3.577 \(\int \frac{1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=139 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{\left (1-x^3\right )^{2/3}}{2 x^2}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

[Out]

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

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Rubi [A]  time = 0.185035, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{\left (1-x^3\right )^{2/3}}{2 x^2}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.9584, size = 121, normalized size = 0.87 \[ - \frac{2^{\frac{2}{3}} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} + \frac{2^{\frac{2}{3}} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} - \frac{\left (- x^{3} + 1\right )^{\frac{2}{3}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(2/3)*log(2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/6 + 2**(2/3)*log(2**(2/3)*x**2/
(-x**3 + 1)**(2/3) - 2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/12 + 2**(2/3)*sqrt(3)*at
an(sqrt(3)*(-2*2**(1/3)*x/(3*(-x**3 + 1)**(1/3)) + 1/3))/6 - (-x**3 + 1)**(2/3)/
(2*x**2)

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Mathematica [A]  time = 0.154516, size = 120, normalized size = 0.86 \[ \frac{1}{12} \left (2^{2/3} \left (-2 \log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt{3}}\right )+\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac{2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )\right )-\frac{6 \left (1-x^3\right )^{2/3}}{x^2}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ({x}^{3}+1 \right ) }{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-x^3+1)^(1/3)/(x^3+1),x)

[Out]

int(1/x^3/(-x^3+1)^(1/3)/(x^3+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^3), x)

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Fricas [A]  time = 1.84154, size = 386, normalized size = 2.78 \[ \frac{\sqrt{3} 2^{\frac{2}{3}}{\left (2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{2} \log \left (\frac{6 \, \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 3 \cdot 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - 2^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) - \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{2} \log \left (\frac{2^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 6 \, \left (-1\right )^{\frac{1}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \cdot 2^{\frac{1}{3}}{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 6 \, \left (-1\right )^{\frac{1}{3}} x^{2} \arctan \left (-\frac{6 \, \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 6 \, \sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - \sqrt{3} 2^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} + 1\right )}}{3 \,{\left (6 \, \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 2^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} + 1\right )}\right )}}\right ) - 9 \, \sqrt{3} 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right )}}{108 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/108*sqrt(3)*2^(2/3)*(2*sqrt(3)*(-1)^(1/3)*x^2*log((6*(-1)^(2/3)*(-x^3 + 1)^(1/
3)*x^2 + 3*2^(2/3)*(-x^3 + 1)^(2/3)*x - 2^(1/3)*(-1)^(1/3)*(x^3 + 1))/(x^3 + 1))
- sqrt(3)*(-1)^(1/3)*x^2*log((2^(2/3)*(-1)^(2/3)*(19*x^6 - 16*x^3 + 1) - 6*(-1)
^(1/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) - 12*2^(1/3)*(2*x^5 - x^2)*(-x^3 + 1)^(1/3))
/(x^6 + 2*x^3 + 1)) - 6*(-1)^(1/3)*x^2*arctan(-1/3*(6*sqrt(3)*(-1)^(2/3)*(-x^3 +
1)^(1/3)*x^2 - 6*sqrt(3)*2^(2/3)*(-x^3 + 1)^(2/3)*x - sqrt(3)*2^(1/3)*(-1)^(1/3
)*(x^3 + 1))/(6*(-1)^(2/3)*(-x^3 + 1)^(1/3)*x^2 + 2^(1/3)*(-1)^(1/3)*(x^3 + 1)))
- 9*sqrt(3)*2^(1/3)*(-x^3 + 1)^(2/3))/x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*(-(x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^3), x)