Optimal. Leaf size=26 \[ -\frac{F_1\left (-\frac{2}{3};\frac{2}{3},1;\frac{1}{3};x^3,-x^3\right )}{2 x^2} \]
[Out]
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Rubi [A] time = 0.0630453, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{F_1\left (-\frac{2}{3};\frac{2}{3},1;\frac{1}{3};x^3,-x^3\right )}{2 x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(1 - x^3)^(2/3)*(1 + x^3)),x]
[Out]
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Rubi in Sympy [A] time = 6.27822, size = 20, normalized size = 0.77 \[ - \frac{\operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{2}{3},1,\frac{1}{3},x^{3},- x^{3} \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(-x**3+1)**(2/3)/(x**3+1),x)
[Out]
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Mathematica [B] time = 0.181723, size = 120, normalized size = 4.62 \[ \frac{\sqrt [3]{1-x^3} \left (\frac{4 x^3 F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};x^3,-x^3\right )+F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )\right )}-1\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^3*(1 - x^3)^(2/3)*(1 + x^3)),x]
[Out]
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Maple [F] time = 0.08, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ({x}^{3}+1 \right ) } \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(-x^3+1)^(2/3)/(x^3+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.81875, size = 474, normalized size = 18.23 \[ \frac{4^{\frac{2}{3}} \sqrt{3}{\left (2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{2} \log \left (-\frac{3 \cdot 4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} + 3 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \left (-1\right )^{\frac{2}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{2} \log \left (\frac{6 \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (x^{8} - 4 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 3 \cdot 4^{\frac{1}{3}}{\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \left (-1\right )^{\frac{1}{3}}{\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) + 6 \, \left (-1\right )^{\frac{1}{3}} x^{2} \arctan \left (\frac{6 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} - 3 \cdot 4^{\frac{1}{3}} \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}}{3 \,{\left (3 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \left (-1\right )^{\frac{2}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}\right )}}\right ) - 18 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}}{432 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(-x**3+1)**(2/3)/(x**3+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^3),x, algorithm="giac")
[Out]