3.1493 \(\int \frac{1}{x \left (1+x^8\right )} \, dx\)

Optimal. Leaf size=13 \[ \log (x)-\frac{1}{8} \log \left (x^8+1\right ) \]

[Out]

Log[x] - Log[1 + x^8]/8

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Rubi [A]  time = 0.0197897, 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 \[ \log (x)-\frac{1}{8} \log \left (x^8+1\right ) \]

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 = 3.26237, size = 14, normalized size = 1.08 \[ \frac{\log{\left (x^{8} \right )}}{8} - \frac{\log{\left (x^{8} + 1 \right )}}{8} \]

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 = 0.00545347, size = 13, normalized size = 1. \[ \log (x)-\frac{1}{8} \log \left (x^8+1\right ) \]

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 [A]  time = 0.006, size = 12, normalized size = 0.9 \[ \ln \left ( x \right ) -{\frac{\ln \left ({x}^{8}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^8+1),x)

[Out]

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

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Maxima [A]  time = 1.4313, size = 20, normalized size = 1.54 \[ -\frac{1}{8} \, \log \left (x^{8} + 1\right ) + \frac{1}{8} \, \log \left (x^{8}\right ) \]

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 = 0.214796, size = 15, normalized size = 1.15 \[ -\frac{1}{8} \, \log \left (x^{8} + 1\right ) + \log \left (x\right ) \]

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 = 0.264828, size = 10, normalized size = 0.77 \[ \log{\left (x \right )} - \frac{\log{\left (x^{8} + 1 \right )}}{8} \]

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 = 0.233752, size = 20, normalized size = 1.54 \[ -\frac{1}{8} \,{\rm ln}\left (x^{8} + 1\right ) + \frac{1}{8} \,{\rm ln}\left (x^{8}\right ) \]

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) + 1/8*ln(x^8)