3.912 \(\int \frac{1}{x^2 \left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{3 \sqrt{1-x^4}}{2 x}+\frac{1}{2 x \sqrt{1-x^4}}+\frac{3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

1/(2*x*Sqrt[1 - x^4]) - (3*Sqrt[1 - x^4])/(2*x) - (3*EllipticE[ArcSin[x], -1])/2
+ (3*EllipticF[ArcSin[x], -1])/2

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Rubi [A]  time = 0.0712506, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{3 \sqrt{1-x^4}}{2 x}+\frac{1}{2 x \sqrt{1-x^4}}+\frac{3}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{3}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(1 - x^4)^(3/2)),x]

[Out]

1/(2*x*Sqrt[1 - x^4]) - (3*Sqrt[1 - x^4])/(2*x) - (3*EllipticE[ArcSin[x], -1])/2
+ (3*EllipticF[ArcSin[x], -1])/2

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Rubi in Sympy [A]  time = 12.1127, size = 46, normalized size = 0.87 \[ - \frac{3 E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{2} + \frac{3 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{2} - \frac{3 \sqrt{- x^{4} + 1}}{2 x} + \frac{1}{2 x \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(-x**4+1)**(3/2),x)

[Out]

-3*elliptic_e(asin(x), -1)/2 + 3*elliptic_f(asin(x), -1)/2 - 3*sqrt(-x**4 + 1)/(
2*x) + 1/(2*x*sqrt(-x**4 + 1))

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Mathematica [A]  time = 0.0641217, size = 49, normalized size = 0.92 \[ \frac{1}{2} \left (-\frac{2}{\sqrt{1-x^4} x}+\frac{3 x^3}{\sqrt{1-x^4}}+3 F\left (\left .\sin ^{-1}(x)\right |-1\right )-3 E\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(1 - x^4)^(3/2)),x]

[Out]

(-2/(x*Sqrt[1 - x^4]) + (3*x^3)/Sqrt[1 - x^4] - 3*EllipticE[ArcSin[x], -1] + 3*E
llipticF[ArcSin[x], -1])/2

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Maple [A]  time = 0.019, size = 68, normalized size = 1.3 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{x}\sqrt{-{x}^{4}+1}}+{\frac{3\,{\it EllipticF} \left ( x,i \right ) -3\,{\it EllipticE} \left ( x,i \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-x^4+1)^(3/2),x)

[Out]

1/2*x^3/(-x^4+1)^(1/2)-(-x^4+1)^(1/2)/x+3/2*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1
)^(1/2)*(EllipticF(x,I)-EllipticE(x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (x^{6} - x^{2}\right )} \sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^2),x, algorithm="fricas")

[Out]

integral(-1/((x^6 - x^2)*sqrt(-x^4 + 1)), x)

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Sympy [A]  time = 2.45739, size = 32, normalized size = 0.6 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(-x**4+1)**(3/2),x)

[Out]

gamma(-1/4)*hyper((-1/4, 3/2), (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. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^2), x)