inttau[r_]=-(1/Sqrt[0.0345106153943703 - ((37.3042 - r) (-25.578 + r) (62.8822 + r))/(3000 r)])
This is my function of r, now I integrated it w r t r
tauanalytical[r_] = Integrate[inttau[r], r]
This is what I got after integration, I got this result.
-((2.3094 Sqrt[1. + 62.1447/r] (-31.0723 + r) EllipticPi[3.,ArcSin[0.57735 Sqrt[1. + 62.1447/r]], 1.])/Sqrt[-0.965489 + 20./r + 0.000333333 r^2])
later I tried to find the numerical value of;
tauanalytical // N
I got this, no the exact number , which I was expecting
-126.491 EllipticPi[3., -1.5708 + 0.153761 I, 1.]
Here the second argument of Ellepticpi comes out to be imaginary , it should be real . Please provide necessary assistance.