3.1372 \(\int \frac{1}{x^5 \left (1+x^6\right )} \, dx\)

Optimal. Leaf size=56 \[ -\frac{1}{4 x^4}-\frac{1}{6} \log \left (x^2+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{12} \log \left (x^4-x^2+1\right ) \]

[Out]

-1/(4*x^4) + ArcTan[(1 - 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - Log[1 + x^2]/6 + Log[1 -
x^2 + x^4]/12

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Rubi [A]  time = 0.0808821, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727 \[ -\frac{1}{4 x^4}-\frac{1}{6} \log \left (x^2+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{12} \log \left (x^4-x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 + x^6)),x]

[Out]

-1/(4*x^4) + ArcTan[(1 - 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - Log[1 + x^2]/6 + Log[1 -
x^2 + x^4]/12

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Rubi in Sympy [A]  time = 10.2153, size = 49, normalized size = 0.88 \[ - \frac{\log{\left (x^{2} + 1 \right )}}{6} + \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} - \frac{1}{3}\right ) \right )}}{6} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**2 + 1)/6 + log(x**4 - x**2 + 1)/12 - sqrt(3)*atan(sqrt(3)*(2*x**2/3 - 1/
3))/6 - 1/(4*x**4)

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Mathematica [A]  time = 0.040434, size = 79, normalized size = 1.41 \[ \frac{1}{12} \left (-\frac{3}{x^4}-2 \log \left (x^2+1\right )+\log \left (x^2-\sqrt{3} x+1\right )+\log \left (x^2+\sqrt{3} x+1\right )+2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+2 \sqrt{3} \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(1 + x^6)),x]

[Out]

(-3/x^4 + 2*Sqrt[3]*ArcTan[Sqrt[3] - 2*x] + 2*Sqrt[3]*ArcTan[Sqrt[3] + 2*x] - 2*
Log[1 + x^2] + Log[1 - Sqrt[3]*x + x^2] + Log[1 + Sqrt[3]*x + x^2])/12

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Maple [A]  time = 0.013, size = 46, normalized size = 0.8 \[{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{4\,{x}^{4}}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^6+1),x)

[Out]

1/12*ln(x^4-x^2+1)-1/6*3^(1/2)*arctan(1/3*(2*x^2-1)*3^(1/2))-1/4/x^4-1/6*ln(x^2+
1)

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Maxima [A]  time = 1.58611, size = 61, normalized size = 1.09 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{4 \, x^{4}} + \frac{1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac{1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)) - 1/4/x^4 + 1/12*log(x^4 - x^2 + 1)
- 1/6*log(x^2 + 1)

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Fricas [A]  time = 0.220564, size = 86, normalized size = 1.54 \[ \frac{\sqrt{3}{\left (\sqrt{3} x^{4} \log \left (x^{4} - x^{2} + 1\right ) - 2 \, \sqrt{3} x^{4} \log \left (x^{2} + 1\right ) - 6 \, x^{4} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - 3 \, \sqrt{3}\right )}}{36 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*sqrt(3)*(sqrt(3)*x^4*log(x^4 - x^2 + 1) - 2*sqrt(3)*x^4*log(x^2 + 1) - 6*x^
4*arctan(1/3*sqrt(3)*(2*x^2 - 1)) - 3*sqrt(3))/x^4

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Sympy [A]  time = 0.538572, size = 53, normalized size = 0.95 \[ - \frac{\log{\left (x^{2} + 1 \right )}}{6} + \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} - \frac{\sqrt{3}}{3} \right )}}{6} - \frac{1}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**6+1),x)

[Out]

-log(x**2 + 1)/6 + log(x**4 - x**2 + 1)/12 - sqrt(3)*atan(2*sqrt(3)*x**2/3 - sqr
t(3)/3)/6 - 1/(4*x**4)

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GIAC/XCAS [A]  time = 0.229357, size = 61, normalized size = 1.09 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{4 \, x^{4}} + \frac{1}{12} \,{\rm ln}\left (x^{4} - x^{2} + 1\right ) - \frac{1}{6} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)) - 1/4/x^4 + 1/12*ln(x^4 - x^2 + 1)
- 1/6*ln(x^2 + 1)