7.60 bug in elliptic integration in Maple 5.1 (2.5.01)

7.60.1 Nathan Sokalski

When using Maple V and Maple 6 I am recieving two different answers when using the int() function with a diff() inside of it. The entered lines are exactly the same. A book of mine gives the answer shown by Maple 6, but I have never recieved any different any between the 2 versions when using int() or diff() before.

Where is the problem, or is it just a special situation?

Maple V:

>f:=x->x^3; 
 
                                        3 
                             f := x -> x 
 
>int(sqrt(1+(diff(f(x),x))^2),x=0..4); evalf(%); 
 
  4/3 sqrt(2305) + 2/3 EllipticK(1/2 sqrt(2)) 
 
                         24 
         - 1/3 EllipticF(---, 1/2 sqrt(2)) 
                         145 
 
                             65.19438187
 

Maple 6:

>f:=x->x^3; 
 
                                        3 
                             f := x -> x 
 
>int(sqrt(1+(diff(f(x),x))^2),x=0..4); evalf(%); 
 
  4/3 sqrt(2305) + 2/9 sqrt(3) EllipticK(1/2 sqrt(2)) 
 
         - 1/9 sqrt(3) EllipticF(8/49 sqrt(3), 1/2 sqrt(2)) 
 
                             64.67196791
                                                                                    
                                                                                    
 

7.60.2 Robert Israel (3.5.01)

Maple 6 is correct here. It’s a bug in previous releases, which has now been corrected. To explore it further:

> assume(P > 4, P < 6); 
> JP:= int(sqrt(1+(diff(f(x),x))^2),x=0..P);
 

Release 5.1 has

  JP := 1/3 P sqrt(1 + 9 P ) 
 
         + 2/3 EllipticK(1/2 sqrt(2)) - 1/3 
 
                       P 
        EllipticF(6 --------, 1/2 sqrt(2)) 
                       2 
                    9 P  + 1
 

But:

> simplify(diff(JP, P) - sqrt(1+diff(f(x),x))^2)); 
 
                  4                      4 
        -sqrt(81 P  + 1) + 3 sqrt(1 + 9 P ) 
    2/3 ----------------------------------- 
                      4           4 
          sqrt(1 + 9 P ) sqrt(81 P  + 1)
 

This should be 0.

Maple 6 has

                          4 
  JP := 1/3 P sqrt(1 + 9 P ) 
 
         + 2/9 sqrt(3) EllipticK(1/2 sqrt(2)) 
 
                                   sqrt(3) P 
         - 1/9 sqrt(3) EllipticF(2 ---------, 1/2 sqrt(2)) 
                                      2 
                                   3 P  + 1 
 
> simplify(diff(JP, P) - sqrt(1+diff(f(x),x))^2)); 
 
               0