3.815 \(\int \frac{1}{\sqrt{x} \left (1-x^2\right )} \, dx\)

Optimal. Leaf size=13 \[ \tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

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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]

ArcTan[Sqrt[x]] + ArcTanh[Sqrt[x]]

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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]

atan(sqrt(x)) + atanh(sqrt(x))

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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]

ArcTan[Sqrt[x]] - Log[1 - Sqrt[x]]/2 + Log[1 + Sqrt[x]]/2

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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]

arctan(x^(1/2))+arctanh(x^(1/2))

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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]

arctan(sqrt(x)) + 1/2*log(sqrt(x) + 1) - 1/2*log(sqrt(x) - 1)

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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]

arctan(sqrt(x)) + 1/2*log(sqrt(x) + 1) - 1/2*log(sqrt(x) - 1)

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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]

-log(sqrt(x) - 1)/2 + log(sqrt(x) + 1)/2 + atan(sqrt(x))

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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]

arctan(sqrt(x)) + 1/2*ln(sqrt(x) + 1) - 1/2*ln(abs(sqrt(x) - 1))