3.964 \(\int \frac{1}{x^5 \sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{\sqrt{16-x^4}}{64 x^4}-\frac{1}{256} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right ) \]

[Out]

-Sqrt[16 - x^4]/(64*x^4) - ArcTanh[Sqrt[16 - x^4]/4]/256

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Rubi [A]  time = 0.0538851, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt{16-x^4}}{64 x^4}-\frac{1}{256} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(64*x^4) - ArcTanh[Sqrt[16 - x^4]/4]/256

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Rubi in Sympy [A]  time = 4.76968, size = 27, normalized size = 0.69 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{- x^{4} + 16}}{4} \right )}}{256} - \frac{\sqrt{- x^{4} + 16}}{64 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(-x**4+16)**(1/2),x)

[Out]

-atanh(sqrt(-x**4 + 16)/4)/256 - sqrt(-x**4 + 16)/(64*x**4)

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Mathematica [A]  time = 0.0450386, size = 39, normalized size = 1. \[ -\frac{\sqrt{16-x^4}}{64 x^4}-\frac{1}{256} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(64*x^4) - ArcTanh[Sqrt[16 - x^4]/4]/256

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Maple [A]  time = 0.016, size = 30, normalized size = 0.8 \[ -{\frac{1}{64\,{x}^{4}}\sqrt{-{x}^{4}+16}}-{\frac{1}{256}{\it Artanh} \left ( 4\,{\frac{1}{\sqrt{-{x}^{4}+16}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(-x^4+16)^(1/2),x)

[Out]

-1/64*(-x^4+16)^(1/2)/x^4-1/256*arctanh(4/(-x^4+16)^(1/2))

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Maxima [A]  time = 1.43463, size = 58, normalized size = 1.49 \[ -\frac{\sqrt{-x^{4} + 16}}{64 \, x^{4}} - \frac{1}{512} \, \log \left (\sqrt{-x^{4} + 16} + 4\right ) + \frac{1}{512} \, \log \left (\sqrt{-x^{4} + 16} - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^5),x, algorithm="maxima")

[Out]

-1/64*sqrt(-x^4 + 16)/x^4 - 1/512*log(sqrt(-x^4 + 16) + 4) + 1/512*log(sqrt(-x^4
+ 16) - 4)

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Fricas [A]  time = 0.280225, size = 68, normalized size = 1.74 \[ -\frac{x^{4} \log \left (\sqrt{-x^{4} + 16} + 4\right ) - x^{4} \log \left (\sqrt{-x^{4} + 16} - 4\right ) + 8 \, \sqrt{-x^{4} + 16}}{512 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^5),x, algorithm="fricas")

[Out]

-1/512*(x^4*log(sqrt(-x^4 + 16) + 4) - x^4*log(sqrt(-x^4 + 16) - 4) + 8*sqrt(-x^
4 + 16))/x^4

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Sympy [A]  time = 6.69003, size = 75, normalized size = 1.92 \[ \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{4}{x^{2}} \right )}}{256} - \frac{\sqrt{-1 + \frac{16}{x^{4}}}}{64 x^{2}} & \text{for}\: 16 \left |{\frac{1}{x^{4}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{4}{x^{2}} \right )}}{256} - \frac{i}{64 x^{2} \sqrt{1 - \frac{16}{x^{4}}}} + \frac{i}{4 x^{6} \sqrt{1 - \frac{16}{x^{4}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(-x**4+16)**(1/2),x)

[Out]

Piecewise((-acosh(4/x**2)/256 - sqrt(-1 + 16/x**4)/(64*x**2), 16*Abs(x**(-4)) >
1), (I*asin(4/x**2)/256 - I/(64*x**2*sqrt(1 - 16/x**4)) + I/(4*x**6*sqrt(1 - 16/
x**4)), True))

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GIAC/XCAS [A]  time = 0.215477, size = 61, normalized size = 1.56 \[ -\frac{\sqrt{-x^{4} + 16}}{64 \, x^{4}} - \frac{1}{512} \,{\rm ln}\left (\sqrt{-x^{4} + 16} + 4\right ) + \frac{1}{512} \,{\rm ln}\left (-\sqrt{-x^{4} + 16} + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 16)*x^5),x, algorithm="giac")

[Out]

-1/64*sqrt(-x^4 + 16)/x^4 - 1/512*ln(sqrt(-x^4 + 16) + 4) + 1/512*ln(-sqrt(-x^4
+ 16) + 4)