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I was trying to (re)calculate a problem of an older Wolfram blog post (Problem 11457, by M. L. Glasser) with Mathematica 9.0.0.0 (on OS X 10.8.2).

Assuming[0 < a < b, Integrate[ArcCos[x/Sqrt[(a + b) x - a b]], {x, a, b}]]

Instead of the expected solution, it just returns the integral unevaluated. Is this a regression?

More details: As pointed out in the commentes, the indefinite integral

Integrate[ArcCos[x/Sqrt[(a + b) x - a b]], x]

still gives the same result in Mathematica 8 and 9.

The next two each returned ConditionalExpression in Mathematica 8 but return unevaluated in Mathematica 9:

Integrate[ArcCos[x/Sqrt[(a + b) x - a b]], {x, a, b}]
Integrate[ArcCos[x/Sqrt[(a + b) x - a b]], {x, a, b}, Assumptions -> 0 <= a <= b]

The actual problem

Integrate[ArcCos[x/Sqrt[(a + b) x - a b]], {x, a, b}, Assumptions -> 0 < a < b]

computes correctly to ((a - b)^2 \[Pi])/(4 (a + b)) in Mathematica 8 but still returns unevaulated in Mathematica 9.

J. M.'s missing motivation
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Norbert Fabritius
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    I can confirm that it does not work in MMA 9 win 7 64 bit, but works in MMA 8.0.1. In MMA8 I get ((a - b)^2 \[Pi])/(4 (a + b)) – Ajasja Jan 24 '13 at 12:16
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    The topic is misleading. I thought it was about Regression in Statistics – asim Jan 24 '13 at 14:35
  • It works as expected in Mma 8.0.0 too – Dr. belisarius Jan 24 '13 at 14:54
  • You can also send to this support@wolfram.com and report it as an issue with 9. – Searke Jan 24 '13 at 15:37
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    Mathematica 8 and 9 give the same correct indefinite integral. The difference is in the calculation for the limits of integration. – Searke Jan 24 '13 at 15:45
  • So solving the indefinite integral must use a different branch cut. All the limits looks like they're being calculated correctly, but ultimately ends up giving a different result. The limits and indefinite integral are correct and identical in 8 and 9. So this looks like a regression with integrate itself. – Searke Jan 24 '13 at 16:11
  • As suggested, I emailed support@wolfram.com about this. – Norbert Fabritius Jan 24 '13 at 16:30
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    Here are another instances of definite integrals which are unevaluated in ver.9 while they are in ver.8 http://mathematica.stackexchange.com/questions/18327/definite-integral-takes-a-very-long-time/18328#18328 – Artes Jan 24 '13 at 17:18
  • @Artes I may have missed something at your link. I did not see a regression example therein. If you post an explicit Integrate[...] I'll take a look. – Daniel Lichtblau Jan 25 '13 at 01:17
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    [I am NOT putting this into a response.] Yes, this appears to be a regression. Investigating... – Daniel Lichtblau Jan 25 '13 at 01:18
  • just to experiment, what if you split the Assumptions into 0 < a && 0 < b && a < b – alfC May 16 '13 at 02:54
  • Separating the conditions does not help. Setting GenerateConditions -> False does however. – Norbert Fabritius Oct 18 '13 at 11:38

1 Answers1

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This works in V9.0.1:

Assuming[0 < a < b, 
  Integrate[ArcCos[x/Sqrt[(a + b) x - a b]], {x, a, b}, 
   GenerateConditions -> False]] // Timing
(* {3.835651, ((a - b)^2 π)/(4 (a + b))} *)
Michael E2
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