Optimal. Leaf size=84 \[ 3 \sqrt [3]{1-x}+\frac{3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac{\log (x+1)}{2^{2/3}}-\sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x}+1}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.0908086, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ 3 \sqrt [3]{1-x}+\frac{3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac{\log (x+1)}{2^{2/3}}-\sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x}+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 - x)^(1/3)/(1 + x),x]
[Out]
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Rubi in Sympy [A] time = 5.43206, size = 75, normalized size = 0.89 \[ 3 \sqrt [3]{- x + 1} - \frac{\sqrt [3]{2} \log{\left (x + 1 \right )}}{2} + \frac{3 \sqrt [3]{2} \log{\left (- \sqrt [3]{- x + 1} + \sqrt [3]{2} \right )}}{2} - \sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x + 1}}{3} + \frac{1}{3}\right ) \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)**(1/3)/(1+x),x)
[Out]
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Mathematica [A] time = 0.0590676, size = 113, normalized size = 1.35 \[ 3 \sqrt [3]{1-x}+\sqrt [3]{2} \log \left (2-2^{2/3} \sqrt [3]{1-x}\right )-\frac{\log \left (\sqrt [3]{2} (1-x)^{2/3}+2^{2/3} \sqrt [3]{1-x}+2\right )}{2^{2/3}}-\sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x}+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x)^(1/3)/(1 + x),x]
[Out]
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Maple [A] time = 0.01, size = 84, normalized size = 1. \[ 3\,\sqrt [3]{1-x}+\sqrt [3]{2}\ln \left ( \sqrt [3]{1-x}-\sqrt [3]{2} \right ) -{\frac{\sqrt [3]{2}}{2}\ln \left ( \left ( 1-x \right ) ^{{\frac{2}{3}}}+\sqrt [3]{1-x}\sqrt [3]{2}+{2}^{{\frac{2}{3}}} \right ) }-\sqrt [3]{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+{2}^{{\frac{2}{3}}}\sqrt [3]{1-x} \right ) } \right ) \sqrt{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)^(1/3)/(1+x),x)
[Out]
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Maxima [A] time = 1.5109, size = 116, normalized size = 1.38 \[ -\sqrt{3} 2^{\frac{1}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{2} \cdot 2^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}} +{\left (-x + 1\right )}^{\frac{2}{3}}\right ) + 2^{\frac{1}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x + 1\right )}^{\frac{1}{3}}\right ) + 3 \,{\left (-x + 1\right )}^{\frac{1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^(1/3)/(x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233858, size = 116, normalized size = 1.38 \[ -\sqrt{3} 2^{\frac{1}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{2} \cdot 2^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x + 1\right )}^{\frac{1}{3}} +{\left (-x + 1\right )}^{\frac{2}{3}}\right ) + 2^{\frac{1}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x + 1\right )}^{\frac{1}{3}}\right ) + 3 \,{\left (-x + 1\right )}^{\frac{1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^(1/3)/(x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.05226, size = 172, normalized size = 2.05 \[ \frac{4 \sqrt [3]{-1} \sqrt [3]{x - 1} \Gamma \left (\frac{4}{3}\right )}{\Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{-2} e^{\frac{5 i \pi }{3}} \log{\left (- \frac{2^{\frac{2}{3}} \sqrt [3]{x - 1} e^{\frac{i \pi }{3}}}{2} + 1 \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} - \frac{4 \sqrt [3]{-2} \log{\left (- \frac{2^{\frac{2}{3}} \sqrt [3]{x - 1} e^{i \pi }}{2} + 1 \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{-2} e^{\frac{i \pi }{3}} \log{\left (- \frac{2^{\frac{2}{3}} \sqrt [3]{x - 1} e^{\frac{5 i \pi }{3}}}{2} + 1 \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x)**(1/3)/(1+x),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 + 1)^(1/3)/(x + 1),x, algorithm="giac")
[Out]