3.1353 \(\int \frac{1}{x^3 \left (1-x^6\right )} \, dx\)

Optimal. Leaf size=56 \[ -\frac{1}{2 x^2}-\frac{1}{6} \log \left (1-x^2\right )-\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{12} \log \left (x^4+x^2+1\right ) \]

[Out]

-1/(2*x^2) - ArcTan[(1 + 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - Log[1 - x^2]/6 + Log[1 +
x^2 + x^4]/12

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Rubi [A]  time = 0.0846454, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{1}{2 x^2}-\frac{1}{6} \log \left (1-x^2\right )-\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{12} \log \left (x^4+x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 - x^6)),x]

[Out]

-1/(2*x^2) - ArcTan[(1 + 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - Log[1 - x^2]/6 + Log[1 +
x^2 + x^4]/12

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Rubi in Sympy [A]  time = 10.4688, size = 49, normalized size = 0.88 \[ - \frac{\log{\left (- x^{2} + 1 \right )}}{6} + \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{6} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(-x**6+1),x)

[Out]

-log(-x**2 + 1)/6 + log(x**4 + x**2 + 1)/12 - sqrt(3)*atan(sqrt(3)*(2*x**2/3 + 1
/3))/6 - 1/(2*x**2)

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Mathematica [A]  time = 0.0409975, size = 78, normalized size = 1.39 \[ \frac{1}{12} \left (-\frac{6}{x^2}+\log \left (x^2-x+1\right )+\log \left (x^2+x+1\right )-2 \log (1-x)-2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(1 - x^6)),x]

[Out]

(-6/x^2 - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt
[3]] - 2*Log[1 - x] - 2*Log[1 + x] + Log[1 - x + x^2] + Log[1 + x + x^2])/12

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Maple [A]  time = 0.014, size = 71, normalized size = 1.3 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( -1+x \right ) }{6}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{1}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-x^6+1),x)

[Out]

1/12*ln(x^2+x+1)+1/6*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/6*ln(-1+x)+1/12*ln(x^
2-x+1)-1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))-1/6*ln(1+x)-1/2/x^2

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Maxima [A]  time = 1.57701, size = 58, normalized size = 1.04 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) - \frac{1}{2 \, x^{2}} + \frac{1}{12} \, \log \left (x^{4} + x^{2} + 1\right ) - \frac{1}{6} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1)) - 1/2/x^2 + 1/12*log(x^4 + x^2 + 1)
- 1/6*log(x^2 - 1)

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Fricas [A]  time = 0.228073, size = 84, normalized size = 1.5 \[ \frac{\sqrt{3}{\left (\sqrt{3} x^{2} \log \left (x^{4} + x^{2} + 1\right ) - 2 \, \sqrt{3} x^{2} \log \left (x^{2} - 1\right ) - 6 \, x^{2} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) - 6 \, \sqrt{3}\right )}}{36 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*sqrt(3)*(sqrt(3)*x^2*log(x^4 + x^2 + 1) - 2*sqrt(3)*x^2*log(x^2 - 1) - 6*x^
2*arctan(1/3*sqrt(3)*(2*x^2 + 1)) - 6*sqrt(3))/x^2

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Sympy [A]  time = 0.514444, size = 53, normalized size = 0.95 \[ - \frac{\log{\left (x^{2} - 1 \right )}}{6} + \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} + \frac{\sqrt{3}}{3} \right )}}{6} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(-x**6+1),x)

[Out]

-log(x**2 - 1)/6 + log(x**4 + x**2 + 1)/12 - sqrt(3)*atan(2*sqrt(3)*x**2/3 + sqr
t(3)/3)/6 - 1/(2*x**2)

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GIAC/XCAS [A]  time = 0.224435, size = 59, normalized size = 1.05 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) - \frac{1}{2 \, x^{2}} + \frac{1}{12} \,{\rm ln}\left (x^{4} + x^{2} + 1\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1)) - 1/2/x^2 + 1/12*ln(x^4 + x^2 + 1)
- 1/6*ln(abs(x^2 - 1))