3.1342 \(\int \frac{x^8}{1-x^6} \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{3} \tanh ^{-1}\left (x^3\right )-\frac{x^3}{3} \]

[Out]

-x^3/3 + ArcTanh[x^3]/3

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Rubi [A]  time = 0.0254175, 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 \[ \frac{1}{3} \tanh ^{-1}\left (x^3\right )-\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(1 - x^6),x]

[Out]

-x^3/3 + ArcTanh[x^3]/3

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Rubi in Sympy [A]  time = 5.41662, size = 10, normalized size = 0.62 \[ - \frac{x^{3}}{3} + \frac{\operatorname{atanh}{\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(-x**6+1),x)

[Out]

-x**3/3 + atanh(x**3)/3

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Mathematica [A]  time = 0.00890097, size = 30, normalized size = 1.88 \[ -\frac{x^3}{3}-\frac{1}{6} \log \left (1-x^3\right )+\frac{1}{6} \log \left (x^3+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(1 - x^6),x]

[Out]

-x^3/3 - Log[1 - x^3]/6 + Log[1 + x^3]/6

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Maple [A]  time = 0.005, size = 23, normalized size = 1.4 \[ -{\frac{{x}^{3}}{3}}-{\frac{\ln \left ({x}^{3}-1 \right ) }{6}}+{\frac{\ln \left ({x}^{3}+1 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(-x^6+1),x)

[Out]

-1/3*x^3-1/6*ln(x^3-1)+1/6*ln(x^3+1)

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Maxima [A]  time = 1.44016, size = 30, normalized size = 1.88 \[ -\frac{1}{3} \, x^{3} + \frac{1}{6} \, \log \left (x^{3} + 1\right ) - \frac{1}{6} \, \log \left (x^{3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^8/(x^6 - 1),x, algorithm="maxima")

[Out]

-1/3*x^3 + 1/6*log(x^3 + 1) - 1/6*log(x^3 - 1)

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Fricas [A]  time = 0.218559, size = 30, normalized size = 1.88 \[ -\frac{1}{3} \, x^{3} + \frac{1}{6} \, \log \left (x^{3} + 1\right ) - \frac{1}{6} \, \log \left (x^{3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^8/(x^6 - 1),x, algorithm="fricas")

[Out]

-1/3*x^3 + 1/6*log(x^3 + 1) - 1/6*log(x^3 - 1)

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Sympy [A]  time = 0.22793, size = 20, normalized size = 1.25 \[ - \frac{x^{3}}{3} - \frac{\log{\left (x^{3} - 1 \right )}}{6} + \frac{\log{\left (x^{3} + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(-x**6+1),x)

[Out]

-x**3/3 - log(x**3 - 1)/6 + log(x**3 + 1)/6

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GIAC/XCAS [A]  time = 0.223745, size = 32, normalized size = 2. \[ -\frac{1}{3} \, x^{3} + \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} + 1 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^8/(x^6 - 1),x, algorithm="giac")

[Out]

-1/3*x^3 + 1/6*ln(abs(x^3 + 1)) - 1/6*ln(abs(x^3 - 1))