3.971 \(\int \frac{1}{x^4 \sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{96} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-\frac{\sqrt{16-x^4}}{48 x^3} \]

[Out]

-Sqrt[16 - x^4]/(48*x^3) + EllipticF[ArcSin[x/2], -1]/96

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Rubi [A]  time = 0.0217035, antiderivative size = 31, 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 \[ \frac{1}{96} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-\frac{\sqrt{16-x^4}}{48 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(48*x^3) + EllipticF[ArcSin[x/2], -1]/96

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Rubi in Sympy [A]  time = 2.94358, size = 22, normalized size = 0.71 \[ \frac{F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -1\right )}{96} - \frac{\sqrt{- x^{4} + 16}}{48 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(-x**4+16)**(1/2),x)

[Out]

elliptic_f(asin(x/2), -1)/96 - sqrt(-x**4 + 16)/(48*x**3)

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Mathematica [A]  time = 0.0363894, size = 48, normalized size = 1.55 \[ \frac{2 x^4+\sqrt{16-x^4} x^3 F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-32}{96 x^3 \sqrt{16-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[16 - x^4]),x]

[Out]

(-32 + 2*x^4 + x^3*Sqrt[16 - x^4]*EllipticF[ArcSin[x/2], -1])/(96*x^3*Sqrt[16 -
x^4])

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Maple [B]  time = 0.016, size = 49, normalized size = 1.6 \[ -{\frac{1}{48\,{x}^{3}}\sqrt{-{x}^{4}+16}}+{\frac{1}{96}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4}{\it EllipticF} \left ({\frac{x}{2}},i \right ){\frac{1}{\sqrt{-{x}^{4}+16}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-x^4+16)^(1/2),x)

[Out]

-1/48*(-x^4+16)^(1/2)/x^3+1/96*(-x^2+4)^(1/2)*(x^2+4)^(1/2)/(-x^4+16)^(1/2)*Elli
pticF(1/2*x,I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + 16} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^4),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^4 + 16)*x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{4} + 16} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^4),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-x^4 + 16)*x^4), x)

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Sympy [A]  time = 2.36044, size = 36, normalized size = 1.16 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(-x**4+16)**(1/2),x)

[Out]

gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), x**4*exp_polar(2*I*pi)/16)/(16*x**3*gamma
(1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + 16} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^4),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + 16)*x^4), x)