3.175 \(\int \frac{1}{-x^3+x^6} \, dx\)

Optimal. Leaf size=48 \[ \frac{1}{2 x^2}-\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0490998, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727 \[ \frac{1}{2 x^2}-\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(-x^3 + x^6)^(-1),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.47222, size = 44, normalized size = 0.92 \[ \frac{\log{\left (- x + 1 \right )}}{3} - \frac{\log{\left (x^{2} + x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} + \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**6-x**3),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0191526, size = 48, normalized size = 1. \[ \frac{1}{2 x^2}-\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-x^3 + x^6)^(-1),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 38, normalized size = 0.8 \[{\frac{1}{2\,{x}^{2}}}+{\frac{\ln \left ( -1+x \right ) }{3}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^6-x^3),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.51673, size = 50, normalized size = 1.04 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2 \, x^{2}} - \frac{1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{3} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 - x^3),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.202438, size = 76, normalized size = 1.58 \[ -\frac{\sqrt{3}{\left (\sqrt{3} x^{2} \log \left (x^{2} + x + 1\right ) - 2 \, \sqrt{3} x^{2} \log \left (x - 1\right ) + 6 \, x^{2} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 3 \, \sqrt{3}\right )}}{18 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 - x^3),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(sqrt(3)*x^2*log(x^2 + x + 1) - 2*sqrt(3)*x^2*log(x - 1) + 6*x^2*a
rctan(1/3*sqrt(3)*(2*x + 1)) - 3*sqrt(3))/x^2

_______________________________________________________________________________________

Sympy [A]  time = 0.214274, size = 48, normalized size = 1. \[ \frac{\log{\left (x - 1 \right )}}{3} - \frac{\log{\left (x^{2} + x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} + \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**6-x**3),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211882, size = 51, normalized size = 1.06 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2 \, x^{2}} - \frac{1}{6} \,{\rm ln}\left (x^{2} + x + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 - x^3),x, algorithm="giac")

[Out]

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