3.1476 \(\int \frac{1}{x^5 \left (1-x^8\right )} \, dx\)

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

[Out]

-1/(4*x^4) + ArcTanh[x^4]/4

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Rubi [A]  time = 0.0245837, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{4} \tanh ^{-1}\left (x^4\right )-\frac{1}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 - x^8)),x]

[Out]

-1/(4*x^4) + ArcTanh[x^4]/4

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Rubi in Sympy [A]  time = 5.37335, size = 12, normalized size = 0.75 \[ \frac{\operatorname{atanh}{\left (x^{4} \right )}}{4} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(-x**8+1),x)

[Out]

atanh(x**4)/4 - 1/(4*x**4)

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Mathematica [A]  time = 0.00719386, size = 30, normalized size = 1.88 \[ -\frac{1}{4 x^4}-\frac{1}{8} \log \left (1-x^4\right )+\frac{1}{8} \log \left (x^4+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(1 - x^8)),x]

[Out]

-1/(4*x^4) - Log[1 - x^4]/8 + Log[1 + x^4]/8

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Maple [B]  time = 0.016, size = 35, normalized size = 2.2 \[ -{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{1}{4\,{x}^{4}}}+{\frac{\ln \left ({x}^{4}+1 \right ) }{8}}-{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(-x^8+1),x)

[Out]

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

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Maxima [A]  time = 1.44005, size = 30, normalized size = 1.88 \[ -\frac{1}{4 \, x^{4}} + \frac{1}{8} \, \log \left (x^{4} + 1\right ) - \frac{1}{8} \, \log \left (x^{4} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.214106, size = 38, normalized size = 2.38 \[ \frac{x^{4} \log \left (x^{4} + 1\right ) - x^{4} \log \left (x^{4} - 1\right ) - 2}{8 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.39724, size = 22, normalized size = 1.38 \[ - \frac{\log{\left (x^{4} - 1 \right )}}{8} + \frac{\log{\left (x^{4} + 1 \right )}}{8} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(-x**8+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219388, size = 31, normalized size = 1.94 \[ -\frac{1}{4 \, x^{4}} + \frac{1}{8} \,{\rm ln}\left (x^{4} + 1\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x^{4} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^5),x, algorithm="giac")

[Out]

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