertices.

Can Table Values Be Modified in maple tables()...

Asked by: ianmccr 165
table
May 03 2023
1  2

The example worksheet uses table to determine the covering relations in a POSET. This is old (Maple 11 and earlier code) that I have been trying to update into a package to explore calculations in an algebraic structure.  The example procedure seems to work well, but because of problems with similar procedures elsewhere, I have some concerns about the validity of using tables in this fashion. Specifically, this procedure initializes a table, then proceeds to modify the table entries, and then reformats the sequences into sets.  I have not been able to find documentation for modifying entries in tables after they have been defined. The documentation for tables only covers adding entries, removing entries, but not modifying entries.

Is modifying tables as my  procedure does an undocumented feature?

In addition, the documentation does not explain how to clear a table. It only describes how to clear a table entry. Older code sometimes purported to clear a table by assigning its name (with uneval quotes) to itself, but this does not seem to work.

exampletableoperations.mw

Unable to reduce these (specific) primitive functi...

Asked by: sursumCorda 922
integration simplify int
May 03 2023
1  4

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: 
 

> 

restart;
> 

interface(version)

`Standard Worksheet Interface, Maple 2023.0, Windows 10, March 6 2023 Build ID 1689885`

 (1)
> 

timelimit(0.1e4, `assuming`([simplify(int(x^(1/2)/(x^2+1)^(3/4), x))], [x > 0]))

(2/3)*x^(3/2)*hypergeom([3/4, 3/4], [7/4], -x^2)

 (2)
> 

timelimit(0.1e4, `assuming`([simplify(int((x*(-x^2+1))^(1/3), x))], [`or`(`and`(x <= 1, x >= 0), x <= -1)]))

(3/4)*x^(4/3)*hypergeom([-1/3, 2/3], [5/3], x^2)

 (3)
> 

timelimit(0.1e4, `assuming`([simplify(int(1/(x*(x^6+1)^(1/6)), x))], [x <> 0]))

-(1/36)*x^6*hypergeom([1, 1, 7/6], [2, 2], -x^6)-(1/12)*Pi*3^(1/2)-(1/3)*ln(2)-(1/4)*ln(3)+ln(x)

 (4)
> 

timelimit(0.1e4, `assuming`([simplify(int(x^(17/2)*(x^2+1)^(1/4), x))], [x > 0]))

(1/81920)*(8192*x^10*(x^2+1)^(1/4)+512*x^8*(x^2+1)^(1/4)-640*x^6*(x^2+1)^(1/4)+880*x^4*(x^2+1)^(1/4)-1155*ln(2*RootOf(_Z^2+1)*x^(1/2)*(x^2+1)^(3/4)-2*RootOf(_Z^2+1)*x^(3/2)*(x^2+1)^(1/4)+2*(x^2+1)^(1/2)*x-2*x^2-1)*x^(1/2)*RootOf(_Z^2+1)-1540*x^2*(x^2+1)^(1/4)-1155*x^(1/2)*ln(-2*x^(1/2)*(x^2+1)^(3/4)-2*x^(3/2)*(x^2+1)^(1/4)+2*(x^2+1)^(1/2)*x+2*x^2+1))/x^(1/2)

 (5)
> 

gc()


 

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?

Nearly fails to evaluate 'int(1/(z^2 - sqrt(z^2*(z...

Asked by: sursumCorda 922
integration ode dsolve
May 02 2023
1  3

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.) 
 

> 

restart;
> 

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

dsolve(ode__1, y(x), explicit)
> 

expr__1 := convert(useInt(eval(selectfun([%], Intat), _Z = z)), list)

[Int(1/(z^2+(z^4+4*a*z^2)^(1/2)+4*a), z), Int(-1/(z^2-(z^4+4*a*z^2)^(1/2)+4*a), z)]

 (1)
> 

value(expr__1)

[(z^4+4*a*z^2)^(1/2)*((1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)+2*(-a)^(1/2)*ln(z+((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)-2*(-a)^(1/2)*ln(z+((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)-2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)^2*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2))*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z*(z^2+4*a)^(1/2))+(1/2)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/2)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(1/2))-4*a*(-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*a*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/8)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(3/2))), -(1/2)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))+(1/2)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(1/2))+(z^4+4*a*z^2)^(1/2)*((1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)+2*(-a)^(1/2)*ln(z+((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)-2*(-a)^(1/2)*ln(z+((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)-2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)^2*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2))*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z*(z^2+4*a)^(1/2))+4*a*(-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*a*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/8)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(3/2)))]

