3.1474 \(\int \frac{1}{x \left (1-x^8\right )} \, dx\)

Optimal. Leaf size=15 \[ \log (x)-\frac{1}{8} \log \left (1-x^8\right ) \]

[Out]

Log[x] - Log[1 - x^8]/8

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Rubi [A]  time = 0.0249033, 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 \[ \log (x)-\frac{1}{8} \log \left (1-x^8\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 - x^8)),x]

[Out]

Log[x] - Log[1 - x^8]/8

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Rubi in Sympy [A]  time = 3.94034, size = 14, normalized size = 0.93 \[ \frac{\log{\left (x^{8} \right )}}{8} - \frac{\log{\left (- x^{8} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-x**8+1),x)

[Out]

log(x**8)/8 - log(-x**8 + 1)/8

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Mathematica [A]  time = 0.00622879, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{8} \log \left (1-x^8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(1 - x^8)),x]

[Out]

Log[x] - Log[1 - x^8]/8

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Maple [B]  time = 0.015, size = 32, normalized size = 2.1 \[ \ln \left ( x \right ) -{\frac{\ln \left ( -1+x \right ) }{8}}-{\frac{\ln \left ({x}^{4}+1 \right ) }{8}}-{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^8+1),x)

[Out]

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 = 1.41509, size = 20, normalized size = 1.33 \[ -\frac{1}{8} \, \log \left (x^{8} - 1\right ) + \frac{1}{8} \, \log \left (x^{8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*log(x^8 - 1) + 1/8*log(x^8)

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Fricas [A]  time = 0.213585, size = 15, normalized size = 1. \[ -\frac{1}{8} \, \log \left (x^{8} - 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*log(x^8 - 1) + log(x)

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Sympy [A]  time = 0.276096, size = 10, normalized size = 0.67 \[ \log{\left (x \right )} - \frac{\log{\left (x^{8} - 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-x**8+1),x)

[Out]

log(x) - log(x**8 - 1)/8

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GIAC/XCAS [A]  time = 0.221763, size = 22, normalized size = 1.47 \[ \frac{1}{8} \,{\rm ln}\left (x^{8}\right ) - \frac{1}{8} \,{\rm ln}\left ({\left | x^{8} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*ln(x^8) - 1/8*ln(abs(x^8 - 1))