3.1374 \(\int \frac{1}{x^7 \left (1+x^6\right )} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{6 x^6}+\frac{1}{6} \log \left (x^6+1\right )-\log (x) \]

[Out]

-1/(6*x^6) - Log[x] + Log[1 + x^6]/6

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Rubi [A]  time = 0.0275422, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{1}{6 x^6}+\frac{1}{6} \log \left (x^6+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(1 + x^6)),x]

[Out]

-1/(6*x^6) - Log[x] + Log[1 + x^6]/6

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Rubi in Sympy [A]  time = 3.90876, size = 20, normalized size = 0.91 \[ - \frac{\log{\left (x^{6} \right )}}{6} + \frac{\log{\left (x^{6} + 1 \right )}}{6} - \frac{1}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(x**6+1),x)

[Out]

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

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Mathematica [A]  time = 0.0061104, size = 22, normalized size = 1. \[ -\frac{1}{6 x^6}+\frac{1}{6} \log \left (x^6+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(1 + x^6)),x]

[Out]

-1/(6*x^6) - Log[x] + Log[1 + x^6]/6

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Maple [A]  time = 0.012, size = 32, normalized size = 1.5 \[{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{6}}-{\frac{1}{6\,{x}^{6}}}-\ln \left ( x \right ) +{\frac{\ln \left ({x}^{2}+1 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^6+1),x)

[Out]

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

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Maxima [A]  time = 1.43214, size = 27, normalized size = 1.23 \[ -\frac{1}{6 \, x^{6}} + \frac{1}{6} \, \log \left (x^{6} + 1\right ) - \frac{1}{6} \, \log \left (x^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.211627, size = 32, normalized size = 1.45 \[ \frac{x^{6} \log \left (x^{6} + 1\right ) - 6 \, x^{6} \log \left (x\right ) - 1}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^7),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.389663, size = 17, normalized size = 0.77 \[ - \log{\left (x \right )} + \frac{\log{\left (x^{6} + 1 \right )}}{6} - \frac{1}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(x**6+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229271, size = 34, normalized size = 1.55 \[ \frac{x^{6} - 1}{6 \, x^{6}} + \frac{1}{6} \,{\rm ln}\left (x^{6} + 1\right ) - \frac{1}{6} \,{\rm ln}\left (x^{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^7),x, algorithm="giac")

[Out]

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