 (2)
> 

verify(diff([-z/(z^2+sqrt(z^2*(z^2+4*a))), z/(z^2-sqrt(z^2*(z^2+4*a)))], z), `~`[op](1, expr__1), simplify)

true

 (3)
> 

dsolve(ode__4, y(x), explicit)
> 

expr__4 := convert(useInt(eval(selectfun([%], Intat), _Z = z)), list)

[Int((z^2-4*a*z+(-4*a*z^3+z^4-8*a*z^2+4*z^3-4*a*z+6*z^2+4*z+1)^(1/2)+2*z+1)/(z*(-4*a*z+z^2+2*z+1)), z), Int(-(z^2-4*a*z+2*z+1-((-4*a*z+z^2+2*z+1)*(z+1)^2)^(1/2))/(z*(-4*a*z+z^2+2*z+1)), z)]

 (4)
> 

value(expr__4)

[(-4*a*z^3+z^4-8*a*z^2+4*z^3-4*a*z+6*z^2+4*z+1)^(1/2)*(ln(z-2*a+1+(-4*a*z+z^2+2*z+1)^(1/2))+arctanh((2*a*z-z-1)/(-4*a*z+z^2+2*z+1)^(1/2)))/((z+1)*(-4*a*z+z^2+2*z+1)^(1/2))+ln(z), ((-4*a*z+z^2+2*z+1)*(z+1)^2)^(1/2)*(ln(z-2*a+1+(-4*a*z+z^2+2*z+1)^(1/2))+arctanh((2*a*z-z-1)/(-4*a*z+z^2+2*z+1)^(1/2)))/((z+1)*(-4*a*z+z^2+2*z+1)^(1/2))-ln(z)]

 (5)
> 

verify(diff([2*arctanh(sqrt((z+1)^2*(z*(z-2*(2*a-1))+1))/(z^2-1))+ln(z), 2*arctanh(sqrt((z+1)^2*(z*(z-2*(2*a-1))+1))/(z^2-1))-ln(z)], z), `~`[op](1, expr__4), simplify)

true

 (6)
> 

NULL


 

Download senseless_results_of_int.mw
 

> 

restart;
> 

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

dsolve(ode__1, y(x), explicit)
> 

expr__1 := convert(useInt(eval(selectfun([%], Intat), _Z = z)), list)

[Int(1/(z^2+(z^4+4*a*z^2)^(1/2)+4*a), z), Int(-1/(z^2-(z^4+4*a*z^2)^(1/2)+4*a), z)]

 (1)
> 

value(expr__1)

[(z^4+4*a*z^2)^(1/2)*((1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)+2*(-a)^(1/2)*ln(z+((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)-2*(-a)^(1/2)*ln(z+((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)-2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)^2*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2))*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z*(z^2+4*a)^(1/2))+(1/2)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/2)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(1/2))-4*a*(-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*a*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/8)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(3/2))), -(1/2)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))+(1/2)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(1/2))+(z^4+4*a*z^2)^(1/2)*((1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)+2*(-a)^(1/2)*ln(z+((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)-2*(-a)^(1/2)*ln(z+((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)-2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)^2*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2))*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z*(z^2+4*a)^(1/2))+4*a*(-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*a*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/8)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(3/2)))]

 (2)
> 

verify(diff([-z/(z^2+sqrt(z^2*(z^2+4*a))), z/(z^2-sqrt(z^2*(z^2+4*a)))], z), `~`[op](1, expr__1), simplify)

true

 (3)
> 

dsolve(ode__4, y(x), explicit)
> 

expr__4 := convert(useInt(eval(selectfun([%], Intat), _Z = z)), list)

[Int((z^2-4*a*z+(-4*a*z^3+z^4-8*a*z^2+4*z^3-4*a*z+6*z^2+4*z+1)^(1/2)+2*z+1)/(z*(-4*a*z+z^2+2*z+1)), z), Int(-(z^2-4*a*z+2*z+1-((-4*a*z+z^2+2*z+1)*(z+1)^2)^(1/2))/(z*(-4*a*z+z^2+2*z+1)), z)]

 (4)
> 

value(expr__4)

