Optimal. Leaf size=98 \[ -\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.163835, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[x^5/((1 - x^3)^(1/3)*(1 + x^3)),x]
[Out]
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Rubi in Sympy [A] time = 9.84204, size = 82, normalized size = 0.84 \[ - \frac{\left (- x^{3} + 1\right )^{\frac{2}{3}}}{2} + \frac{2^{\frac{2}{3}} \log{\left (x^{3} + 1 \right )}}{12} - \frac{2^{\frac{2}{3}} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(-x**3+1)**(1/3)/(x**3+1),x)
[Out]
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Mathematica [C] time = 0.0384303, size = 58, normalized size = 0.59 \[ \frac{2 \sqrt [3]{\frac{x^3-1}{x^3+1}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{2}{x^3+1}\right )+x^3-1}{2 \sqrt [3]{1-x^3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/((1 - x^3)^(1/3)*(1 + x^3)),x]
[Out]
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Maple [F] time = 0.029, size = 0, normalized size = 0. \[ \int{\frac{{x}^{5}}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(-x^3+1)^(1/3)/(x^3+1),x)
[Out]
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Maxima [A] time = 1.5219, size = 131, normalized size = 1.34 \[ -\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) - \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) - \frac{1}{2} \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.214897, size = 178, normalized size = 1.82 \[ -\frac{1}{36} \, \sqrt{3} 2^{\frac{2}{3}}{\left (\sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 2 \, \left (-1\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2 \, \left (-1\right )^{\frac{2}{3}}\right ) + 3 \, \sqrt{3} 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 6 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{3} \, \left (-1\right )^{\frac{1}{3}}{\left (\sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \sqrt{3} \left (-1\right )^{\frac{2}{3}}\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(-x**3+1)**(1/3)/(x**3+1),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="giac")
[Out]