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

Optimal. Leaf size=12 \[ \frac{1}{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

[Out]

EllipticF[ArcSin[x/2], -1]/2

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Rubi [A]  time = 0.00638878, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[16 - x^4],x]

[Out]

EllipticF[ArcSin[x/2], -1]/2

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Rubi in Sympy [A]  time = 1.03983, size = 8, normalized size = 0.67 \[ \frac{F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | -1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

elliptic_f(asin(x/2), -1)/2

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Mathematica [A]  time = 0.0209384, size = 12, normalized size = 1. \[ \frac{1}{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[16 - x^4],x]

[Out]

EllipticF[ArcSin[x/2], -1]/2

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Maple [B]  time = 0.007, size = 34, normalized size = 2.8 \[{\frac{1}{2}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4}{\it EllipticF} \left ({\frac{x}{2}},i \right ){\frac{1}{\sqrt{-{x}^{4}+16}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^4+16)^(1/2),x)

[Out]

1/2*(-x^2+4)^(1/2)*(x^2+4)^(1/2)/(-x^4+16)^(1/2)*EllipticF(1/2*x,I)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(-x^4 + 16), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(-x^4 + 16), x)

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Sympy [A]  time = 1.73025, size = 31, normalized size = 2.58 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), x**4*exp_polar(2*I*pi)/16)/(16*gamma(5/4)
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(-x^4 + 16), x)