0
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Bug introduced in 8 or earlier and fixed in 10


I was working on this question. On Mathematica 9 for the query

N[Integrate[(ArcSin[x])/Sqrt[2*x - x^2], {x, 0, 1}], 50]

I've got the answer

4.3490873650146277599014049625438707032244466030283

which seems to be incorrect. It seems that the query without the N[] part gives an incorrect analytic solution too. For

NIntegrate[(ArcSin[x])/Sqrt[2*x - x^2], {x, 0, 1}, 
 WorkingPrecision -> 50]

I've got the answer

0.64368050414630466770404234158695708461111179796355

which seems to be correct. Any idea what happened here?

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  • 1
    $\begingroup$ On 11.1.0 for Microsoft Windows (64-bit) (March 13, 2017) in both cases you get 0.64... $\endgroup$ – TeM Aug 20 '17 at 12:34
  • $\begingroup$ Can confirm; bug is no longer present in versions 10 and 11. Interestingly version 5 also returns a different result: 5.578... $\endgroup$ – J. M. will be back soon Aug 20 '17 at 12:57
  • $\begingroup$ @J.M. Does version 10 or 11 give a correct analytic solution too? $\endgroup$ – user153012 Aug 20 '17 at 13:44
  • $\begingroup$ The result of Integrate[] in version 11 gives a result consistent with that of NIntegrate[]. $\endgroup$ – J. M. will be back soon Aug 20 '17 at 13:53
  • 1
    $\begingroup$ After simplification I get π^2/8 + π/6 (3 ArcCoth[Sqrt[3]] + Hypergeometric2F1[3/4, 3/4, 7/4, (2 - Sqrt[3])/4]/(2 + Sqrt[3])^(3/4)) - (2 Gamma[3/4]^2 HypergeometricPFQ[{1/4, 1/4, 3/4, 3/4}, {1/2, 5/4, 5/4}, 1/4] + Gamma[1/4]^2/72 HypergeometricPFQ[{3/4, 3/4, 5/4, 5/4}, {3/2, 7/4, 7/4}, 1/4])/Sqrt[π]. $\endgroup$ – J. M. will be back soon Aug 20 '17 at 14:02
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$Version

(*  "11.1.1 for Mac OS X x86 (64-bit) (April 18, 2017)"  *)

f[x_] = ArcSin[x]/Sqrt[2*x - x^2];

Plot[f[x], {x, 0, 1}]

enter image description here

int = Integrate[f[x], {x, 0, 1}] // FullSimplify

(*  π^2/8 + 
 1/6 π (3 ArcCoth[Sqrt[3]] + 
    Hypergeometric2F1[3/4, 3/4, 7/4, 
     1/4 (2 - Sqrt[3])]/(2 + Sqrt[3])^(3/4)) - (1/(9 Sqrt[π]))
 2 (9 Gamma[3/
      4]^2 HypergeometricPFQ[{1/4, 1/4, 3/4, 3/4}, {1/2, 5/4, 5/4}, 1/
      4] + Gamma[5/
      4]^2 HypergeometricPFQ[{3/4, 3/4, 5/4, 5/4}, {3/2, 7/4, 7/4}, 1/
      4])  *)

int // N[#, 50] &

(*  0.64368050414630466770404234158695708461111179796355  *)

NIntegrate[f[x], {x, 0, 1}, WorkingPrecision -> 50]

(*  0.64368050414630466770404234158695708461111179796355  *)
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