7.56 Bug in Elliptic Integral, Maple 6 to Maple 8 (24.9.01)

7.56.1 James R. FitzSimons

Maple is giving the wrong answer using elliptic integrals.

> evalf(Int(sqrt((x^2+1)/(x+2)), x=-1..5)); 
 
                             7.277500982 
 
> evalf(int(sqrt((x^2+1)/(x+2)), x=-1..5)); 
 
                     14.18222461 + 4.701625434 I
 

This is the numeric result using DERIVE. 7.2775009819508996444

7.56.2 Koch-Beuttenmueller (26.9.01)

In MapleVr5.1 the results were still identical, but since Maple6 they are different:

> i1:=evalf(Int(sqrt((x^2+1)/(x+2)), x=-1..5)); 
 
                          i1 := 7.277500982 
 
> i2:=int(sqrt((x^2+1)/(x+2)), x=-1..5); 
 
  i2 := 2/3 sqrt(182) - 4/1365 I sqrt(182) sqrt(2 - 10 I) 
 
        sqrt(70 + 35 I) sqrt(2) sqrt(1 + 5 I) 
 
        EllipticE(1/2 sqrt(2 - 10 I), %1) + 1/273 I sqrt(182) 
 
        sqrt(2 - 10 I) sqrt(70 + 35 I) sqrt(2) sqrt(1 + 5 I) 
 
        EllipticF(1/2 sqrt(2 - 10 I), %1) - 2/1365 sqrt(182) 
 
        sqrt(2 - 10 I) sqrt(70 + 35 I) sqrt(2) sqrt(1 + 5 I) 
 
        EllipticE(1/2 sqrt(2 - 10 I), %1) - 2/3 sqrt(2) + 8/15 I 
 
        sqrt(2 + 2 I) sqrt(10 + 5 I) sqrt(1 - I) 
 
        EllipticE(1/2 sqrt(2 + 2 I), %1) - 2/3 I sqrt(2 + 2 I) 
 
        sqrt(10 + 5 I) sqrt(1 - I) EllipticF(1/2 sqrt(2 + 2 I), %1) 
 
         + 4/15 sqrt(2 + 2 I) sqrt(10 + 5 I) sqrt(1 - I) 
 
        EllipticE(1/2 sqrt(2 + 2 I), %1) 
 
  %1 := 1/5 sqrt(10 - 20 I) 
 
> evalf(%); 
 
                                          -8 
                       7.277500973 + .3 10   I 
 
> interface(version); 
 
  Maple Worksheet Interface, Release 5.1, SUN SPARC SOLARIS, Jan 7\ 
         1999 
 
> i2:=int(sqrt((x^2+1)/(x+2)), x=-1..5); 
  i2 := -2/6825 I sqrt(-45 - 35 I) sqrt(-45 + 35 I) sqrt(70 + 35 I) 
 
        sqrt(182) EllipticF(1/5 sqrt(70 + 35 I), %1) + 2/3 sqrt(182) 
 
         + 4/6825 sqrt(-45 - 35 I) sqrt(-45 + 35 I) sqrt(70 + 35 I) 
 
        sqrt(182) EllipticE(1/5 sqrt(70 + 35 I), %1) - 2/3 sqrt(2) - 
 
        4/75 sqrt(15 - 5 I) sqrt(15 + 5 I) sqrt(10 + 5 I) sqrt(2) 
 
        EllipticE(1/5 sqrt(10 + 5 I), %1) + 2/75 I sqrt(15 - 5 I) 
 
        sqrt(15 + 5 I) sqrt(10 + 5 I) sqrt(2) 
 
        EllipticF(1/5 sqrt(10 + 5 I), %1) 
 
  %1 := 2/5 sqrt(5) - 1/5 I sqrt(5) 
 
> evalf(%); 
 
                     14.18222461 + 4.701625434 I 
 
> interface(version); 
 
  Maple Worksheet Interface, Maple 6.01, SUN SPARC SOLARIS, June 9  2000 Build ID 79514
 

Until now I told my students it is good to use different CAS to test if an integral is right. Now I can even tell them use different Maple versions to see if the results can be correct ?

7.56.3 Gerald A. Edgar (26.9.01)

It may be related to choice of square-roots...

> evalf(int(sqrt((x^2+1)/(x+2)),x=-1..5)); 
 
                     14.18222460 + 4.701625434 I 
 
> evalf(int(sqrt(x^2+1)/sqrt(x+2),x=-1..5)); 
 
                             7.277500972