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I'm trying to compute the following definite integral (μ is a parameter):

Integrate[Sqrt[3]/Sqrt[1 + Sqrt[1 + 12*u^2 - 24*μ]], {u, -Sqrt[1 + 8*μ]/2, Sqrt[1 + 8*μ]/2}]

Only for some specific case of μ Mathematica seems to be able to compute it. The "funny" things are that:

  • if I keep the integral indefinite, it returns me a solution (quite ugly, but at least...): enter image description here

  • if I give precise values for the boundary (e.g, u = ±1/2), after a long time it just returns the definite integral without any result.

  • if I additionally specify a precise value of μ (so it knows μ + boundary of integration), in one lucky case it is able to directly give me the result for the definite integral; this does not match with the value I would obtain by using the fundamental theorem of calculus (i.e. substituting the values of μ and u in the ugly formula and taking the difference).

Has any of you an idea about what the problem could be? I would like to also point out that the square roots are always well definite for the values I consider.

Thank you.

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  • $\begingroup$ Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Read the faq! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! $\endgroup$ – Michael E2 Jan 15 '15 at 14:44
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    $\begingroup$ Possibly it fails in searching for path singularities. Elliptics can be quite troublesome that way. $\endgroup$ – Daniel Lichtblau Jan 15 '15 at 15:38
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$Version

"10.0 for Mac OS X x86 (64-bit) (December 4, 2014)"

int[mu_] = Integrate[
   Sqrt[3]/Sqrt[1 + Sqrt[1 + 12*u^2 - 24*mu]],
   {u, -Sqrt[1 + 8*mu]/2, Sqrt[1 + 8*mu]/2}] //
  FullSimplify

ConditionalExpression[ (12*Sqrt[mu/(-1 + Sqrt[ 1 - 24*mu])]*(1 + 8*mu) - I*Sqrt[6 + 48*mu]* ((1 + Sqrt[1 - 24*mu])* EllipticE[(1/Sqrt[mu])* (I*Sqrt[-1 + Sqrt[ 1 - 24*mu]]Sqrt[ mu/(-1 + Sqrt[1 - 24 mu])]*ArcSinh[ (2*Sqrt[2]*Sqrt[mu])/ Sqrt[-1 + Sqrt[1 - 24*mu]]]), -((-1 + Sqrt[1 - 24*mu] + 12*mu)/(12*mu))] - Sqrt[1 - 24*mu]EllipticF[ (1/Sqrt[mu]) (I*Sqrt[-1 + Sqrt[ 1 - 24*mu]]Sqrt[ mu/(-1 + Sqrt[1 - 24 mu])]*ArcSinh[ (2*Sqrt[2]*Sqrt[mu])/ Sqrt[-1 + Sqrt[1 - 24*mu]]]), -((-1 + Sqrt[1 - 24*mu] + 12*mu)/(12*mu))]))/ (6*Sqrt[mu/(-1 + Sqrt[ 1 - 24*mu])]* Sqrt[1 + 8*mu]), Element[Sqrt[1 + 8*mu], Reals] && (Sqrt[3] + Re[Sqrt[-1 + 24*mu]/ Sqrt[1 + 8*mu]] < 0 || Re[Sqrt[-1 + 24*mu]/ Sqrt[1 + 8*mu]] > Sqrt[3] || NotElement[Sqrt[-1 + 24*mu]/ Sqrt[1 + 8*mu], Reals])]

The stated conditions for the integral to exist are

int[mu][[-1]]

Element[Sqrt[1 + 8*mu], Reals] && (Sqrt[3] + Re[Sqrt[-1 + 24*mu]/ Sqrt[1 + 8*mu]] < 0 || Re[Sqrt[-1 + 24*mu]/ Sqrt[1 + 8*mu]] > Sqrt[3] || NotElement[Sqrt[-1 + 24*mu]/ Sqrt[1 + 8*mu], Reals])

This requires that

Reduce[int[mu][[-1]], mu]

-(1/8) < mu < 1/24

Consequently, the integral is undefined for mu equal to -1/2 or 1/2

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  • $\begingroup$ This is exactly the range of values I am considering for Mu, +-1/2 were the values I was giving to u. I will try to do the computations using a more advanced version of Mathematica, hoping it works. $\endgroup$ – annuk89 Jan 15 '15 at 15:43

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