Optimal. Leaf size=175 \[ -\frac{1}{3 x^3 \sqrt [3]{1-x^3}}+\frac{5}{6 \sqrt [3]{1-x^3}}-\frac{\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac{1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}-\frac{\log (x)}{6} \]
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Rubi [A] time = 0.115659, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {446, 103, 156, 51, 55, 618, 204, 31, 617} \[ -\frac{1}{3 x^3 \sqrt [3]{1-x^3}}+\frac{5}{6 \sqrt [3]{1-x^3}}-\frac{\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac{1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}-\frac{\log (x)}{6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 103
Rule 156
Rule 51
Rule 55
Rule 618
Rule 204
Rule 31
Rule 617
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{4/3} x^2 (1+x)} \, dx,x,x^3\right )\\ &=-\frac{1}{3 x^3 \sqrt [3]{1-x^3}}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{-\frac{1}{3}-\frac{4 x}{3}}{(1-x)^{4/3} x (1+x)} \, dx,x,x^3\right )\\ &=-\frac{1}{3 x^3 \sqrt [3]{1-x^3}}+\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{4/3} x} \, dx,x,x^3\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(1-x)^{4/3} (1+x)} \, dx,x,x^3\right )\\ &=\frac{5}{6 \sqrt [3]{1-x^3}}-\frac{1}{3 x^3 \sqrt [3]{1-x^3}}+\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} x} \, dx,x,x^3\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac{5}{6 \sqrt [3]{1-x^3}}-\frac{1}{3 x^3 \sqrt [3]{1-x^3}}-\frac{\log (x)}{6}-\frac{\log \left (1+x^3\right )}{12 \sqrt [3]{2}}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{2^{2/3}+\sqrt [3]{2} 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 )}{4 \sqrt [3]{2}}\\ &=\frac{5}{6 \sqrt [3]{1-x^3}}-\frac{1}{3 x^3 \sqrt [3]{1-x^3}}-\frac{\log (x)}{6}-\frac{\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac{1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^3}\right )-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=\frac{5}{6 \sqrt [3]{1-x^3}}-\frac{1}{3 x^3 \sqrt [3]{1-x^3}}+\frac{\tan ^{-1}\left (\frac{1+2 \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}-\frac{\log (x)}{6}-\frac{\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac{1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ \end{align*}
Mathematica [C] time = 0.0209776, size = 64, normalized size = 0.37 \[ \frac{3 x^3 \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{1}{2} \left (1-x^3\right )\right )+2 x^3 \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};1-x^3\right )-2}{6 x^3 \sqrt [3]{1-x^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({x}^{3}+1 \right ){x}^{4}} \left ( -{x}^{3}+1 \right ) ^{-{\frac{4}{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{4}{3}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83717, size = 633, normalized size = 3.62 \begin{align*} \frac{6 \, \sqrt{6} 2^{\frac{1}{6}}{\left (x^{6} - x^{3}\right )} \arctan \left (\frac{1}{6} \cdot 2^{\frac{1}{6}}{\left (\sqrt{6} 2^{\frac{1}{3}} + 2 \, \sqrt{6}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) - 3 \cdot 2^{\frac{2}{3}}{\left (x^{6} - x^{3}\right )} \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 ) + 6 \cdot 2^{\frac{2}{3}}{\left (x^{6} - x^{3}\right )} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) + 8 \, \sqrt{3}{\left (x^{6} - x^{3}\right )} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 4 \,{\left (x^{6} - x^{3}\right )} \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + 8 \,{\left (x^{6} - x^{3}\right )} \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) - 12 \,{\left (5 \, x^{3} - 2\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{72 \,{\left (x^{6} - x^{3}\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^{4} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{4}{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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