Optimal. Leaf size=135 \[ \frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{2 \sqrt [3]{2}}-\frac{\log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{\sqrt [3]{2}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2}} \]
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Rubi [C] time = 0.300503, antiderivative size = 409, normalized size of antiderivative = 3.03, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1586, 6728, 2148} \[ -\frac{3 \left (-\sqrt{3}+i\right ) \log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x-i \sqrt{3}+1\right )}{4 \sqrt [3]{2} \left (\sqrt{3}+i\right )}-\frac{3 \left (\sqrt{3}+i\right ) \log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x+i \sqrt{3}+1\right )}{4 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}-\frac{\left (3-i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (2 x-i \sqrt{3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (\sqrt{3}+i\right )}+\frac{\left (3+i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (2 x+i \sqrt{3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}+\frac{\left (-\sqrt{3}+i\right ) \log \left (-\left (-2 x-i \sqrt{3}+1\right )^2 \left (2 x-i \sqrt{3}+1\right )\right )}{4 \sqrt [3]{2} \left (\sqrt{3}+i\right )}+\frac{\left (\sqrt{3}+i\right ) \log \left (-\left (-2 x+i \sqrt{3}+1\right )^2 \left (2 x+i \sqrt{3}+1\right )\right )}{4 \sqrt [3]{2} \left (-\sqrt{3}+i\right )} \]
Antiderivative was successfully verified.
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Rule 1586
Rule 6728
Rule 2148
Rubi steps
\begin{align*} \int \frac{(1+x)^2}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\int \frac{1+x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx\\ &=\int \left (\frac{1-i \sqrt{3}}{\left (-1-i \sqrt{3}+2 x\right ) \sqrt [3]{1-x^3}}+\frac{1+i \sqrt{3}}{\left (-1+i \sqrt{3}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=\left (1-i \sqrt{3}\right ) \int \frac{1}{\left (-1-i \sqrt{3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\left (1+i \sqrt{3}\right ) \int \frac{1}{\left (-1+i \sqrt{3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx\\ &=-\frac{\left (3-i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (1-i \sqrt{3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (i+\sqrt{3}\right )}+\frac{\left (3+i \sqrt{3}\right ) \tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (1+i \sqrt{3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right )}{2 \sqrt [3]{2} \left (i-\sqrt{3}\right )}+\frac{\left (i-\sqrt{3}\right ) \log \left (-\left (1-i \sqrt{3}-2 x\right )^2 \left (1-i \sqrt{3}+2 x\right )\right )}{4 \sqrt [3]{2} \left (i+\sqrt{3}\right )}+\frac{\left (i+\sqrt{3}\right ) \log \left (-\left (1+i \sqrt{3}-2 x\right )^2 \left (1+i \sqrt{3}+2 x\right )\right )}{4 \sqrt [3]{2} \left (i-\sqrt{3}\right )}-\frac{3 \left (i-\sqrt{3}\right ) \log \left (1-i \sqrt{3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i+\sqrt{3}\right )}-\frac{3 \left (i+\sqrt{3}\right ) \log \left (1+i \sqrt{3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i-\sqrt{3}\right )}\\ \end{align*}
Mathematica [C] time = 0.440747, size = 150, normalized size = 1.11 \[ \frac{1}{3} x^3 F_1\left (1;\frac{1}{3},1;2;x^3,-x^3\right )+x^2 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right )+\frac{-\log \left (\frac{2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt{3}}\right )}{6 \sqrt [3]{2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 1+x \right ) ^{2}}{{x}^{3}+1}{\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{{\left (x + 1\right )}^{2}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 99.4621, size = 882, normalized size = 6.53 \begin{align*} \frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{1}{6}}{\left (4 \cdot 2^{\frac{1}{6}} \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} - 4 \, x + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 4 \, \sqrt{2} \left (-1\right )^{\frac{1}{3}}{\left (x^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + x - 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2^{\frac{5}{6}}{\left (x^{6} - 7 \, x^{5} + 10 \, x^{4} - 7 \, x^{3} + 10 \, x^{2} - 7 \, x + 1\right )}\right )}}{6 \,{\left (3 \, x^{6} - 9 \, x^{5} + 6 \, x^{4} - x^{3} + 6 \, x^{2} - 9 \, x + 3\right )}}\right ) - \frac{1}{12} \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}{\left (x^{2} - 3 \, x + 1\right )} + 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - 3 \, x^{2} + 1\right )} + 4 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - x\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{2 \cdot 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )} + 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{2} - x + 1\right )} - 2 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2} - x + 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{x + 1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + 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{{\left (x + 1\right )}^{2}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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