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

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

[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 = 0.125172, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ \frac{x^2}{2}+\frac{1}{2} \log \left (x^2-x+1\right )+\log (x+1)+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

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 + Integ
ral(x, x)

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Mathematica [A]  time = 0.024239, size = 54, normalized size = 1.26 \[ \frac{1}{6} \left (4 \log \left (x^3+1\right )+3 x^2-\log \left (x^2-x+1\right )+2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

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 = 0.007, size = 38, normalized size = 0.9 \[{\frac{{x}^{2}}{2}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+\ln \left ( 1+x \right ) \]

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

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Maxima [A]  time = 1.52392, size = 50, normalized size = 1.16 \[ \frac{1}{2} \, x^{2} - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} - x + 1\right ) + \log \left (x + 1\right ) \]

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) + log
(x + 1)

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Fricas [A]  time = 0.225107, size = 65, normalized size = 1.51 \[ \frac{1}{6} \, \sqrt{3}{\left (\sqrt{3} x^{2} + \sqrt{3} \log \left (x^{2} - x + 1\right ) + 2 \, \sqrt{3} \log \left (x + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

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*a
rctan(1/3*sqrt(3)*(2*x - 1)))

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Sympy [A]  time = 0.153301, size = 44, normalized size = 1.02 \[ \frac{x^{2}}{2} + \log{\left (x + 1 \right )} + \frac{\log{\left (x^{2} - x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

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 = 0.212748, size = 51, normalized size = 1.19 \[ \frac{1}{2} \, x^{2} - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} - x + 1\right ) +{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

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(a
bs(x + 1))