3.970 \(\int \frac{1}{x^2 \sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{\sqrt{16-x^4}}{16 x}+\frac{1}{8} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-\frac{1}{8} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

[Out]

-Sqrt[16 - x^4]/(16*x) - EllipticE[ArcSin[x/2], -1]/8 + EllipticF[ArcSin[x/2], -
1]/8

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Rubi [A]  time = 0.0664928, antiderivative size = 43, 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 \[ -\frac{\sqrt{16-x^4}}{16 x}+\frac{1}{8} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-\frac{1}{8} E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(16*x) - EllipticE[ArcSin[x/2], -1]/8 + EllipticF[ArcSin[x/2], -
1]/8

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Rubi in Sympy [A]  time = 11.4579, size = 31, normalized size = 0.72 \[ - \frac{E\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -1\right )}{8} + \frac{F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -1\right )}{8} - \frac{\sqrt{- x^{4} + 16}}{16 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-elliptic_e(asin(x/2), -1)/8 + elliptic_f(asin(x/2), -1)/8 - sqrt(-x**4 + 16)/(1
6*x)

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Mathematica [A]  time = 0.0749275, size = 57, normalized size = 1.33 \[ \frac{1}{16} \left (\frac{x^4+2 \sqrt{16-x^4} x F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-16}{x \sqrt{16-x^4}}-2 E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*Sqrt[16 - x^4]),x]

[Out]

(-2*EllipticE[ArcSin[x/2], -1] + (-16 + x^4 + 2*x*Sqrt[16 - x^4]*EllipticF[ArcSi
n[x/2], -1])/(x*Sqrt[16 - x^4]))/16

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Maple [A]  time = 0.014, size = 58, normalized size = 1.4 \[ -{\frac{1}{16\,x}\sqrt{-{x}^{4}+16}}+{\frac{1}{8}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4} \left ({\it EllipticF} \left ({\frac{x}{2}},i \right ) -{\it EllipticE} \left ({\frac{x}{2}},i \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+16}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-x^4+16)^(1/2),x)

[Out]

-1/16*(-x^4+16)^(1/2)/x+1/8*(-x^2+4)^(1/2)*(x^2+4)^(1/2)/(-x^4+16)^(1/2)*(Ellipt
icF(1/2*x,I)-EllipticE(1/2*x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + 16} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^4 + 16)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{4} + 16} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^2),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-x^4 + 16)*x^2), x)

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Sympy [A]  time = 2.06529, size = 34, normalized size = 0.79 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + 16} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + 16)*x^2), x)