Optimal. Leaf size=95 \[ \frac{\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac{3 \log \left ((1-x)^{2/3}+\sqrt [3]{2} \sqrt [3]{x+1}\right )}{2\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} (1-x)^{2/3}}{\sqrt{3} \sqrt [3]{x+1}}\right )}{2^{2/3}} \]
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Rubi [A] time = 0.0171041, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1008} \[ \frac{\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac{3 \log \left ((1-x)^{2/3}+\sqrt [3]{2} \sqrt [3]{x+1}\right )}{2\ 2^{2/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} (1-x)^{2/3}}{\sqrt{3} \sqrt [3]{x+1}}\right )}{2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 1008
Rubi steps
\begin{align*} \int \frac{3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} (1-x)^{2/3}}{\sqrt{3} \sqrt [3]{1+x}}\right )}{2^{2/3}}+\frac{\log \left (3+x^2\right )}{2\ 2^{2/3}}-\frac{3 \log \left ((1-x)^{2/3}+\sqrt [3]{2} \sqrt [3]{1+x}\right )}{2\ 2^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0978211, size = 143, normalized size = 1.51 \[ \frac{1}{6} x^2 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )-\frac{27 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{\sqrt [3]{1-x^2} \left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{\frac{3+x}{{x}^{2}+3}{\frac{1}{\sqrt [3]{-{x}^{2}+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{x + 3}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 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 = 55.3616, size = 911, normalized size = 9.59 \begin{align*} -\frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3} \left (-1\right )^{\frac{1}{3}} \arctan \left (\frac{4^{\frac{1}{6}} \sqrt{3}{\left (12 \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - 3 \, x^{3} + 3 \, x^{2} - 9 \, x\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 12 \, \left (-1\right )^{\frac{1}{3}}{\left (x^{5} - 19 \, x^{4} + 42 \, x^{3} - 6 \, x^{2} - 27 \, x + 9\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (x^{6} + 18 \, x^{5} - 117 \, x^{4} + 36 \, x^{3} + 207 \, x^{2} - 54 \, x - 27\right )}\right )}}{6 \,{\left (x^{6} - 54 \, x^{5} + 171 \, x^{4} - 108 \, x^{3} - 81 \, x^{2} + 162 \, x - 27\right )}}\right ) - \frac{1}{24} \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{6 \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{2} - 3 \, x\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 4^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (x^{4} - 18 \, x^{3} + 24 \, x^{2} + 18 \, x - 9\right )} - 6 \,{\left (x^{3} - 7 \, x^{2} + 3 \, x + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}{x^{4} + 6 \, x^{2} + 9}\right ) + \frac{1}{12} \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{6 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}{\left (x - 1\right )} + 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{2} + 3\right )} - 12 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{x^{2} + 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{x + 3}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\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{x + 3}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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