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

Optimal. Leaf size=35 \[ -\frac{\sqrt{1-x^4}}{4 x^4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

[Out]

-Sqrt[1 - x^4]/(4*x^4) - ArcTanh[Sqrt[1 - x^4]]/4

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Rubi [A]  time = 0.0456465, antiderivative size = 35, 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 \[ -\frac{\sqrt{1-x^4}}{4 x^4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(4*x^4) - ArcTanh[Sqrt[1 - x^4]]/4

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Rubi in Sympy [A]  time = 4.69523, size = 26, normalized size = 0.74 \[ - \frac{\operatorname{atanh}{\left (\sqrt{- x^{4} + 1} \right )}}{4} - \frac{\sqrt{- x^{4} + 1}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(-x**4 + 1))/4 - sqrt(-x**4 + 1)/(4*x**4)

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Mathematica [A]  time = 0.0417946, size = 35, normalized size = 1. \[ -\frac{\sqrt{1-x^4}}{4 x^4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 - x^4]/(4*x^4) - ArcTanh[Sqrt[1 - x^4]]/4

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Maple [A]  time = 0.015, size = 28, normalized size = 0.8 \[ -{\frac{1}{4\,{x}^{4}}\sqrt{-{x}^{4}+1}}-{\frac{1}{4}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{4}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.53975, size = 58, normalized size = 1.66 \[ -\frac{\sqrt{-x^{4} + 1}}{4 \, x^{4}} - \frac{1}{8} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{-x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239126, size = 68, normalized size = 1.94 \[ -\frac{x^{4} \log \left (\sqrt{-x^{4} + 1} + 1\right ) - x^{4} \log \left (\sqrt{-x^{4} + 1} - 1\right ) + 2 \, \sqrt{-x^{4} + 1}}{8 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(x^4*log(sqrt(-x^4 + 1) + 1) - x^4*log(sqrt(-x^4 + 1) - 1) + 2*sqrt(-x^4 +
1))/x^4

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Sympy [A]  time = 6.43017, size = 73, normalized size = 2.09 \[ \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{\sqrt{-1 + \frac{1}{x^{4}}}}{4 x^{2}} & \text{for}\: \left |{\frac{1}{x^{4}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{i}{4 x^{2} \sqrt{1 - \frac{1}{x^{4}}}} + \frac{i}{4 x^{6} \sqrt{1 - \frac{1}{x^{4}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-acosh(x**(-2))/4 - sqrt(-1 + x**(-4))/(4*x**2), Abs(x**(-4)) > 1), (
I*asin(x**(-2))/4 - I/(4*x**2*sqrt(1 - 1/x**4)) + I/(4*x**6*sqrt(1 - 1/x**4)), T
rue))

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GIAC/XCAS [A]  time = 0.215293, size = 61, normalized size = 1.74 \[ -\frac{\sqrt{-x^{4} + 1}}{4 \, x^{4}} - \frac{1}{8} \,{\rm ln}\left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{8} \,{\rm ln}\left (-\sqrt{-x^{4} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(-x^4 + 1)/x^4 - 1/8*ln(sqrt(-x^4 + 1) + 1) + 1/8*ln(-sqrt(-x^4 + 1) +
1)