[(-4*a*z^3+z^4-8*a*z^2+4*z^3-4*a*z+6*z^2+4*z+1)^(1/2)*(ln(z-2*a+1+(-4*a*z+z^2+2*z+1)^(1/2))+arctanh((2*a*z-z-1)/(-4*a*z+z^2+2*z+1)^(1/2)))/((z+1)*(-4*a*z+z^2+2*z+1)^(1/2))+ln(z), ((-4*a*z+z^2+2*z+1)*(z+1)^2)^(1/2)*(ln(z-2*a+1+(-4*a*z+z^2+2*z+1)^(1/2))+arctanh((2*a*z-z-1)/(-4*a*z+z^2+2*z+1)^(1/2)))/((z+1)*(-4*a*z+z^2+2*z+1)^(1/2))-ln(z)]

 (5)
> 

verify(diff([2*arctanh(sqrt((z+1)^2*(z*(z-2*(2*a-1))+1))/(z^2-1))+ln(z), 2*arctanh(sqrt((z+1)^2*(z*(z-2*(2*a-1))+1))/(z^2-1))-ln(z)], z), `~`[op](1, expr__4), simplify)

true

 (6)
> 

NULL


 

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.

How do I obtain the solutions to this ODE in one g...

Asked by: sursumCorda 922
ode dsolve differential-equations
April 30 2023
4  5

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: 
 

> 

restart;
> 

eqn := y(x)*(2*x*(diff(y(x), x))+y(x)*((diff(y(x), x))^2-1)) = -1
> 

dsolve(eqn, {y(x)}, 'parametric', 'singsol' = all)

y(x) = (c__1^2+2*c__1*x+1)^(1/2), y(x) = -(c__1^2+2*c__1*x+1)^(1/2)

 (1)
> 

dsolve(eqn, {y(x)}, 'singsol' = all)

y(x) = (-x^2+1)^(1/2), y(x) = -(-x^2+1)^(1/2), Int(-((_a^2+y(x)^2-1)^(1/2)*_a*y(x)^2-_a^2*y(x)^2-y(x)^4+2*_a^2+3*y(x)^2-2)/((y(x)^2+2*_a-2)*(-y(x)^2+2*_a+2)*(_a^2+y(x)^2-1)), _a = _b .. x)+Intat(-_f/(2*(_f^2+x^2-1)^(1/2)*x^2+(_f^2+x^2-1)^(1/2)*_f^2+2*x^3+2*x*_f^2-2*(_f^2+x^2-1)^(1/2)-2*x)-(Int(-(_a*_f^3/(_a^2+_f^2-1)^(1/2)+2*(_a^2+_f^2-1)^(1/2)*_a*_f-2*_a^2*_f-4*_f^3+6*_f)/((_f^2+2*_a-2)*(-_f^2+2*_a+2)*(_a^2+_f^2-1))+2*((_a^2+_f^2-1)^(1/2)*_a*_f^2-_a^2*_f^2-_f^4+2*_a^2+3*_f^2-2)*_f/((_f^2+2*_a-2)^2*(-_f^2+2*_a+2)*(_a^2+_f^2-1))-2*((_a^2+_f^2-1)^(1/2)*_a*_f^2-_a^2*_f^2-_f^4+2*_a^2+3*_f^2-2)*_f/((_f^2+2*_a-2)*(-_f^2+2*_a+2)^2*(_a^2+_f^2-1))+2*((_a^2+_f^2-1)^(1/2)*_a*_f^2-_a^2*_f^2-_f^4+2*_a^2+3*_f^2-2)*_f/((_f^2+2*_a-2)*(-_f^2+2*_a+2)*(_a^2+_f^2-1)^2), _a = _b .. x)), _f = y(x))+c__1 = 0, Int((_a^2*y(x)^2-2*_a^2+y(x)^4-3*y(x)^2+(_a^2+y(x)^2-1)^(1/2)*_a*y(x)^2+2)/((y(x)^2+2*_a-2)*(-y(x)^2+2*_a+2)*(_a^2+y(x)^2-1)), _a = _b .. x)+Intat(_f/(2*(_f^2+x^2-1)^(1/2)*x^2+(_f^2+x^2-1)^(1/2)*_f^2-2*x^3-2*x*_f^2-2*(_f^2+x^2-1)^(1/2)+2*x)-(Int((2*_a^2*_f+4*_f^3-6*_f+_a*_f^3/(_a^2+_f^2-1)^(1/2)+2*(_a^2+_f^2-1)^(1/2)*_a*_f)/((_f^2+2*_a-2)*(-_f^2+2*_a+2)*(_a^2+_f^2-1))-2*(_a^2*_f^2-2*_a^2+_f^4-3*_f^2+(_a^2+_f^2-1)^(1/2)*_a*_f^2+2)*_f/((_f^2+2*_a-2)^2*(-_f^2+2*_a+2)*(_a^2+_f^2-1))+2*(_a^2*_f^2-2*_a^2+_f^4-3*_f^2+(_a^2+_f^2-1)^(1/2)*_a*_f^2+2)*_f/((_f^2+2*_a-2)*(-_f^2+2*_a+2)^2*(_a^2+_f^2-1))-2*(_a^2*_f^2-2*_a^2+_f^4-3*_f^2+(_a^2+_f^2-1)^(1/2)*_a*_f^2+2)*_f/((_f^2+2*_a-2)*(-_f^2+2*_a+2)*(_a^2+_f^2-1)^2), _a = _b .. x)), _f = y(x))+c__1 = 0

