Optimal. Leaf size=55 \[ \frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]
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Rubi [A] time = 0.0332979, antiderivative size = 55, 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, Rules used = {266, 55, 618, 204, 31} \[ \frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 55
Rule 618
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt [3]{1-x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x} \, dx,x,x^3\right )\\ &=-\frac{\log (x)}{2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )\\ &=-\frac{\log (x)}{2}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^3}\right )\\ &=\frac{\tan ^{-1}\left (\frac{1+2 \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )\\ \end{align*}
Mathematica [A] time = 0.012324, size = 55, normalized size = 1. \[ \frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.037, size = 65, normalized size = 1.2 \begin{align*}{\frac{\sqrt{3}\Gamma \left ({\frac{2}{3}} \right ) }{6\,\pi } \left ({\frac{2\,\pi \,\sqrt{3}}{3\,\Gamma \left ( 2/3 \right ) } \left ( -{\frac{\pi \,\sqrt{3}}{6}}-{\frac{3\,\ln \left ( 3 \right ) }{2}}+3\,\ln \left ( x \right ) +i\pi \right ) }+{\frac{2\,\pi \,\sqrt{3}{x}^{3}}{9\,\Gamma \left ( 2/3 \right ) }{\mbox{$_3$F$_2$}(1,1,{\frac{4}{3}};\,2,2;\,{x}^{3})}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43444, size = 84, normalized size = 1.53 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02553, size = 198, normalized size = 3.6 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - \frac{1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.914684, size = 32, normalized size = 0.58 \begin{align*} - \frac{e^{- \frac{i \pi }{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{1}{x^{3}}} \right )}}{3 x \Gamma \left (\frac{4}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08466, size = 85, normalized size = 1.55 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \, \log \left ({\left |{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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