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

Optimal. Leaf size=39 \[ -\frac{1}{2} \log \left (x^2+x+1\right )-\log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0832375, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{1}{2} \log \left (x^2+x+1\right )-\log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(1 + 2*x))/(1 - x^3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.1168, size = 37, normalized size = 0.95 \[ - \log{\left (- x + 1 \right )} - \frac{\log{\left (x^{2} + x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} \]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0206975, size = 53, normalized size = 1.36 \[ -\frac{2}{3} \log \left (1-x^3\right )+\frac{1}{6} \log \left (x^2+x+1\right )-\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 33, normalized size = 0.9 \[ -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(1+2*x)/(-x^3+1),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.52304, size = 43, normalized size = 1.1 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{2} \, \log \left (x^{2} + x + 1\right ) - \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.238685, size = 53, normalized size = 1.36 \[ -\frac{1}{6} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + x + 1\right ) + 2 \, \sqrt{3} \log \left (x - 1\right ) + 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.155739, size = 41, normalized size = 1.05 \[ - \log{\left (x - 1 \right )} - \frac{\log{\left (x^{2} + x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211462, size = 45, normalized size = 1.15 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{2} \,{\rm ln}\left (x^{2} + x + 1\right ) -{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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