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

Optimal. Leaf size=97 \[ -\frac{x}{6 \left (x^3+1\right )}+\frac{1}{72} \log \left (x^2-x+1\right )+\frac{1}{24} \log \left (x^2+x+1\right )-\frac{1}{12} \log (1-x)-\frac{1}{36} \log (x+1)+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

[Out]

-x/(6*(1 + x^3)) + ArcTan[(1 - 2*x)/Sqrt[3]]/(12*Sqrt[3]) + ArcTan[(1 + 2*x)/Sqr
t[3]]/(4*Sqrt[3]) - Log[1 - x]/12 - Log[1 + x]/36 + Log[1 - x + x^2]/72 + Log[1
+ x + x^2]/24

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Rubi [A]  time = 0.184403, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{x}{6 \left (x^3+1\right )}+\frac{1}{72} \log \left (x^2-x+1\right )+\frac{1}{24} \log \left (x^2+x+1\right )-\frac{1}{12} \log (1-x)-\frac{1}{36} \log (x+1)+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{12 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-x/(6*(1 + x^3)) + ArcTan[(1 - 2*x)/Sqrt[3]]/(12*Sqrt[3]) + ArcTan[(1 + 2*x)/Sqr
t[3]]/(4*Sqrt[3]) - Log[1 - x]/12 - Log[1 + x]/36 + Log[1 - x + x^2]/72 + Log[1
+ x + x^2]/24

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Rubi in Sympy [A]  time = 25.7174, size = 85, normalized size = 0.88 \[ - \frac{x}{6 \left (x^{3} + 1\right )} - \frac{\log{\left (- x + 1 \right )}}{12} - \frac{\log{\left (x + 1 \right )}}{36} + \frac{\log{\left (x^{2} - x + 1 \right )}}{72} + \frac{\log{\left (x^{2} + x + 1 \right )}}{24} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{36} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(6*(x**3 + 1)) - log(-x + 1)/12 - log(x + 1)/36 + log(x**2 - x + 1)/72 + log(
x**2 + x + 1)/24 - sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))/36 + sqrt(3)*atan(sqrt(3)
*(2*x/3 + 1/3))/12

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Mathematica [A]  time = 0.0825719, size = 85, normalized size = 0.88 \[ \frac{1}{72} \left (-\frac{12 x}{x^3+1}+\log \left (x^2-x+1\right )+3 \log \left (x^2+x+1\right )-6 \log (1-x)-2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-12*x)/(1 + x^3) - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 6*Sqrt[3]*ArcTan[(1
+ 2*x)/Sqrt[3]] - 6*Log[1 - x] - 2*Log[1 + x] + Log[1 - x + x^2] + 3*Log[1 + x +
x^2])/72

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Maple [A]  time = 0.019, size = 90, normalized size = 0.9 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{24}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( -1+x \right ) }{12}}+{\frac{1}{18+18\,x}}-{\frac{\ln \left ( 1+x \right ) }{36}}+{\frac{-2\,x-2}{36\,{x}^{2}-36\,x+36}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{72}}-{\frac{\sqrt{3}}{36}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(-x^3+1)/(x^3+1)^2,x)

[Out]

1/24*ln(x^2+x+1)+1/12*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/12*ln(-1+x)+1/18/(1+
x)-1/36*ln(1+x)+1/36*(-2*x-2)/(x^2-x+1)+1/72*ln(x^2-x+1)-1/36*3^(1/2)*arctan(1/3
*(2*x-1)*3^(1/2))

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Maxima [A]  time = 0.905521, size = 101, normalized size = 1.04 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{36} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{x}{6 \,{\left (x^{3} + 1\right )}} + \frac{1}{24} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{72} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{36} \, \log \left (x + 1\right ) - \frac{1}{12} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/36*sqrt(3)*arctan(1/3*sqrt(3)*(2*
x - 1)) - 1/6*x/(x^3 + 1) + 1/24*log(x^2 + x + 1) + 1/72*log(x^2 - x + 1) - 1/36
*log(x + 1) - 1/12*log(x - 1)

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Fricas [A]  time = 0.260667, size = 159, normalized size = 1.64 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3}{\left (x^{3} + 1\right )} \log \left (x^{2} + x + 1\right ) + \sqrt{3}{\left (x^{3} + 1\right )} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3}{\left (x^{3} + 1\right )} \log \left (x + 1\right ) - 6 \, \sqrt{3}{\left (x^{3} + 1\right )} \log \left (x - 1\right ) + 18 \,{\left (x^{3} + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 6 \,{\left (x^{3} + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 12 \, \sqrt{3} x\right )}}{216 \,{\left (x^{3} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/216*sqrt(3)*(3*sqrt(3)*(x^3 + 1)*log(x^2 + x + 1) + sqrt(3)*(x^3 + 1)*log(x^2
- x + 1) - 2*sqrt(3)*(x^3 + 1)*log(x + 1) - 6*sqrt(3)*(x^3 + 1)*log(x - 1) + 18*
(x^3 + 1)*arctan(1/3*sqrt(3)*(2*x + 1)) - 6*(x^3 + 1)*arctan(1/3*sqrt(3)*(2*x -
1)) - 12*sqrt(3)*x)/(x^3 + 1)

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Sympy [A]  time = 1.15848, size = 92, normalized size = 0.95 \[ - \frac{x}{6 x^{3} + 6} - \frac{\log{\left (x - 1 \right )}}{12} - \frac{\log{\left (x + 1 \right )}}{36} + \frac{\log{\left (x^{2} - x + 1 \right )}}{72} + \frac{\log{\left (x^{2} + x + 1 \right )}}{24} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{36} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(6*x**3 + 6) - log(x - 1)/12 - log(x + 1)/36 + log(x**2 - x + 1)/72 + log(x**
2 + x + 1)/24 - sqrt(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/36 + sqrt(3)*atan(2*sqrt
(3)*x/3 + sqrt(3)/3)/12

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GIAC/XCAS [A]  time = 0.264014, size = 104, normalized size = 1.07 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{36} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{x}{6 \,{\left (x^{3} + 1\right )}} + \frac{1}{24} \,{\rm ln}\left (x^{2} + x + 1\right ) + \frac{1}{72} \,{\rm ln}\left (x^{2} - x + 1\right ) - \frac{1}{36} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{12} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/36*sqrt(3)*arctan(1/3*sqrt(3)*(2*
x - 1)) - 1/6*x/(x^3 + 1) + 1/24*ln(x^2 + x + 1) + 1/72*ln(x^2 - x + 1) - 1/36*l
n(abs(x + 1)) - 1/12*ln(abs(x - 1))