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