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

Optimal. Leaf size=27 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]

[Out]

Sqrt[(1 - x^2)^(-1)]*Sqrt[1 - x^2]*ArcSin[x]

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Rubi [A]  time = 0.0401559, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[(1 - x^2)^(-1)]*Sqrt[1 - x^2]*ArcSin[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x^{2} + 1}{- x^{4} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x**2 + 1)/(-x**4 + 1)), x)

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Mathematica [A]  time = 0.00774935, size = 27, normalized size = 1. \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[(1 - x^2)^(-1)]*Sqrt[1 - x^2]*ArcSin[x]

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Maple [A]  time = 0.005, size = 30, normalized size = 1.1 \[ \sqrt{- \left ({x}^{2}-1 \right ) ^{-1}}\sqrt{{x}^{2}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((x^2+1)/(-x^4+1))^(1/2),x)

[Out]

(-1/(x^2-1))^(1/2)*(x^2-1)^(1/2)*ln(x+(x^2-1)^(1/2))

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Maxima [A]  time = 0.792059, size = 3, normalized size = 0.11 \[ \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(x)

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Fricas [A]  time = 0.2736, size = 42, normalized size = 1.56 \[ 2 \, \arctan \left (\frac{\sqrt{-\frac{1}{x^{2} - 1}} - 1}{x \sqrt{-\frac{1}{x^{2} - 1}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan((sqrt(-1/(x^2 - 1)) - 1)/(x*sqrt(-1/(x^2 - 1))))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x^{2} + 1}{- x^{4} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x**2 + 1)/(-x**4 + 1)), x)

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GIAC/XCAS [A]  time = 0.268806, size = 14, normalized size = 0.52 \[ -\arcsin \left (x\right ){\rm sign}\left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(x)*sign(x^2 - 1)