Optimal. Leaf size=122 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]
[Out]
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Rubi [A] time = 0.155538, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[x/((1 - x^3)^(2/3)*(1 + x^3)),x]
[Out]
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Rubi in Sympy [A] time = 16.7135, size = 107, normalized size = 0.88 \[ - \frac{\sqrt [3]{2} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} + \frac{\sqrt [3]{2} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(-x**3+1)**(2/3)/(x**3+1),x)
[Out]
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Mathematica [C] time = 0.105468, size = 59, normalized size = 0.48 \[ \frac{x^2 \left (\frac{1-x^3}{x^3+1}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{2 x^3}{x^3+1}\right )}{2 \left (1-x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x/((1 - x^3)^(2/3)*(1 + x^3)),x]
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Maple [F] time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{x}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(-x^3+1)^(2/3)/(x^3+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.44773, size = 348, normalized size = 2.85 \[ -\frac{1}{216} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (\frac{3 \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \cdot 4^{\frac{1}{3}}{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2 \, \left (-1\right )^{\frac{1}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - 2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{6 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 3 \cdot 4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - 2 \, \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 6 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{3 \cdot 4^{\frac{1}{3}} \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 3 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )}}{3 \,{\left (3 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )}\right )}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(-x**3+1)**(2/3)/(x**3+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="giac")
[Out]