Optimal. Leaf size=137 \[ \frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2} \]
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Rubi [A] time = 0.0896032, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {446, 86, 57, 618, 204, 31, 617} \[ \frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 86
Rule 57
Rule 618
Rule 204
Rule 31
Rule 617
Rubi steps
\begin{align*} \int \frac{1}{x \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{2/3} x (1+x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{2/3} x} \, dx,x,x^3\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{2/3} (1+x)} \, dx,x,x^3\right )\\ &=-\frac{\log (x)}{2}+\frac{\log \left (1+x^3\right )}{6\ 2^{2/3}}-\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{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=-\frac{\log (x)}{2}+\frac{\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{2^{2/3}}+\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{\tan ^{-1}\left (\frac{1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2}+\frac{\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.03803, size = 179, normalized size = 1.31 \[ \frac{1}{12} \left (4 \log \left (1-\sqrt [3]{1-x^3}\right )-2 \sqrt [3]{2} \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )+\sqrt [3]{2} \log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{2-2 x^3}+2^{2/3}\right )-2 \log \left (\left (1-x^3\right )^{2/3}+\sqrt [3]{1-x^3}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )+2 \sqrt [3]{2} \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({x}^{3}+1 \right ) } \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5104, size = 605, normalized size = 4.42 \begin{align*} -\frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{1}{6} \cdot 4^{\frac{1}{6}}{\left (4^{\frac{2}{3}} \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}} \sqrt{3}\right )}\right ) - \frac{1}{24} \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{12} \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) - \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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