3.1351 \(\int \frac{1}{x \left (1-x^6\right )} \, dx\)

Optimal. Leaf size=15 \[ \log (x)-\frac{1}{6} \log \left (1-x^6\right ) \]

[Out]

Log[x] - Log[1 - x^6]/6

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Rubi [A]  time = 0.021795, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \log (x)-\frac{1}{6} \log \left (1-x^6\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 - x^6)),x]

[Out]

Log[x] - Log[1 - x^6]/6

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Rubi in Sympy [A]  time = 3.78962, size = 14, normalized size = 0.93 \[ \frac{\log{\left (x^{6} \right )}}{6} - \frac{\log{\left (- x^{6} + 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-x**6+1),x)

[Out]

log(x**6)/6 - log(-x**6 + 1)/6

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Mathematica [A]  time = 0.00585633, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{6} \log \left (1-x^6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(1 - x^6)),x]

[Out]

Log[x] - Log[1 - x^6]/6

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Maple [B]  time = 0.013, size = 36, normalized size = 2.4 \[ \ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{6}}-{\frac{\ln \left ( -1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}-{\frac{\ln \left ( 1+x \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^6+1),x)

[Out]

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

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Maxima [A]  time = 1.41747, size = 20, normalized size = 1.33 \[ -\frac{1}{6} \, \log \left (x^{6} - 1\right ) + \frac{1}{6} \, \log \left (x^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*log(x^6 - 1) + 1/6*log(x^6)

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Fricas [A]  time = 0.218518, size = 15, normalized size = 1. \[ -\frac{1}{6} \, \log \left (x^{6} - 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*log(x^6 - 1) + log(x)

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Sympy [A]  time = 0.243783, size = 10, normalized size = 0.67 \[ \log{\left (x \right )} - \frac{\log{\left (x^{6} - 1 \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-x**6+1),x)

[Out]

log(x) - log(x**6 - 1)/6

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GIAC/XCAS [A]  time = 0.235845, size = 22, normalized size = 1.47 \[ \frac{1}{6} \,{\rm ln}\left (x^{6}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x^{6} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*ln(x^6) - 1/6*ln(abs(x^6 - 1))