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

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

[Out]

ArcSin[x]

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Rubi [A]  time = 0.00537891, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcSin[x]

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Rubi in Sympy [A]  time = 1.29857, size = 2, normalized size = 1. \[ \operatorname{asin}{\left (x \right )} \]

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]

asin(x)

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Mathematica [B]  time = 0.0169729, size = 32, normalized size = 16. \[ -\tan ^{-1}\left (\frac{x \sqrt{x^2+1} \sqrt{1-x^4}}{x^4-1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(x*Sqrt[1 + x^2]*Sqrt[1 - x^4])/(-1 + x^4)]

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Maple [B]  time = 0.017, size = 29, normalized size = 14.5 \[{\arcsin \left ( x \right ) \sqrt{-{x}^{4}+1}{\frac{1}{\sqrt{{x}^{2}+1}}}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

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

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Maxima [A]  time = 0.777977, size = 3, normalized size = 1.5 \[ \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(x)

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Fricas [A]  time = 0.26998, size = 36, normalized size = 18. \[ -\arctan \left (\frac{\sqrt{-x^{4} + 1} \sqrt{x^{2} + 1}}{x^{3} + x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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**2 + 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)