Using SageMathCloud I was surprised to find this bug:

integral(sqrt(cos(x)-cos(x)^3), x, 0, pi/2)
-2/3

The correct result is 2/3 (luckily I noticed that something was wrong 
because the result must be positive!)

numerical_integral gets it right:

numerical_integral(sqrt(cos(x)-cos(x)^3), 0, pi/2)
(0.6666666687973078, 4.409680787972339e-07)

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