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

Optimal. Leaf size=20 \[ -\frac{1}{8} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right ) \]

[Out]

-ArcTanh[Sqrt[16 - x^4]/4]/8

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Rubi [A]  time = 0.0312655, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{1}{8} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[16 - x^4]),x]

[Out]

-ArcTanh[Sqrt[16 - x^4]/4]/8

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Rubi in Sympy [A]  time = 3.9284, size = 14, normalized size = 0.7 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{- x^{4} + 16}}{4} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(-x**4 + 16)/4)/8

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Mathematica [A]  time = 0.0365142, size = 20, normalized size = 1. \[ -\frac{1}{8} \tanh ^{-1}\left (\frac{\sqrt{16-x^4}}{4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTanh[Sqrt[16 - x^4]/4]/8

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Maple [A]  time = 0.013, size = 15, normalized size = 0.8 \[ -{\frac{1}{8}{\it Artanh} \left ( 4\,{\frac{1}{\sqrt{-{x}^{4}+16}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^4+16)^(1/2),x)

[Out]

-1/8*arctanh(4/(-x^4+16)^(1/2))

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Maxima [A]  time = 1.41982, size = 39, normalized size = 1.95 \[ -\frac{1}{16} \, \log \left (\sqrt{-x^{4} + 16} + 4\right ) + \frac{1}{16} \, \log \left (\sqrt{-x^{4} + 16} - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.281048, size = 39, normalized size = 1.95 \[ -\frac{1}{16} \, \log \left (\sqrt{-x^{4} + 16} + 4\right ) + \frac{1}{16} \, \log \left (\sqrt{-x^{4} + 16} - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.41409, size = 26, normalized size = 1.3 \[ \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{4}{x^{2}} \right )}}{8} & \text{for}\: 16 \left |{\frac{1}{x^{4}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{4}{x^{2}} \right )}}{8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-acosh(4/x**2)/8, 16*Abs(x**(-4)) > 1), (I*asin(4/x**2)/8, True))

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GIAC/XCAS [A]  time = 0.213667, size = 42, normalized size = 2.1 \[ -\frac{1}{16} \,{\rm ln}\left (\sqrt{-x^{4} + 16} + 4\right ) + \frac{1}{16} \,{\rm ln}\left (-\sqrt{-x^{4} + 16} + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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