3.1368 \(\int \frac{1}{x \left (1+x^6\right )} \, dx\)

Optimal. Leaf size=13 \[ \log (x)-\frac{1}{6} \log \left (x^6+1\right ) \]

[Out]

Log[x] - Log[1 + x^6]/6

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Rubi [A]  time = 0.0175683, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \log (x)-\frac{1}{6} \log \left (x^6+1\right ) \]

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 = 3.3332, size = 14, normalized size = 1.08 \[ \frac{\log{\left (x^{6} \right )}}{6} - \frac{\log{\left (x^{6} + 1 \right )}}{6} \]

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 = 0.00550371, size = 13, normalized size = 1. \[ \log (x)-\frac{1}{6} \log \left (x^6+1\right ) \]

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 = 0.01, size = 25, normalized size = 1.9 \[ -{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{6}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+1 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^6+1),x)

[Out]

-1/6*ln(x^4-x^2+1)+ln(x)-1/6*ln(x^2+1)

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Maxima [A]  time = 1.43667, size = 20, normalized size = 1.54 \[ -\frac{1}{6} \, \log \left (x^{6} + 1\right ) + \frac{1}{6} \, \log \left (x^{6}\right ) \]

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 = 0.211558, size = 15, normalized size = 1.15 \[ -\frac{1}{6} \, \log \left (x^{6} + 1\right ) + \log \left (x\right ) \]

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 = 0.225741, size = 10, normalized size = 0.77 \[ \log{\left (x \right )} - \frac{\log{\left (x^{6} + 1 \right )}}{6} \]

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 = 0.227176, size = 20, normalized size = 1.54 \[ -\frac{1}{6} \,{\rm ln}\left (x^{6} + 1\right ) + \frac{1}{6} \,{\rm ln}\left (x^{6}\right ) \]

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) + 1/6*ln(x^6)