3.77 \(\int \frac{2 x^2+x^4}{1-x^3} \, dx\)

Optimal. Leaf size=46 \[ -\frac{x^2}{2}-\frac{1}{2} \log \left (x^2+x+1\right )-\log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.125808, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{x^2}{2}-\frac{1}{2} \log \left (x^2+x+1\right )-\log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(2*x^2 + x^4)/(1 - x^3),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \log{\left (- x + 1 \right )} - \frac{\log{\left (x^{2} + x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} - \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+2*x**2)/(-x**3+1),x)

[Out]

-log(-x + 1) - log(x**2 + x + 1)/2 - sqrt(3)*atan(sqrt(3)*(2*x/3 + 1/3))/3 - Int
egral(x, x)

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Mathematica [A]  time = 0.0233239, size = 54, normalized size = 1.17 \[ \frac{1}{6} \left (-4 \log \left (1-x^3\right )-3 x^2+\log \left (x^2+x+1\right )-2 \log (1-x)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2*x^2 + x^4)/(1 - x^3),x]

[Out]

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

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Maple [A]  time = 0.008, size = 38, normalized size = 0.8 \[ -{\frac{{x}^{2}}{2}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+2*x^2)/(-x^3+1),x)

[Out]

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

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Maxima [A]  time = 1.51471, size = 50, normalized size = 1.09 \[ -\frac{1}{2} \, x^{2} - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{2} \, \log \left (x^{2} + x + 1\right ) - \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 + 2*x^2)/(x^3 - 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.244687, size = 62, normalized size = 1.35 \[ -\frac{1}{6} \, \sqrt{3}{\left (\sqrt{3} x^{2} + \sqrt{3} \log \left (x^{2} + x + 1\right ) + 2 \, \sqrt{3} \log \left (x - 1\right ) + 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 + 2*x^2)/(x^3 - 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.152768, size = 46, normalized size = 1. \[ - \frac{x^{2}}{2} - \log{\left (x - 1 \right )} - \frac{\log{\left (x^{2} + x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+2*x**2)/(-x**3+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211878, size = 51, normalized size = 1.11 \[ -\frac{1}{2} \, x^{2} - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + x + 1\right ) -{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 + 2*x^2)/(x^3 - 1),x, algorithm="giac")

[Out]

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