3.1297 \(\int \frac{1}{x^6 \left (1+x^5\right )} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{5 x^5}+\frac{1}{5} \log \left (x^5+1\right )-\log (x) \]

[Out]

-1/(5*x^5) - Log[x] + Log[1 + x^5]/5

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Rubi [A]  time = 0.026797, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{1}{5 x^5}+\frac{1}{5} \log \left (x^5+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 + x^5)),x]

[Out]

-1/(5*x^5) - Log[x] + Log[1 + x^5]/5

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Rubi in Sympy [A]  time = 3.85463, size = 20, normalized size = 0.91 \[ - \frac{\log{\left (x^{5} \right )}}{5} + \frac{\log{\left (x^{5} + 1 \right )}}{5} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(x**5+1),x)

[Out]

-log(x**5)/5 + log(x**5 + 1)/5 - 1/(5*x**5)

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Mathematica [A]  time = 0.00633246, size = 22, normalized size = 1. \[ -\frac{1}{5 x^5}+\frac{1}{5} \log \left (x^5+1\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 + x^5)),x]

[Out]

-1/(5*x^5) - Log[x] + Log[1 + x^5]/5

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Maple [A]  time = 0.011, size = 36, normalized size = 1.6 \[ -{\frac{1}{5\,{x}^{5}}}-\ln \left ( x \right ) +{\frac{\ln \left ( 1+x \right ) }{5}}+{\frac{\ln \left ({x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) }{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^5+1),x)

[Out]

-1/5/x^5-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 = 1.44966, size = 27, normalized size = 1.23 \[ -\frac{1}{5 \, x^{5}} + \frac{1}{5} \, \log \left (x^{5} + 1\right ) - \frac{1}{5} \, \log \left (x^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^5 + 1)*x^6),x, algorithm="maxima")

[Out]

-1/5/x^5 + 1/5*log(x^5 + 1) - 1/5*log(x^5)

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Fricas [A]  time = 0.210938, size = 32, normalized size = 1.45 \[ \frac{x^{5} \log \left (x^{5} + 1\right ) - 5 \, x^{5} \log \left (x\right ) - 1}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^5 + 1)*x^6),x, algorithm="fricas")

[Out]

1/5*(x^5*log(x^5 + 1) - 5*x^5*log(x) - 1)/x^5

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Sympy [A]  time = 0.320354, size = 17, normalized size = 0.77 \[ - \log{\left (x \right )} + \frac{\log{\left (x^{5} + 1 \right )}}{5} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(x**5+1),x)

[Out]

-log(x) + log(x**5 + 1)/5 - 1/(5*x**5)

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GIAC/XCAS [A]  time = 0.226598, size = 34, normalized size = 1.55 \[ \frac{x^{5} - 1}{5 \, x^{5}} + \frac{1}{5} \,{\rm ln}\left ({\left | x^{5} + 1 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^5 + 1)*x^6),x, algorithm="giac")

[Out]

1/5*(x^5 - 1)/x^5 + 1/5*ln(abs(x^5 + 1)) - ln(abs(x))