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

Optimal. Leaf size=44 \[ \frac{1}{6} \log \left (x^2-x+1\right )-x+\frac{2}{3} \log (x+1)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.112368, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{1}{6} \log \left (x^2-x+1\right )-x+\frac{2}{3} \log (x+1)-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - x)*x^2)/(1 + x^3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.7968, size = 41, normalized size = 0.93 \[ - x + \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.015404, size = 53, normalized size = 1.2 \[ \frac{1}{3} \log \left (x^3+1\right )-\frac{1}{6} \log \left (x^2-x+1\right )-x+\frac{1}{3} \log (x+1)+\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 38, normalized size = 0.9 \[ -x+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}+{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{2\,\ln \left ( 1+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)*x^2/(x^3+1),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.53376, size = 50, normalized size = 1.14 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - x + \frac{1}{6} \, \log \left (x^{2} - x + 1\right ) + \frac{2}{3} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.21263, size = 65, normalized size = 1.48 \[ -\frac{1}{18} \, \sqrt{3}{\left (6 \, \sqrt{3} x - \sqrt{3} \log \left (x^{2} - x + 1\right ) - 4 \, \sqrt{3} \log \left (x + 1\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(6*sqrt(3)*x - sqrt(3)*log(x^2 - x + 1) - 4*sqrt(3)*log(x + 1) - 6
*arctan(1/3*sqrt(3)*(2*x - 1)))

_______________________________________________________________________________________

Sympy [A]  time = 0.172118, size = 44, normalized size = 1. \[ - x + \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209723, size = 51, normalized size = 1.16 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - x + \frac{1}{6} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{2}{3} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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