Optimal. Leaf size=24 \[ 2 \sqrt{x-2}-4 \tan ^{-1}\left (\frac{\sqrt{x-2}}{2}\right ) \]
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Rubi [A] time = 0.0196514, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 2 \sqrt{x-2}-4 \tan ^{-1}\left (\frac{\sqrt{x-2}}{2}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-2 + x]/(2 + x),x]
[Out]
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Rubi in Sympy [A] time = 1.60992, size = 19, normalized size = 0.79 \[ 2 \sqrt{x - 2} - 4 \operatorname{atan}{\left (\frac{\sqrt{x - 2}}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-2+x)**(1/2)/(2+x),x)
[Out]
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Mathematica [A] time = 0.0110829, size = 24, normalized size = 1. \[ 2 \sqrt{x-2}-4 \tan ^{-1}\left (\frac{\sqrt{x-2}}{2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-2 + x]/(2 + x),x]
[Out]
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Maple [A] time = 0.009, size = 19, normalized size = 0.8 \[ -4\,\arctan \left ( 1/2\,\sqrt{-2+x} \right ) +2\,\sqrt{-2+x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-2+x)^(1/2)/(2+x),x)
[Out]
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Maxima [A] time = 1.49346, size = 24, normalized size = 1. \[ 2 \, \sqrt{x - 2} - 4 \, \arctan \left (\frac{1}{2} \, \sqrt{x - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 2)/(x + 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206545, size = 24, normalized size = 1. \[ 2 \, \sqrt{x - 2} - 4 \, \arctan \left (\frac{1}{2} \, \sqrt{x - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 2)/(x + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.5413, size = 109, normalized size = 4.54 \[ \begin{cases} - 4 i \operatorname{acosh}{\left (\frac{2}{\sqrt{x + 2}} \right )} - \frac{2 i \sqrt{x + 2}}{\sqrt{-1 + \frac{4}{x + 2}}} + \frac{8 i}{\sqrt{-1 + \frac{4}{x + 2}} \sqrt{x + 2}} & \text{for}\: 4 \left |{\frac{1}{x + 2}}\right | > 1 \\4 \operatorname{asin}{\left (\frac{2}{\sqrt{x + 2}} \right )} + \frac{2 \sqrt{x + 2}}{\sqrt{1 - \frac{4}{x + 2}}} - \frac{8}{\sqrt{1 - \frac{4}{x + 2}} \sqrt{x + 2}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2+x)**(1/2)/(2+x),x)
[Out]
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GIAC/XCAS [A] time = 0.19962, size = 24, normalized size = 1. \[ 2 \, \sqrt{x - 2} - 4 \, \arctan \left (\frac{1}{2} \, \sqrt{x - 2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x - 2)/(x + 2),x, algorithm="giac")
[Out]