Optimal. Leaf size=14 \[ -\tanh ^{-1}\left (\frac{\sqrt{x+2}}{2}\right ) \]
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Rubi [A] time = 0.0141759, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\tanh ^{-1}\left (\frac{\sqrt{x+2}}{2}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/((-2 + x)*Sqrt[2 + x]),x]
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Rubi in Sympy [A] time = 2.40351, size = 10, normalized size = 0.71 \[ - \operatorname{atanh}{\left (\frac{\sqrt{x + 2}}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-2+x)/(2+x)**(1/2),x)
[Out]
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Mathematica [B] time = 0.00629183, size = 31, normalized size = 2.21 \[ \frac{1}{2} \log \left (2-\sqrt{x+2}\right )-\frac{1}{2} \log \left (\sqrt{x+2}+2\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((-2 + x)*Sqrt[2 + x]),x]
[Out]
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Maple [B] time = 0.011, size = 22, normalized size = 1.6 \[ -{\frac{1}{2}\ln \left ( 2+\sqrt{2+x} \right ) }+{\frac{1}{2}\ln \left ( -2+\sqrt{2+x} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-2+x)/(2+x)^(1/2),x)
[Out]
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Maxima [A] time = 1.32925, size = 28, normalized size = 2. \[ -\frac{1}{2} \, \log \left (\sqrt{x + 2} + 2\right ) + \frac{1}{2} \, \log \left (\sqrt{x + 2} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x + 2)*(x - 2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207266, size = 28, normalized size = 2. \[ -\frac{1}{2} \, \log \left (\sqrt{x + 2} + 2\right ) + \frac{1}{2} \, \log \left (\sqrt{x + 2} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x + 2)*(x - 2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.74843, size = 27, normalized size = 1.93 \[ \begin{cases} - \operatorname{acoth}{\left (\frac{\sqrt{x + 2}}{2} \right )} & \text{for}\: \frac{\left |{x + 2}\right |}{4} > 1 \\- \operatorname{atanh}{\left (\frac{\sqrt{x + 2}}{2} \right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-2+x)/(2+x)**(1/2),x)
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GIAC/XCAS [A] time = 0.219974, size = 30, normalized size = 2.14 \[ -\frac{1}{2} \,{\rm ln}\left (\sqrt{x + 2} + 2\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | \sqrt{x + 2} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x + 2)*(x - 2)),x, algorithm="giac")
[Out]