3.299 \(\int \frac{1}{-8+x^3} \, dx\)

Optimal. Leaf size=43 \[ -\frac{1}{24} \log \left (x^2+2 x+4\right )+\frac{1}{12} \log (2-x)-\frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

[Out]

-ArcTan[(1 + x)/Sqrt[3]]/(4*Sqrt[3]) + Log[2 - x]/12 - Log[4 + 2*x + x^2]/24

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Rubi [A]  time = 0.044413, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857 \[ -\frac{1}{24} \log \left (x^2+2 x+4\right )+\frac{1}{12} \log (2-x)-\frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(-8 + x^3)^(-1),x]

[Out]

-ArcTan[(1 + x)/Sqrt[3]]/(4*Sqrt[3]) + Log[2 - x]/12 - Log[4 + 2*x + x^2]/24

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Rubi in Sympy [A]  time = 2.86422, size = 37, normalized size = 0.86 \[ \frac{\log{\left (- x + 2 \right )}}{12} - \frac{\log{\left (x^{2} + 2 x + 4 \right )}}{24} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{x}{3} + \frac{1}{3}\right ) \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3-8),x)

[Out]

log(-x + 2)/12 - log(x**2 + 2*x + 4)/24 - sqrt(3)*atan(sqrt(3)*(x/3 + 1/3))/12

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Mathematica [A]  time = 0.0127126, size = 43, normalized size = 1. \[ -\frac{1}{24} \log \left (x^2+2 x+4\right )+\frac{1}{12} \log (2-x)-\frac{\tan ^{-1}\left (\frac{x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-8 + x^3)^(-1),x]

[Out]

-ArcTan[(1 + x)/Sqrt[3]]/(4*Sqrt[3]) + Log[2 - x]/12 - Log[4 + 2*x + x^2]/24

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Maple [A]  time = 0.009, size = 35, normalized size = 0.8 \[{\frac{\ln \left ( -2+x \right ) }{12}}-{\frac{\ln \left ({x}^{2}+2\,x+4 \right ) }{24}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x+2 \right ) \sqrt{3}}{6}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3-8),x)

[Out]

1/12*ln(-2+x)-1/24*ln(x^2+2*x+4)-1/12*3^(1/2)*arctan(1/6*(2*x+2)*3^(1/2))

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Maxima [A]  time = 1.48308, size = 43, normalized size = 1. \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x + 1\right )}\right ) - \frac{1}{24} \, \log \left (x^{2} + 2 \, x + 4\right ) + \frac{1}{12} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^3 - 8),x, algorithm="maxima")

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(x + 1)) - 1/24*log(x^2 + 2*x + 4) + 1/12*log(x
- 2)

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Fricas [A]  time = 0.213994, size = 53, normalized size = 1.23 \[ -\frac{1}{72} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + 2 \, x + 4\right ) - 2 \, \sqrt{3} \log \left (x - 2\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^3 - 8),x, algorithm="fricas")

[Out]

-1/72*sqrt(3)*(sqrt(3)*log(x^2 + 2*x + 4) - 2*sqrt(3)*log(x - 2) + 6*arctan(1/3*
sqrt(3)*(x + 1)))

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Sympy [A]  time = 0.188309, size = 41, normalized size = 0.95 \[ \frac{\log{\left (x - 2 \right )}}{12} - \frac{\log{\left (x^{2} + 2 x + 4 \right )}}{24} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3-8),x)

[Out]

log(x - 2)/12 - log(x**2 + 2*x + 4)/24 - sqrt(3)*atan(sqrt(3)*x/3 + sqrt(3)/3)/1
2

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GIAC/XCAS [A]  time = 0.219035, size = 45, normalized size = 1.05 \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x + 1\right )}\right ) - \frac{1}{24} \,{\rm ln}\left (x^{2} + 2 \, x + 4\right ) + \frac{1}{12} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^3 - 8),x, algorithm="giac")

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(x + 1)) - 1/24*ln(x^2 + 2*x + 4) + 1/12*ln(abs
(x - 2))