3.1354 \(\int \frac{1}{x^4 \left (1-x^6\right )} \, dx\)

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

[Out]

-1/(3*x^3) + ArcTanh[x^3]/3

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Rubi [A]  time = 0.0233069, 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 \[ \frac{1}{3} \tanh ^{-1}\left (x^3\right )-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(1 - x^6)),x]

[Out]

-1/(3*x^3) + ArcTanh[x^3]/3

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Rubi in Sympy [A]  time = 5.13423, size = 12, normalized size = 0.75 \[ \frac{\operatorname{atanh}{\left (x^{3} \right )}}{3} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(-x**6+1),x)

[Out]

atanh(x**3)/3 - 1/(3*x**3)

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Mathematica [A]  time = 0.00761368, size = 30, normalized size = 1.88 \[ -\frac{1}{3 x^3}-\frac{1}{6} \log \left (1-x^3\right )+\frac{1}{6} \log \left (x^3+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(1 - x^6)),x]

[Out]

-1/(3*x^3) - Log[1 - x^3]/6 + Log[1 + x^3]/6

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Maple [B]  time = 0.013, size = 39, normalized size = 2.4 \[ -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{6}}-{\frac{\ln \left ( -1+x \right ) }{6}}-{\frac{1}{3\,{x}^{3}}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}+{\frac{\ln \left ( 1+x \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-x^6+1),x)

[Out]

-1/6*ln(x^2+x+1)-1/6*ln(-1+x)-1/3/x^3+1/6*ln(x^2-x+1)+1/6*ln(1+x)

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Maxima [A]  time = 1.44008, 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(-1/((x^6 - 1)*x^4),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 = 0.226442, size = 38, normalized size = 2.38 \[ \frac{x^{3} \log \left (x^{3} + 1\right ) - x^{3} \log \left (x^{3} - 1\right ) - 2}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^6 - 1)*x^4),x, algorithm="fricas")

[Out]

1/6*(x^3*log(x^3 + 1) - x^3*log(x^3 - 1) - 2)/x^3

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Sympy [A]  time = 0.319197, size = 22, normalized size = 1.38 \[ - \frac{\log{\left (x^{3} - 1 \right )}}{6} + \frac{\log{\left (x^{3} + 1 \right )}}{6} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(-x**6+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223971, 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(-1/((x^6 - 1)*x^4),x, algorithm="giac")

[Out]

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