 (2)
> 

NULL


 

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?

The function IsIntervalGraph needs an additional ...

Asked by: lcz 994
graphtheory
April 28 2023
1  1

An interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. Recognizing interval graphs  is in linear time. 

Seven intervals on the real line and the corresponding seven-vertex interval graph.

 

 

IsIntervalGraph(G) (was introduced in Maple 2022) tests whether the graph G could be expressed as an interval graph for some collection of intervals. If a graph is an interval graph, then the intervals corresponding to its vertices should be given. However,  IsIntervalGraphdoes not provide such an option, which makes it impossible for me to verify the correctness of the results or see more information.

with(GraphTheory):
G:=Graph({{1,2},{1,3},{1,4}, {4,2},{4,3}});
IsIntervalGraph(G)

true

Therefore, an option like the "certificate" option in SageMath needs to be provided.

g = Graph({1: [2, 3, 4], 4: [2, 3]})
g.show()
g.is_interval()
g.is_interval(certificate=True)

(True, {1: (0, 5), 2: (4, 6), 3: (1, 3), 4: (2, 7)})

 

 

I have looked at the source code of IsIntervalGraphand it seems to be checking whether the complement graph is comparability. I am not sure if this transformation can still find the corresponding intervals.

print(IsIntervalGraph)
proc(G::GRAPHLN)::truefalse;
    local G2;
    G2 := GraphTheory:-GraphComplement(G);
    return GraphTheory:-IsComparabilityGraph(G2);
end proc

print(IsComparabilityGraph)
proc (G::GRAPHLN, { transitiveorientation::truefalse := false, 

   usecached::truefalseFAIL := FAIL }, ` $`)::truefalse; local 

   iscomparability, L, A, result, V; A := op(4, G); result := 

   FindTransitiveOrientation(A, transitiveorientation); if 

   result = NULL then false elif transitiveorientation then V 

   := op(3, G); true, GraphTheory:-Graph(V, result) else true 

   end if end proc

 

By the way, can the  "FindTransitiveOrientation "  in the function IsComparabilityGraph be used by the user?

Is it possible to upgrade some external library to...

Asked by: sursumCorda 922
external
April 28 2023
2  0

https://www.maplesoft.com/support/help/Maple/view.aspx?path=copyright lists some external packages used by Maple, but it appears that certain libraries are of outdated (albeit not obsolete) versions. For example, Maple 2023 uses FLINT 2.6.3 (released in 2020), but the newest stable version of FLINT is 2.9.0. Also, Maple 2023 uses Z3 4.5.0 (released in 2016), but the newest stable version of Z3 is 4.12.1. In addition, Maple 2023 uses GCC 10.2.0 (released in 2020), but the newest stable version of GCC is 13.1. Since they are distributed under free licenses, I can download the most recent (or even nightly) release's source code, but how can I replace the old components that Maple uses by the latest ones by myself?

Does the expression sequence operator `$` work in ...

Asked by: ianmccr 165
2dmath sequence
April 26 2023
0  3

If there is a list of operations that do not work properly in 2D notation, please add the expression sequence operator `$` which does not work with strings as demonstrated in the attached worksheet . sequenceoperastionfailure.mw

It seems to work properly with numeric sequences. I only noticed this in maple 2023, but it likely applies to earlier versions as well

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