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

Optimal. Leaf size=41 \[ \frac{5}{6} \log \left (x^2-x+1\right )+\frac{1}{3} \log (x+1)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-(ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3]) + Log[1 + x]/3 + (5*Log[1 - x + x^2])/6

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Rubi [A]  time = 0.0814405, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{5}{6} \log \left (x^2-x+1\right )+\frac{1}{3} \log (x+1)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(1 + 2*x))/(1 + x^3),x]

[Out]

-(ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3]) + Log[1 + x]/3 + (5*Log[1 - x + x^2])/6

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Rubi in Sympy [A]  time = 11.5199, size = 39, normalized size = 0.95 \[ \frac{\log{\left (x + 1 \right )}}{3} + \frac{5 \log{\left (x^{2} - x + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + 1)/3 + 5*log(x**2 - x + 1)/6 + sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))/3

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Mathematica [A]  time = 0.0139612, size = 47, normalized size = 1.15 \[ \frac{1}{6} \left (4 \log \left (x^3+1\right )+\log \left (x^2-x+1\right )-2 \log (x+1)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(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 = 35, normalized size = 0.9 \[{\frac{5\,\ln \left ({x}^{2}-x+1 \right ) }{6}}+{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(1+2*x)/(x^3+1),x)

[Out]

5/6*ln(x^2-x+1)+1/3*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+1/3*ln(1+x)

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Maxima [A]  time = 1.58207, size = 46, normalized size = 1.12 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{5}{6} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{3} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 5/6*log(x^2 - x + 1) + 1/3*log(x + 1
)

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Fricas [A]  time = 0.23471, size = 57, normalized size = 1.39 \[ \frac{1}{18} \, \sqrt{3}{\left (5 \, \sqrt{3} \log \left (x^{2} - x + 1\right ) + 2 \, \sqrt{3} \log \left (x + 1\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(5*sqrt(3)*log(x^2 - x + 1) + 2*sqrt(3)*log(x + 1) + 6*arctan(1/3*s
qrt(3)*(2*x - 1)))

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Sympy [A]  time = 0.166632, size = 42, normalized size = 1.02 \[ \frac{\log{\left (x + 1 \right )}}{3} + \frac{5 \log{\left (x^{2} - x + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(1+2*x)/(x**3+1),x)

[Out]

log(x + 1)/3 + 5*log(x**2 - x + 1)/6 + sqrt(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/3

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GIAC/XCAS [A]  time = 0.211482, size = 47, normalized size = 1.15 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{5}{6} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 5/6*ln(x^2 - x + 1) + 1/3*ln(abs(x +
1))