3.1296 \(\int \frac{1}{x \left (1+x^5\right )} \, dx\)

Optimal. Leaf size=13 \[ \log (x)-\frac{1}{5} \log \left (x^5+1\right ) \]

[Out]

Log[x] - Log[1 + x^5]/5

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Rubi [A]  time = 0.0168762, 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 \[ \log (x)-\frac{1}{5} \log \left (x^5+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 + x^5)),x]

[Out]

Log[x] - Log[1 + x^5]/5

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Rubi in Sympy [A]  time = 3.22916, size = 14, normalized size = 1.08 \[ \frac{\log{\left (x^{5} \right )}}{5} - \frac{\log{\left (x^{5} + 1 \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(x**5+1),x)

[Out]

log(x**5)/5 - log(x**5 + 1)/5

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Mathematica [A]  time = 0.00472231, size = 13, normalized size = 1. \[ \log (x)-\frac{1}{5} \log \left (x^5+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(1 + x^5)),x]

[Out]

Log[x] - Log[1 + x^5]/5

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Maple [B]  time = 0.01, size = 29, normalized size = 2.2 \[ \ln \left ( x \right ) -{\frac{\ln \left ( 1+x \right ) }{5}}-{\frac{\ln \left ({x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) }{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^5+1),x)

[Out]

ln(x)-1/5*ln(1+x)-1/5*ln(x^4-x^3+x^2-x+1)

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Maxima [A]  time = 1.44393, size = 20, normalized size = 1.54 \[ -\frac{1}{5} \, \log \left (x^{5} + 1\right ) + \frac{1}{5} \, \log \left (x^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^5 + 1)*x),x, algorithm="maxima")

[Out]

-1/5*log(x^5 + 1) + 1/5*log(x^5)

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Fricas [A]  time = 0.215933, size = 15, normalized size = 1.15 \[ -\frac{1}{5} \, \log \left (x^{5} + 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^5 + 1)*x),x, algorithm="fricas")

[Out]

-1/5*log(x^5 + 1) + log(x)

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Sympy [A]  time = 0.218647, size = 10, normalized size = 0.77 \[ \log{\left (x \right )} - \frac{\log{\left (x^{5} + 1 \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(x**5+1),x)

[Out]

log(x) - log(x**5 + 1)/5

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GIAC/XCAS [A]  time = 0.228347, size = 18, normalized size = 1.38 \[ -\frac{1}{5} \,{\rm ln}\left ({\left | x^{5} + 1 \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^5 + 1)*x),x, algorithm="giac")

[Out]

-1/5*ln(abs(x^5 + 1)) + ln(abs(x))