#### 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