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

Optimal. Leaf size=137 \[ -\frac{\sqrt [3]{1-x^3}}{x}+\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.193234, antiderivative size = 137, 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 \[ -\frac{\sqrt [3]{1-x^3}}{x}+\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.775, size = 117, normalized size = 0.85 \[ \frac{\sqrt [3]{2} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} - \frac{\sqrt [3]{2} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} - \frac{\sqrt [3]{- x^{3} + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/3)*log(2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/6 - 2**(1/3)*log(2**(2/3)*x**2/(
-x**3 + 1)**(2/3) - 2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/12 + 2**(1/3)*sqrt(3)*ata
n(sqrt(3)*(-2*2**(1/3)*x/(3*(-x**3 + 1)**(1/3)) + 1/3))/6 - (-x**3 + 1)**(1/3)/x

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Mathematica [C]  time = 0.834052, size = 154, normalized size = 1.12 \[ \frac{5 \left (-3 x^6+x^3+2\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right )-12 x^3 \left (x^3+1\right ) \, _2F_1\left (\frac{5}{3},2;\frac{8}{3};\frac{2 x^3}{x^3-1}\right )}{2 x \left (1-x^3\right )^{2/3} \left (15 \left (x^6-1\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right )+18 \left (x^6+x^3\right ) \, _2F_1\left (\frac{5}{3},2;\frac{8}{3};\frac{2 x^3}{x^3-1}\right )+5 \left (3 x^6-5 x^3+2\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5*(2 + x^3 - 3*x^6)*Hypergeometric2F1[2/3, 1, 5/3, (2*x^3)/(-1 + x^3)] - 12*x^3
*(1 + x^3)*Hypergeometric2F1[5/3, 2, 8/3, (2*x^3)/(-1 + x^3)])/(2*x*(1 - x^3)^(2
/3)*(5*(2 - 5*x^3 + 3*x^6) + 15*(-1 + x^6)*Hypergeometric2F1[2/3, 1, 5/3, (2*x^3
)/(-1 + x^3)] + 18*(x^3 + x^6)*Hypergeometric2F1[5/3, 2, 8/3, (2*x^3)/(-1 + x^3)
]))

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ({x}^{3}+1 \right ) } \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x^2/(-x^3+1)^(2/3)/(x^3+1),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^2), x)

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Fricas [A]  time = 1.45293, size = 324, normalized size = 2.36 \[ -\frac{4^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3} x \log \left (\frac{38 \, x^{6} - 32 \, x^{3} + 3 \cdot 4^{\frac{2}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \cdot 4^{\frac{1}{3}}{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2}{x^{6} + 2 \, x^{3} + 1}\right ) - 2 \, \sqrt{3} x \log \left (\frac{2 \, x^{3} + 6 \cdot 4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 3 \cdot 4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x + 2}{x^{3} + 1}\right ) - 6 \, x \arctan \left (\frac{3 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 3 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x + \sqrt{3}{\left (x^{3} + 1\right )}}{3 \,{\left (x^{3} - 3 \cdot 4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 1\right )}}\right ) + 18 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}}{216 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/216*4^(2/3)*sqrt(3)*(sqrt(3)*x*log((38*x^6 - 32*x^3 + 3*4^(2/3)*(5*x^4 - x)*(
-x^3 + 1)^(2/3) - 12*4^(1/3)*(2*x^5 - x^2)*(-x^3 + 1)^(1/3) + 2)/(x^6 + 2*x^3 +
1)) - 2*sqrt(3)*x*log((2*x^3 + 6*4^(1/3)*(-x^3 + 1)^(1/3)*x^2 + 3*4^(2/3)*(-x^3
+ 1)^(2/3)*x + 2)/(x^3 + 1)) - 6*x*arctan(1/3*(3*4^(1/3)*sqrt(3)*(-x^3 + 1)^(1/3
)*x^2 - 3*4^(2/3)*sqrt(3)*(-x^3 + 1)^(2/3)*x + sqrt(3)*(x^3 + 1))/(x^3 - 3*4^(1/
3)*(-x^3 + 1)^(1/3)*x^2 + 1)) + 18*4^(1/3)*sqrt(3)*(-x^3 + 1)^(1/3))/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^2), x)