Mathematica is able to compute an indefinite integral but not the corresponding definite one

Question

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...):

• 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.

Answer 1

$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