3.111 \(\int \frac{-2+2 x+3 x^2}{-1+x^3} \, dx\)

Optimal. Leaf size=28 \[ \log \left (1-x^3\right )+\frac{4 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0491363, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \log \left (1-x^3\right )+\frac{4 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(-2 + 2*x + 3*x^2)/(-1 + x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.14249, size = 29, normalized size = 1.04 \[ \log{\left (- x^{3} + 1 \right )} + \frac{4 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0167873, size = 28, normalized size = 1. \[ \log \left (1-x^3\right )+\frac{4 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-2 + 2*x + 3*x^2)/(-1 + x^3),x]

[Out]

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

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Maple [A]  time = 0.01, size = 29, normalized size = 1. \[ \ln \left ( -1+x \right ) +\ln \left ({x}^{2}+x+1 \right ) +{\frac{4\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2*x-2)/(x^3-1),x)

[Out]

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

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Maxima [A]  time = 1.50273, size = 38, normalized size = 1.36 \[ \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \log \left (x^{2} + x + 1\right ) + \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2*x - 2)/(x^3 - 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.19918, size = 51, normalized size = 1.82 \[ \frac{1}{3} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + x + 1\right ) + \sqrt{3} \log \left (x - 1\right ) + 4 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2*x - 2)/(x^3 - 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.141372, size = 3, normalized size = 0.11 \[ \log{\left (x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2*x-2)/(x**3-1),x)

[Out]

log(x - 1)

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GIAC/XCAS [A]  time = 0.211376, size = 39, normalized size = 1.39 \[ \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) +{\rm ln}\left (x^{2} + x + 1\right ) +{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2*x - 2)/(x^3 - 1),x, algorithm="giac")

[Out]

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