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 = 0.30, 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 2148
Rule 6728
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*}
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Mathematica [C] time = 0.31, size = 150, normalized size = 1.11 \[ \frac {1}{3} x^3 F_1\left (1;\frac {1}{3},1;2;x^3,-x^3\right )+x^2 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^3,-x^3\right )+\frac {2 \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt {3}}\right )-\log \left (-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 12.88, size = 318, normalized size = 2.36 \[ \frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (4 \cdot 2^{\frac {1}{6}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} - 4 \, x + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + x - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2^{\frac {5}{6}} {\left (x^{6} - 7 \, x^{5} + 10 \, x^{4} - 7 \, x^{3} + 10 \, x^{2} - 7 \, x + 1\right )}\right )}}{6 \, {\left (3 \, x^{6} - 9 \, x^{5} + 6 \, x^{4} - x^{3} + 6 \, x^{2} - 9 \, x + 3\right )}}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - 3 \, x + 1\right )} + 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 3 \, x^{2} + 1\right )} + 4 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {2 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} - x + 1\right )} - 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} - x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x + 1\right )}^{2}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 7.50, size = 676, normalized size = 5.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x + 1\right )}^{2}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (x+1\right )}^2}{{\left (1-x^3\right )}^{1/3}\,\left (x^3+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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