3.266 \(\int \frac{\sqrt{-2+x}}{2+x} \, dx\)

Optimal. Leaf size=24 \[ 2 \sqrt{x-2}-4 \tan ^{-1}\left (\frac{\sqrt{x-2}}{2}\right ) \]

[Out]

2*Sqrt[-2 + x] - 4*ArcTan[Sqrt[-2 + x]/2]

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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]

2*Sqrt[-2 + x] - 4*ArcTan[Sqrt[-2 + x]/2]

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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]

2*sqrt(x - 2) - 4*atan(sqrt(x - 2)/2)

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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]

2*Sqrt[-2 + x] - 4*ArcTan[Sqrt[-2 + x]/2]

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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]

-4*arctan(1/2*(-2+x)^(1/2))+2*(-2+x)^(1/2)

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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]

2*sqrt(x - 2) - 4*arctan(1/2*sqrt(x - 2))

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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]

2*sqrt(x - 2) - 4*arctan(1/2*sqrt(x - 2))

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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]

Piecewise((-4*I*acosh(2/sqrt(x + 2)) - 2*I*sqrt(x + 2)/sqrt(-1 + 4/(x + 2)) + 8*
I/(sqrt(-1 + 4/(x + 2))*sqrt(x + 2)), 4*Abs(1/(x + 2)) > 1), (4*asin(2/sqrt(x +
2)) + 2*sqrt(x + 2)/sqrt(1 - 4/(x + 2)) - 8/(sqrt(1 - 4/(x + 2))*sqrt(x + 2)), T
rue))

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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]

2*sqrt(x - 2) - 4*arctan(1/2*sqrt(x - 2))