3.1370 \(\int \frac{1}{x^3 \left (1+x^6\right )} \, dx\)

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

[Out]

-1/(2*x^2) + 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 = 0.0878581, 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 \[ -\frac{1}{2 x^2}+\frac{1}{6} \log \left (x^2+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{12} \log \left (x^4-x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 + x^6)),x]

[Out]

-1/(2*x^2) + 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 = 9.68515, size = 49, normalized size = 0.88 \[ \frac{\log{\left (x^{2} + 1 \right )}}{6} - \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} - \frac{1}{3}\right ) \right )}}{6} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(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/(2*x**2)

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Mathematica [A]  time = 0.0385807, size = 83, normalized size = 1.48 \[ \frac{1}{12} \left (-\frac{6}{x^2}+2 \log \left (x^2+1\right )-\log \left (x^2-\sqrt{3} x+1\right )-\log \left (x^2+\sqrt{3} x+1\right )+2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )+2 \sqrt{3} \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(1 + x^6)),x]

[Out]

(-6/x^2 + 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 \[ -{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{1}{2\,{x}^{2}}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(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))-1/2/x^2+1/6*ln(x^2
+1)

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Maxima [A]  time = 1.60931, size = 61, normalized size = 1.09 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{2 \, x^{2}} - \frac{1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) + \frac{1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219208, size = 86, normalized size = 1.54 \[ -\frac{\sqrt{3}{\left (\sqrt{3} x^{2} \log \left (x^{4} - x^{2} + 1\right ) - 2 \, \sqrt{3} x^{2} \log \left (x^{2} + 1\right ) + 6 \, x^{2} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) + 6 \, \sqrt{3}\right )}}{36 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.478509, size = 53, normalized size = 0.95 \[ \frac{\log{\left (x^{2} + 1 \right )}}{6} - \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} - \frac{\sqrt{3}}{3} \right )}}{6} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(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 - sqrt
(3)/3)/6 - 1/(2*x**2)

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GIAC/XCAS [A]  time = 0.227318, size = 61, normalized size = 1.09 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{2 \, x^{2}} - \frac{1}{12} \,{\rm ln}\left (x^{4} - x^{2} + 1\right ) + \frac{1}{6} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 1)*x^3),x, algorithm="giac")

[Out]

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