3.1348 \(\int \frac{x^2}{1-x^6} \, dx\)

Optimal. Leaf size=8 \[ \frac{1}{3} \tanh ^{-1}\left (x^3\right ) \]

[Out]

ArcTanh[x^3]/3

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Rubi [A]  time = 0.0134175, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{3} \tanh ^{-1}\left (x^3\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 - x^6),x]

[Out]

ArcTanh[x^3]/3

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Rubi in Sympy [A]  time = 2.84695, size = 5, normalized size = 0.62 \[ \frac{\operatorname{atanh}{\left (x^{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-x**6+1),x)

[Out]

atanh(x**3)/3

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Mathematica [B]  time = 0.00585889, size = 23, normalized size = 2.88 \[ \frac{1}{6} \log \left (x^3+1\right )-\frac{1}{6} \log \left (1-x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(1 - x^6),x]

[Out]

-Log[1 - x^3]/6 + Log[1 + x^3]/6

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Maple [B]  time = 0.003, size = 18, normalized size = 2.3 \[ -{\frac{\ln \left ({x}^{3}-1 \right ) }{6}}+{\frac{\ln \left ({x}^{3}+1 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-x^6+1),x)

[Out]

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

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Maxima [A]  time = 1.43538, size = 23, normalized size = 2.88 \[ \frac{1}{6} \, \log \left (x^{3} + 1\right ) - \frac{1}{6} \, \log \left (x^{3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/(x^6 - 1),x, algorithm="maxima")

[Out]

1/6*log(x^3 + 1) - 1/6*log(x^3 - 1)

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Fricas [A]  time = 0.224507, size = 23, normalized size = 2.88 \[ \frac{1}{6} \, \log \left (x^{3} + 1\right ) - \frac{1}{6} \, \log \left (x^{3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/(x^6 - 1),x, algorithm="fricas")

[Out]

1/6*log(x^3 + 1) - 1/6*log(x^3 - 1)

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Sympy [A]  time = 0.218665, size = 15, normalized size = 1.88 \[ - \frac{\log{\left (x^{3} - 1 \right )}}{6} + \frac{\log{\left (x^{3} + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-x**6+1),x)

[Out]

-log(x**3 - 1)/6 + log(x**3 + 1)/6

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GIAC/XCAS [A]  time = 0.231203, size = 26, normalized size = 3.25 \[ \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} + 1 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/(x^6 - 1),x, algorithm="giac")

[Out]

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