3.963 \(\int \frac{1}{x^3 \sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\sqrt{16-x^4}}{32 x^2} \]

[Out]

-Sqrt[16 - x^4]/(32*x^2)

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Rubi [A]  time = 0.0158401, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\sqrt{16-x^4}}{32 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(32*x^2)

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Rubi in Sympy [A]  time = 2.54028, size = 14, normalized size = 0.78 \[ - \frac{\sqrt{- x^{4} + 16}}{32 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(-x**4+16)**(1/2),x)

[Out]

-sqrt(-x**4 + 16)/(32*x**2)

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Mathematica [A]  time = 0.0104609, size = 18, normalized size = 1. \[ -\frac{\sqrt{16-x^4}}{32 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(32*x^2)

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Maple [A]  time = 0.006, size = 26, normalized size = 1.4 \[{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) \left ({x}^{2}+4 \right ) }{32\,{x}^{2}}{\frac{1}{\sqrt{-{x}^{4}+16}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-x^4+16)^(1/2),x)

[Out]

1/32/x^2*(-2+x)*(2+x)*(x^2+4)/(-x^4+16)^(1/2)

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Maxima [A]  time = 1.43896, size = 19, normalized size = 1.06 \[ -\frac{\sqrt{-x^{4} + 16}}{32 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^3),x, algorithm="maxima")

[Out]

-1/32*sqrt(-x^4 + 16)/x^2

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Fricas [A]  time = 0.272571, size = 53, normalized size = 2.94 \[ \frac{x^{4} + 4 \, \sqrt{-x^{4} + 16} - 16}{32 \,{\left (\sqrt{-x^{4} + 16} x^{2} - 4 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^3),x, algorithm="fricas")

[Out]

1/32*(x^4 + 4*sqrt(-x^4 + 16) - 16)/(sqrt(-x^4 + 16)*x^2 - 4*x^2)

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Sympy [A]  time = 1.89374, size = 34, normalized size = 1.89 \[ \begin{cases} - \frac{\sqrt{-1 + \frac{16}{x^{4}}}}{32} & \text{for}\: 16 \left |{\frac{1}{x^{4}}}\right | > 1 \\- \frac{i \sqrt{1 - \frac{16}{x^{4}}}}{32} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(-x**4+16)**(1/2),x)

[Out]

Piecewise((-sqrt(-1 + 16/x**4)/32, 16*Abs(x**(-4)) > 1), (-I*sqrt(1 - 16/x**4)/3
2, True))

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GIAC/XCAS [A]  time = 0.214497, size = 15, normalized size = 0.83 \[ -\frac{1}{32} \, \sqrt{\frac{16}{x^{4}} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^3),x, algorithm="giac")

[Out]

-1/32*sqrt(16/x^4 - 1)