3.153 \(\int \frac{x}{1+x^6} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{6} \log \left (x^2+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{12} \log \left (x^4-x^2+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0730563, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778 \[ \frac{1}{6} \log \left (x^2+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{12} \log \left (x^4-x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.31447, size = 42, normalized size = 0.86 \[ \frac{\log{\left (x^{2} + 1 \right )}}{6} - \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} - \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(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

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Mathematica [A]  time = 0.0242909, size = 78, normalized size = 1.59 \[ \frac{1}{12} \left (2 \log \left (x^2+1\right )-\log \left (x^2-\sqrt{3} x+1\right )-\log \left (x^2+\sqrt{3} x+1\right )-2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-2 x\right )-2 \sqrt{3} \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(1 + x^6),x]

[Out]

(-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 = 0.01, size = 41, normalized size = 0.8 \[{\frac{\ln \left ({x}^{2}+1 \right ) }{6}}-{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^6+1),x)

[Out]

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

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Maxima [A]  time = 1.55928, size = 54, normalized size = 1.1 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) + \frac{1}{6} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256307, size = 63, normalized size = 1.29 \[ -\frac{1}{36} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{4} - x^{2} + 1\right ) - 2 \, \sqrt{3} \log \left (x^{2} + 1\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.192293, size = 46, normalized size = 0.94 \[ \frac{\log{\left (x^{2} + 1 \right )}}{6} - \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} - \frac{\sqrt{3}}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(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

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GIAC/XCAS [A]  time = 0.200641, size = 54, normalized size = 1.1 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{12} \,{\rm ln}\left (x^{4} - x^{2} + 1\right ) + \frac{1}{6} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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