Optimal. Leaf size=13 \[ \tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]
[Out]
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Rubi [A] time = 0.0174192, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[x]*(1 - x^2)),x]
[Out]
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Rubi in Sympy [A] time = 1.90019, size = 12, normalized size = 0.92 \[ \operatorname{atan}{\left (\sqrt{x} \right )} + \operatorname{atanh}{\left (\sqrt{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**2+1)/x**(1/2),x)
[Out]
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Mathematica [B] time = 0.00840691, size = 33, normalized size = 2.54 \[ -\frac{1}{2} \log \left (1-\sqrt{x}\right )+\frac{1}{2} \log \left (\sqrt{x}+1\right )+\tan ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[x]*(1 - x^2)),x]
[Out]
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Maple [A] time = 0.006, size = 10, normalized size = 0.8 \[ \arctan \left ( \sqrt{x} \right ) +{\it Artanh} \left ( \sqrt{x} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^2+1)/x^(1/2),x)
[Out]
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Maxima [A] time = 0.809102, size = 28, normalized size = 2.15 \[ \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((x^2 - 1)*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283119, size = 28, normalized size = 2.15 \[ \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((x^2 - 1)*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.31764, size = 26, normalized size = 2. \[ - \frac{\log{\left (\sqrt{x} - 1 \right )}}{2} + \frac{\log{\left (\sqrt{x} + 1 \right )}}{2} + \operatorname{atan}{\left (\sqrt{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**2+1)/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.263804, size = 30, normalized size = 2.31 \[ \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \,{\rm ln}\left (\sqrt{x} + 1\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | \sqrt{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((x^2 - 1)*sqrt(x)),x, algorithm="giac")
[Out]