3.1478 \(\int \frac{1}{x^9 \left (1-x^8\right )} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{8 x^8}-\frac{1}{8} \log \left (1-x^8\right )+\log (x) \]

[Out]

-1/(8*x^8) + Log[x] - Log[1 - x^8]/8

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Rubi [A]  time = 0.0320517, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{1}{8 x^8}-\frac{1}{8} \log \left (1-x^8\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*(1 - x^8)),x]

[Out]

-1/(8*x^8) + Log[x] - Log[1 - x^8]/8

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Rubi in Sympy [A]  time = 4.95365, size = 20, normalized size = 0.91 \[ \frac{\log{\left (x^{8} \right )}}{8} - \frac{\log{\left (- x^{8} + 1 \right )}}{8} - \frac{1}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(-x**8+1),x)

[Out]

log(x**8)/8 - log(-x**8 + 1)/8 - 1/(8*x**8)

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Mathematica [A]  time = 0.00660765, size = 22, normalized size = 1. \[ -\frac{1}{8 x^8}-\frac{1}{8} \log \left (1-x^8\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^9*(1 - x^8)),x]

[Out]

-1/(8*x^8) + Log[x] - Log[1 - x^8]/8

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Maple [A]  time = 0.018, size = 37, normalized size = 1.7 \[ -{\frac{1}{8\,{x}^{8}}}+\ln \left ( x \right ) -{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\ln \left ({x}^{4}+1 \right ) }{8}}-{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^9/(-x^8+1),x)

[Out]

-1/8/x^8+ln(x)-1/8*ln(-1+x)-1/8*ln(x^4+1)-1/8*ln(1+x)-1/8*ln(x^2+1)

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Maxima [A]  time = 1.43906, size = 27, normalized size = 1.23 \[ -\frac{1}{8 \, x^{8}} - \frac{1}{8} \, \log \left (x^{8} - 1\right ) + \frac{1}{8} \, \log \left (x^{8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^9),x, algorithm="maxima")

[Out]

-1/8/x^8 - 1/8*log(x^8 - 1) + 1/8*log(x^8)

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Fricas [A]  time = 0.214903, size = 32, normalized size = 1.45 \[ -\frac{x^{8} \log \left (x^{8} - 1\right ) - 8 \, x^{8} \log \left (x\right ) + 1}{8 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^9),x, algorithm="fricas")

[Out]

-1/8*(x^8*log(x^8 - 1) - 8*x^8*log(x) + 1)/x^8

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Sympy [A]  time = 0.543971, size = 17, normalized size = 0.77 \[ \log{\left (x \right )} - \frac{\log{\left (x^{8} - 1 \right )}}{8} - \frac{1}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**9/(-x**8+1),x)

[Out]

log(x) - log(x**8 - 1)/8 - 1/(8*x**8)

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GIAC/XCAS [A]  time = 0.2184, size = 35, normalized size = 1.59 \[ -\frac{x^{8} + 1}{8 \, x^{8}} + \frac{1}{8} \,{\rm ln}\left (x^{8}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x^{8} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^8 - 1)*x^9),x, algorithm="giac")

[Out]

-1/8*(x^8 + 1)/x^8 + 1/8*ln(x^8) - 1/8*ln(abs(x^8 - 1))