Triple Integral

[Albert Chan]

I think it is a real bug.

XCas> f := x^2*log(sqrt(y) + sqrt(y-x^2))
XCas> g := int(f, y = x^2 .. 4)         // keep running this single integral, we get
XCas> subst(g, x=1.5)                     // either 3.76506896552 or 13.7445470629

XCas 1.4.9-57 (Win32) gives 2 different answer from the same integral, g = t1 ± t2
XCas> t1 := 4*x^2*ln(sqrt(-x^2+4)+2) - x^4*ln(abs(x))
XCas> t2 := -x^2*sqrt(-x^2+4) - x^4*ln(x^2)/4 + x^4*ln(abs(x^2+4*sqrt(-x^2+4)-8))/4

XCas> int(t1+t2, x=-2..2) * 1.     // 8.06580915733 ok
XCas> int(t1+t2, x= 0..2) * 2.     // 8.06580915733 ok
XCas> int(t1+t2, x=-2..0) * 2.     // 8.06580915733 ok
XCas> int(t1-t2, x=-2..2) * 1.     // 28.1720021403 bad
XCas> int(t1-t2, x= 0..2) * 2.     // 28.1720021403 bad
XCas> int(t1-t2, x=-2..0) * 2.     // 28.1720021403 bad

Edit: numerical confirmation from EMU71

>10 P=.000001
>20 DEF FNF(X,Y)=X^2*LN(SQRT(Y)+SQRT(Y-X*X))
>30 DEF FNG(X)=INTEGRAL(X*X,4,P,FNF(X,IVAR))
>40 DEF FNH(A,B)=INTEGRAL(A,B,P,FNG(IVAR))
>RUN
>FIX 6
>FNH(-2,2), FNH(-2,0)*2, FNH(0,2)*2
  8.065809           8.065809           8.065809