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Why can't Mathematica solve this integral in definite form?

Posted 10 years ago

Hi guys, Could you help me out? Why can't Mathematica solve this integral in definite form? It's just stuck computing.

Integrate[(1/Sqrt[(x - 1) (x^2 - a^2)]) - (1/ Sqrt[(x + 1) (x^2 - a^2)]), {x, a, b}] ->And it stuck

Integrate[(1/Sqrt[(x - 1) (x^2 - a^2)]) - (1/ Sqrt[(x + 1) (x^2 - a^2)]] -> this gets solved

Am I doing something wrong?

Thanks, M.

POSTED BY: M. R.
3 Replies

Even if you use Assumptions->1<a<b I believe Integrate will have trouble on attempting to find Limit of the antiderivative at the endpoints. I do eventually get a result for this.

Integrate[1/Sqrt[(x - 1)*(x^2 - a^2)] - 1/Sqrt[(x + 1)*(x^2 - a^2)], 
    {x, a, b}, Assumptions -> b > a > 1]

(2*((2*I)*Sqrt[(-1 + a^2)*(-1 + b)] - (2*I)*Sqrt[(-1 + a^2)*(1 + b)] + 
   Sqrt[(1 + a)*(-1 + b^2)]*EllipticF[ArcSin[Sqrt[(a + b)/a]/Sqrt[2]], 
     (2*a)/(-1 + a)] - Sqrt[(-1 + a)*(-1 + b^2)]*
    EllipticF[ArcSin[Sqrt[(a + b)/a]/Sqrt[2]], (2*a)/(1 + a)] - 
   Sqrt[(1 + a)*(-1 + b^2)]*EllipticK[(2*a)/(-1 + a)] + 
   Sqrt[(-1 + a)*(-1 + b^2)]*EllipticK[(2*a)/(1 + a)]))/
 Sqrt[(-1 + a^2)*(-1 + b^2)]
POSTED BY: Daniel Lichtblau
Posted 10 years ago

Wow thanks, Marco. Thank you for the detailed answer. This does indeed seem to be the problem (I should have checked that myself).

POSTED BY: M. R.

The problem might be the singularity at the lower integration boundary. When x equals a the denominators become zero. You can also see that when you plot the integrand for some values a and b:

Plot[(1/Sqrt[(x - 1) (x^2 - a^2)]) - (1/Sqrt[(x + 1) (x^2 - a^2)]) /. a -> 1, {x, 1, 2}] 

enter image description here

You can also check that with Wolfram alpha:

WolframAlpha["Domain of (1/Sqrt[(x - 1) (x^2 - a^2)]) - (1/Sqrt[(x + \ 1) (x^2 - a^2)])", {{"Result", 1}, "Output"}]

This gives:

HoldComplete[(a < -1 && (-Sqrt[a^2] < x < -1 || x > Sqrt[a^2])) || (-1 <= a <= 1 && x > 1) || (a > 1 && (-Sqrt[a^2] < x < -1 || x > Sqrt[a^2]))]

When you solve the indefinite integral:

Integrate[(1/Sqrt[(x - 1) (x^2 - a^2)]) - 1/Sqrt[(x + 1) (x^2 - a^2)], x]

you obtain

(2 I Sqrt[1 - (-1 + a)/(-1 + x)] Sqrt[1 + (1 + a)/(-1 + x)] (-1 + x)^(3/2) Sqrt[-a^2 + x^2] EllipticF[I ArcSinh[Sqrt[1 + a]/Sqrt[-1 + x]](1 - a)/(1 + a)])/(Sqrt[1 + a] Sqrt[1 - a^2 + 2 (-1 + x) + (-1 + x)^2]Sqrt[(-1 + x) (-a^2 + x^2)]) - (2 I (1 + x)^(3/2) Sqrt[-a^2 + x^2] Sqrt[1 + (-1 + a)/(1 + x)] Sqrt[1 - (1 + a)/(1 + x)] EllipticF[I ArcSinh[Sqrt[-1 - a]/Sqrt[1 + x]], (1 - a)/(1 + a)])/(Sqrt[-1 - a] Sqrt[(1 + x) (-a^2 + x^2)] Sqrt[1 - a^2 - 2 (1 + x) + (1 + x)^2])

which is also clearly not defined at x==1.

Cheers, Marco

POSTED BY: Marco Thiel
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