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

Optimal. Leaf size=39 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{6} \log \left (x^6-x^3+1\right )+\log (x) \]

[Out]

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

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Rubi [A]  time = 0.116528, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^3}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{6} \log \left (x^6-x^3+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^3)/(x*(1 - x^3 + x^6)),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.6819, size = 41, normalized size = 1.05 \[ \frac{\log{\left (x^{3} \right )}}{3} - \frac{\log{\left (x^{6} - x^{3} + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{3}}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+1)/x/(x**6-x**3+1),x)

[Out]

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

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Mathematica [C]  time = 0.0210754, size = 55, normalized size = 1.41 \[ \log (x)-\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^3)/(x*(1 - x^3 + x^6)),x]

[Out]

Log[x] - RootSum[1 - #1^3 + #1^6 & , (-2*Log[x - #1] + Log[x - #1]*#1^3)/(-1 + 2
*#1^3) & ]/3

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Maple [A]  time = 0.008, size = 35, normalized size = 0.9 \[ \ln \left ( x \right ) -{\frac{\ln \left ({x}^{6}-{x}^{3}+1 \right ) }{6}}+{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,{x}^{3}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+1)/x/(x^6-x^3+1),x)

[Out]

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

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Maxima [A]  time = 0.820072, size = 51, normalized size = 1.31 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) - \frac{1}{6} \, \log \left (x^{6} - x^{3} + 1\right ) + \frac{1}{3} \, \log \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.275647, size = 58, normalized size = 1.49 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{6} - x^{3} + 1\right ) - 6 \, \sqrt{3} \log \left (x\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*log(x^6 - x^3 + 1) - 6*sqrt(3)*log(x) - 6*arctan(1/3*sqrt
(3)*(2*x^3 - 1)))

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Sympy [A]  time = 0.328135, size = 41, normalized size = 1.05 \[ \log{\left (x \right )} - \frac{\log{\left (x^{6} - x^{3} + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{3}}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+1)/x/(x**6-x**3+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.271527, size = 47, normalized size = 1.21 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{3} - 1\right )}\right ) - \frac{1}{6} \,{\rm ln}\left (x^{6} - x^{3} + 1\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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