3.715 \(\int \frac{\sqrt{1-x^2}}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=2 \[ \sinh ^{-1}(x) \]

[Out]

ArcSinh[x]

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Rubi [A]  time = 0.00472519, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^2]/Sqrt[1 - x^4],x]

[Out]

ArcSinh[x]

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Rubi in Sympy [A]  time = 1.51386, size = 2, normalized size = 1. \[ \operatorname{asinh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)**(1/2)/(-x**4+1)**(1/2),x)

[Out]

asinh(x)

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Mathematica [B]  time = 0.0148152, size = 42, normalized size = 21. \[ \log \left (1-x^2\right )-\log \left (x^3+\sqrt{1-x^2} \sqrt{1-x^4}-x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x^2]/Sqrt[1 - x^4],x]

[Out]

Log[1 - x^2] - Log[-x + x^3 + Sqrt[1 - x^2]*Sqrt[1 - x^4]]

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Maple [B]  time = 0.011, size = 29, normalized size = 14.5 \[{{\it Arcsinh} \left ( x \right ) \sqrt{-{x}^{4}+1}{\frac{1}{\sqrt{-{x}^{2}+1}}}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)^(1/2)/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.779472, size = 3, normalized size = 1.5 \[ \operatorname{arsinh}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)/sqrt(-x^4 + 1),x, algorithm="maxima")

[Out]

arcsinh(x)

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Fricas [A]  time = 0.265651, size = 109, normalized size = 54.5 \[ -\frac{1}{2} \, \log \left (\frac{x^{3} + \sqrt{-x^{4} + 1} \sqrt{-x^{2} + 1} - x}{x^{3} - x}\right ) + \frac{1}{2} \, \log \left (-\frac{x^{3} - \sqrt{-x^{4} + 1} \sqrt{-x^{2} + 1} - x}{x^{3} - x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)/sqrt(-x^4 + 1),x, algorithm="fricas")

[Out]

-1/2*log((x^3 + sqrt(-x^4 + 1)*sqrt(-x^2 + 1) - x)/(x^3 - x)) + 1/2*log(-(x^3 -
sqrt(-x^4 + 1)*sqrt(-x^2 + 1) - x)/(x^3 - x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)**(1/2)/(-x**4+1)**(1/2),x)

[Out]

Integral(sqrt(-(x - 1)*(x + 1))/sqrt(-(x - 1)*(x + 1)*(x**2 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{2} + 1}}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)/sqrt(-x^4 + 1),x, algorithm="giac")

[Out]

integrate(sqrt(-x^2 + 1)/sqrt(-x^4 + 1), x)