# Strange integration #2

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]


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]


0.64368050414630466770404234158695708461111179796355


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

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

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

(*  π^2/8 +
1/6 π (3 ArcCoth[Sqrt] +
Hypergeometric2F1[3/4, 3/4, 7/4,
1/4 (2 - Sqrt)]/(2 + Sqrt)^(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  *)