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I was playing with a problem from the Maple NG, one can state it as
  
  Int( arccos(x) / ( 1+x^4) , x=0 .. 1)

Maple 11.02 gives a result, which numerical can not be valid.

Using real (!) partial fractions (Maple uses decomposition over the
complex, no?) I got a similar problem with denominator = parabola
(and continuity over the integration interval):

  Int( arccos(x) / (x^2 - x * 2^(1/2) + 1), x = 0 .. 1)

Some more and time-consuming consuming experiments reduces troubles
to the following example, where symbolics are disproven by numerics:

  Int( arccos(x)/(x - a), x = 0 .. 1), a = (1 - I ) / sqrt(2)

The interesting thing is: for the conjugate a it works.

And the only special situation I see: abs(zero(denom)) = 1.

Hope it is of interest, find a worksheet enclosed showing more details,
and would be happy about comments.

www.mapleprimes.com/files/102_ugly_trig_integrals.mws

 Edited: I shifted that thread into "Tips / Maple Techniques", it was in the wrong section
complex integral

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