Re: Bug in Mathematica 6 - Integrate - 52 (Sqrt, regression bug, invalid value)

On 16 , 13:12, David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:
Vladimir Bondarenko <v...@xxxxxxxxxxxxxxx> wrote:
What about Wolfram Research?

N[Integrate[Sqrt[1 - z^3] Sqrt[1 - z] Sqrt[2 - z], {z, 0, 1}]]
NIntegrate[Sqrt[1 - z^3] Sqrt[1 - z] Sqrt[2 - z], {z, 0, 1}]

-0.38659 - 2.73505 I
0.773266

Mathematica 6.0 defect <--- REGRESSION BUG # 2
Mathematica 5.2 unevaluated <--- REGRESSION BUG # 1
Mathematica 4.2 OK
Mathematica 3.0 unevaluated

In version 5.2,

In[9]:=
antider = Simplify[Integrate[Sqrt[1 - z^3] Sqrt[1 - z] Sqrt[2 - z], z],
0 < z < 1]; defint = (antider/.z -> 1) - (antider/.z -> 0); N[defint]

Out[9]= 0.773266 + 3.52801*10^-17*I

Perhaps the reason version 5.2 fails to evaluate the definite integral is
that it doesn't Simplify the antiderivative first, before using
Newton-Leibniz.

BTW, defint produced above is rather messy. Even if we use FullSimplify, we
still get a messy result:

(1/1470)*((-1)^(1/4)*((28 - 28*I)*(-7 + 8*Sqrt[6]) +
Sqrt[14*(5*I + Sqrt[3])]*(-71*(-5*I + Sqrt[3])*EllipticE[ArcSin[
Sqrt[(1/7)*(5 + I*Sqrt[3])]], (1/14)*(11 - 5*I*Sqrt[3])] +
71*(-5*I + Sqrt[3])*EllipticE[ArcSin[(1 + I)/Sqrt[5*I + Sqrt[3]]],
(1/14)*(11 - 5*I*Sqrt[3])] + (-285*I + 71*Sqrt[3])*
(EllipticF[ArcSin[Sqrt[(1/7)*(5 + I*Sqrt[3])]], (1/14)*(11 - 5*I*
Sqrt[3])] - EllipticF[ArcSin[(1 + I)/Sqrt[5*I + Sqrt[3]]],
(1/14)*(11 - 5*I*Sqrt[3])]))))

I wouldn't be surprised if there were a much neater way to express that
result.

David

I think for integrals like this the NL formula is applied.
For me it sounds quite mysterious why (version 5.2)

In[48]:=
Integrate[Sqrt[1 - z^3]*Sqrt[1 - z]*Sqrt[2 - z], {z, 0, 1}]
Out[48]=
Integrate[Sqrt[1 - z]*Sqrt[2 - z]*Sqrt[1 - z^3], {z, 0, 1}]

whereas

In[53]:=
Limit[Integrate[Sqrt[1 - z^3]*Sqrt[1 - z]*Sqrt[2 - z], z], z -> 1] -
Limit[Integrate[Sqrt[1 - z^3]*Sqrt[1 - z]*Sqrt[2 - z], z],
z -> 0]
Out[53]=
(1/210)*(-28*Sqrt[2] - (1/(7*Sqrt[-(I/(5*I + Sqrt[3]))]))*(Sqrt[2*(2 -
I*Sqrt[3])]*Sqrt[2 + I*Sqrt[3]]*
(71*(-5*I + Sqrt[3])*EllipticE[I*Log[(1/7)*(Sqrt[-35 -
7*I*Sqrt[3]] + Sqrt[14 - 7*I*Sqrt[3]])],
(5*I + Sqrt[3])/(5*I - Sqrt[3])] + (285*I - 71*Sqrt[3])*
EllipticF[I*Log[(1/7)*(Sqrt[-35 - 7*I*Sqrt[3]] + Sqrt[14 -
7*I*Sqrt[3]])], (5*I + Sqrt[3])/(5*I - Sqrt[3])]))) +
(64 + (Sqrt[(1/2)*(9 - I*Sqrt[3])]*Sqrt[9 + I*Sqrt[3]]*
(71*(-5*I + Sqrt[3])*EllipticE[I*Log[Sqrt[(1/14)*(9 -
I*Sqrt[3])] + Sqrt[-((2*I)/(5*I + Sqrt[3]))]],
(5*I + Sqrt[3])/(5*I - Sqrt[3])] + (285*I - 71*Sqrt[3])*
EllipticF[I*Log[Sqrt[(1/14)*(9 - I*Sqrt[3])] + Sqrt[-((2*I)/
(5*I + Sqrt[3]))]], (5*I + Sqrt[3])/(5*I - Sqrt[3])]))/
(21*Sqrt[-(I/(5*I + Sqrt[3]))]))/(70*Sqrt[3])


Dimitris

.

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