Optimal. Leaf size=97 \[ \frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]
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Rubi [A] time = 0.0438348, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2148} \[ \frac{3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\log \left ((1-x) (x+1)^2\right )}{4 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
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Rule 2148
Rubi steps
\begin{align*} \int \frac{1}{(1+x) \sqrt [3]{1-x^3}} \, dx &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\log \left ((1-x) (1+x)^2\right )}{4 \sqrt [3]{2}}+\frac{3 \log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ \end{align*}
Mathematica [F] time = 0.0762007, size = 0, normalized size = 0. \[ \int \frac{1}{(1+x) \sqrt [3]{1-x^3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{1+x}{\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 )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 37.3663, size = 788, normalized size = 8.12 \begin{align*} \frac{1}{12} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{1}{6}}{\left (2^{\frac{5}{6}}{\left (13 \, x^{6} + 2 \, x^{5} + 19 \, x^{4} - 4 \, x^{3} + 19 \, x^{2} + 2 \, x + 13\right )} - 4 \, \sqrt{2}{\left (5 \, x^{5} - 5 \, x^{4} + 6 \, x^{3} - 6 \, x^{2} + 5 \, x - 5\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 16 \cdot 2^{\frac{1}{6}}{\left (x^{4} + 2 \, x^{3} + 2 \, x^{2} + 2 \, x + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right )}}{6 \,{\left (3 \, x^{6} - 18 \, x^{5} - 3 \, x^{4} - 28 \, x^{3} - 3 \, x^{2} - 18 \, x + 3\right )}}\right ) - \frac{1}{24} \cdot 2^{\frac{2}{3}} \log \left (\frac{4 \cdot 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}{\left (x^{2} + 1\right )} + 2^{\frac{1}{3}}{\left (5 \, x^{4} + 6 \, x^{2} + 5\right )} - 2 \,{\left (3 \, x^{3} - x^{2} + x - 3\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right ) + \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (\frac{2^{\frac{2}{3}}{\left (x^{2} + 2 \, x + 1\right )} - 2 \cdot 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )} - 4 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2} + 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{1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (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 )}^{\frac{1}{3}}{\left (x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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