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

Optimal. Leaf size=27 \[ -\frac{\sqrt{1-x^4}}{x}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

-(Sqrt[1 - x^4]/x) - EllipticE[ArcSin[x], -1] + EllipticF[ArcSin[x], -1]

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Rubi [A]  time = 0.059944, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\sqrt{1-x^4}}{x}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[1 - x^4]),x]

[Out]

-(Sqrt[1 - x^4]/x) - EllipticE[ArcSin[x], -1] + EllipticF[ArcSin[x], -1]

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Rubi in Sympy [A]  time = 10.4525, size = 22, normalized size = 0.81 \[ - E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right ) + F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right ) - \frac{\sqrt{- x^{4} + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-elliptic_e(asin(x), -1) + elliptic_f(asin(x), -1) - sqrt(-x**4 + 1)/x

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Mathematica [A]  time = 0.0548265, size = 42, normalized size = 1.56 \[ -\frac{1}{\sqrt{1-x^4} x}+\frac{x^3}{\sqrt{1-x^4}}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(1/(x*Sqrt[1 - x^4])) + x^3/Sqrt[1 - x^4] - EllipticE[ArcSin[x], -1] + Elliptic
F[ArcSin[x], -1]

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Maple [B]  time = 0.014, size = 53, normalized size = 2. \[ -{\frac{1}{x}\sqrt{-{x}^{4}+1}}+{({\it EllipticF} \left ( x,i \right ) -{\it EllipticE} \left ( x,i \right ) )\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-x^4+1)^(1/2),x)

[Out]

-(-x^4+1)^(1/2)/x+(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*(EllipticF(x,I)-El
lipticE(x,I))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{4} + 1} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(-x^4 + 1)*x^2), x)

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Sympy [A]  time = 2.01959, size = 32, normalized size = 1.19 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), x**4*exp_polar(2*I*pi))/(4*x*gamma(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^2), x)