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

Optimal. Leaf size=27 \[ \frac{1}{3} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{\sqrt{1-x^4}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0210414, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{3} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{\sqrt{1-x^4}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[1 - x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.89664, size = 20, normalized size = 0.74 \[ \frac{F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{3} - \frac{\sqrt{- x^{4} + 1}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

elliptic_f(asin(x), -1)/3 - sqrt(-x**4 + 1)/(3*x**3)

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Mathematica [A]  time = 0.0321807, size = 42, normalized size = 1.56 \[ \frac{x^4+\sqrt{1-x^4} x^3 F\left (\left .\sin ^{-1}(x)\right |-1\right )-1}{3 x^3 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 47, normalized size = 1.7 \[ -{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{{\it EllipticF} \left ( x,i \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-x^4+1)^(1/2),x)

[Out]

-1/3*(-x^4+1)^(1/2)/x^3+1/3*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*Elliptic
F(x,I)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(-x^4 + 1)*x^4), x)

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Sympy [A]  time = 2.27922, size = 34, normalized size = 1.26 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^4), x)