3.277 \(\int \frac{1}{x \left (1+2 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=24 \[ \frac{1}{4 \left (x^4+1\right )}-\frac{1}{4} \log \left (x^4+1\right )+\log (x) \]

[Out]

1/(4*(1 + x^4)) + Log[x] - Log[1 + x^4]/4

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Rubi [A]  time = 0.028266, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{4 \left (x^4+1\right )}-\frac{1}{4} \log \left (x^4+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 + 2*x^4 + x^8)),x]

[Out]

1/(4*(1 + x^4)) + Log[x] - Log[1 + x^4]/4

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Rubi in Sympy [A]  time = 5.37042, size = 22, normalized size = 0.92 \[ \frac{\log{\left (x^{4} \right )}}{4} - \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{1}{4 \left (x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(x**8+2*x**4+1),x)

[Out]

log(x**4)/4 - log(x**4 + 1)/4 + 1/(4*(x**4 + 1))

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Mathematica [A]  time = 0.0133068, size = 24, normalized size = 1. \[ \frac{1}{4 \left (x^4+1\right )}-\frac{1}{4} \log \left (x^4+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(1 + 2*x^4 + x^8)),x]

[Out]

1/(4*(1 + x^4)) + Log[x] - Log[1 + x^4]/4

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Maple [A]  time = 0.019, size = 21, normalized size = 0.9 \[{\frac{1}{4\,{x}^{4}+4}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{4}+1 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^8+2*x^4+1),x)

[Out]

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

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Maxima [A]  time = 0.764091, size = 32, normalized size = 1.33 \[ \frac{1}{4 \,{\left (x^{4} + 1\right )}} - \frac{1}{4} \, \log \left (x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x),x, algorithm="maxima")

[Out]

1/4/(x^4 + 1) - 1/4*log(x^4 + 1) + 1/4*log(x^4)

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Fricas [A]  time = 0.253871, size = 43, normalized size = 1.79 \[ -\frac{{\left (x^{4} + 1\right )} \log \left (x^{4} + 1\right ) - 4 \,{\left (x^{4} + 1\right )} \log \left (x\right ) - 1}{4 \,{\left (x^{4} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x),x, algorithm="fricas")

[Out]

-1/4*((x^4 + 1)*log(x^4 + 1) - 4*(x^4 + 1)*log(x) - 1)/(x^4 + 1)

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Sympy [A]  time = 0.301816, size = 19, normalized size = 0.79 \[ \log{\left (x \right )} - \frac{\log{\left (x^{4} + 1 \right )}}{4} + \frac{1}{4 x^{4} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(x**8+2*x**4+1),x)

[Out]

log(x) - log(x**4 + 1)/4 + 1/(4*x**4 + 4)

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GIAC/XCAS [A]  time = 0.303119, size = 39, normalized size = 1.62 \[ \frac{x^{4} + 2}{4 \,{\left (x^{4} + 1\right )}} - \frac{1}{4} \,{\rm ln}\left (x^{4} + 1\right ) + \frac{1}{4} \,{\rm ln}\left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + 2*x^4 + 1)*x),x, algorithm="giac")

[Out]

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