3.1475 \(\int \frac{1}{x^3 \left (1-x^8\right )} \, dx\)

Optimal. Leaf size=24 \[ -\frac{1}{2 x^2}-\frac{1}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]

[Out]

-1/(2*x^2) - ArcTan[x^2]/4 + ArcTanh[x^2]/4

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Rubi [A]  time = 0.0378722, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{1}{2 x^2}-\frac{1}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4} \tanh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 - x^8)),x]

[Out]

-1/(2*x^2) - ArcTan[x^2]/4 + ArcTanh[x^2]/4

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Rubi in Sympy [A]  time = 6.37405, size = 19, normalized size = 0.79 \[ - \frac{\operatorname{atan}{\left (x^{2} \right )}}{4} + \frac{\operatorname{atanh}{\left (x^{2} \right )}}{4} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(-x**8+1),x)

[Out]

-atan(x**2)/4 + atanh(x**2)/4 - 1/(2*x**2)

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Mathematica [A]  time = 0.00950254, size = 38, normalized size = 1.58 \[ -\frac{1}{2 x^2}-\frac{1}{8} \log \left (1-x^2\right )+\frac{1}{8} \log \left (x^2+1\right )+\frac{1}{4} \tan ^{-1}\left (\frac{1}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(1 - x^8)),x]

[Out]

-1/(2*x^2) + ArcTan[x^(-2)]/4 - Log[1 - x^2]/8 + Log[1 + x^2]/8

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Maple [A]  time = 0.016, size = 33, normalized size = 1.4 \[ -{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\arctan \left ({x}^{2} \right ) }{4}}-{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{1}{2\,{x}^{2}}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-x^8+1),x)

[Out]

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

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Maxima [A]  time = 1.61493, size = 38, normalized size = 1.58 \[ -\frac{1}{2 \, x^{2}} - \frac{1}{4} \, \arctan \left (x^{2}\right ) + \frac{1}{8} \, \log \left (x^{2} + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.225356, size = 50, normalized size = 2.08 \[ -\frac{2 \, x^{2} \arctan \left (x^{2}\right ) - x^{2} \log \left (x^{2} + 1\right ) + x^{2} \log \left (x^{2} - 1\right ) + 4}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.505036, size = 29, normalized size = 1.21 \[ - \frac{\log{\left (x^{2} - 1 \right )}}{8} + \frac{\log{\left (x^{2} + 1 \right )}}{8} - \frac{\operatorname{atan}{\left (x^{2} \right )}}{4} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(-x**8+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217994, size = 39, normalized size = 1.62 \[ -\frac{1}{2 \, x^{2}} - \frac{1}{4} \, \arctan \left (x^{2}\right ) + \frac{1}{8} \,{\rm ln}\left (x^{2} + 1\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^3),x, algorithm="giac")

[Out]

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