In Mathematica, the symbolic definite integration
Integrate[Sqrt[1 - x^2] - (1 - Sqrt[3 + 2 x - x^2]), {x, a, b},
Assumptions -> {a ∈ Reals, b ∈ Reals}] // AbsoluteTiming
Takes 21.987258 seconds on my computer.
In Maple, a similar integration
int(sqrt(1 - x^2) - (sqrt(3 + 2x - x^2)), x = a..b)
takes only 0.087 seconds.
This happens because Mathematica tend to give a mathematically rigorous solution. I did not know this before, and I think it's good to share.
The output of the Mathematica code is
ConditionalExpression[
1/2 (2 a - a Sqrt[1 - a^2] + Sqrt[3 + 2 a - a^2] - a Sqrt[3 + 2 a - a^2] - 2 b +
b Sqrt[1 - b^2] - Sqrt[3 + 2 b - b^2]+ b Sqrt[3 + 2 b - b^2] + 4 ArcSin[(1 - a)/2]
*Routine clean-up*-ArcSin[a] - 4 ArcSin[(1 - b)/2] + ArcSin[b]),
-1 < b < 1 && ( -1 < a < b || b < a < 1)]
Which gives not only the integral, but also the condition when the result is valid. The latter task took most of the 20 seconds. If you don't care about the condition, or that you know the condition will be met, simply use indefinite integral, and do the subtraction.
F[x_] = Integrate[Sqrt[1 - x^2] - (1 - Sqrt[3 + 2 x - x^2]), x]
F[b] - F[a]
This takes only 0.152009 seconds. Still slower than Maple, but very much acceptable.
Integratecould tell you for sure. Additionally, it would be better if you would specify what Maple is doing to evaluate the integral and what result it gives. $\endgroup$GenerateConditions -> False. The integration takes about half as long, but is still much slower than if this were the only time-consuming operation. $\endgroup$