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

Optimal. Leaf size=122 \[ \frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

[Out]

-(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)) + Lo
g[1 + (2^(1/3)*x)/(1 - x^3)^(1/3)]/(3*2^(1/3))

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Rubi [A]  time = 0.133205, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

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

[Out]

-(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)) + Lo
g[1 + (2^(1/3)*x)/(1 - x^3)^(1/3)]/(3*2^(1/3))

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Rubi in Sympy [A]  time = 16.6994, size = 107, normalized size = 0.88 \[ \frac{2^{\frac{2}{3}} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} - \frac{2^{\frac{2}{3}} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-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)*ata
n(sqrt(3)*(-2*2**(1/3)*x/(3*(-x**3 + 1)**(1/3)) + 1/3))/6

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Mathematica [A]  time = 0.120617, size = 104, normalized size = 0.85 \[ \frac{2 \log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt{3}}\right )-\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac{2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(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)])/(6*2^(1/3))

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^3+1)^(1/3)/(x^3+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.86098, size = 302, normalized size = 2.48 \[ \frac{1}{108} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2 \, \sqrt{3} \log \left (\frac{3 \cdot 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x + 6 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 2^{\frac{1}{3}}{\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) - \sqrt{3} \log \left (\frac{2^{\frac{2}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \cdot 2^{\frac{1}{3}}{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 6 \,{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 6 \, \arctan \left (\frac{6 \, \sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - 6 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - \sqrt{3} 2^{\frac{1}{3}}{\left (x^{3} + 1\right )}}{3 \,{\left (6 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 2^{\frac{1}{3}}{\left (x^{3} + 1\right )}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((-(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. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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