Optimal. Leaf size=105 \[ -\frac{\left (1-x^3\right )^{2/3}}{2 x^2}+\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
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Rubi [A] time = 0.0715571, antiderivative size = 139, normalized size of antiderivative = 1.32, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {494, 453, 200, 31, 634, 617, 204, 628} \[ -\frac{\left (1-x^3\right )^{2/3}}{2 x^2}+\frac{\log \left (\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 494
Rule 453
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1+x^3}{x^3 \left (1+2 x^3\right )} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac{\left (1-x^3\right )^{2/3}}{2 x^2}-\operatorname{Subst}\left (\int \frac{1}{1+2 x^3} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac{\left (1-x^3\right )^{2/3}}{2 x^2}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{2} x} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{2-\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac{\left (1-x^3\right )^{2/3}}{2 x^2}-\frac{\log \left (1+\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )+\frac{\operatorname{Subst}\left (\int \frac{-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}\\ &=-\frac{\left (1-x^3\right )^{2/3}}{2 x^2}+\frac{\log \left (1+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}-\frac{\log \left (1+\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2}}\\ &=-\frac{\left (1-x^3\right )^{2/3}}{2 x^2}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{\log \left (1+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}-\frac{\log \left (1+\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}\\ \end{align*}
Mathematica [C] time = 0.360042, size = 82, normalized size = 0.78 \[ -\frac{\left (-6 x^6+4 x^3+2\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right )-3 x^3 \left (x^3+1\right ) \, _2F_1\left (\frac{4}{3},2;\frac{7}{3};\frac{2 x^3}{x^3-1}\right )}{4 x^2 \left (1-x^3\right )^{4/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ({x}^{3}+1 \right ) }{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, 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{1}{3}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 22.0639, size = 841, normalized size = 8.01 \begin{align*} -\frac{2 \, \sqrt{6} 2^{\frac{1}{6}} \left (-1\right )^{\frac{1}{3}} x^{2} \arctan \left (\frac{2^{\frac{1}{6}}{\left (6 \, \sqrt{6} 2^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (5 \, x^{7} + 4 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \, \sqrt{6} \left (-1\right )^{\frac{1}{3}}{\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{6} 2^{\frac{1}{3}}{\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \,{\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 2 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} x^{2} \log \left (\frac{6 \cdot 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} + 1\right )} + 6 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x}{x^{3} + 1}\right ) + 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} x^{2} \log \left (-\frac{3 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} + 12 \,{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 18 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{36 \, x^{2}} \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^{3} \sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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