Optimal. Leaf size=17 \[ x+x \log \left (\log \left (\frac {3}{x}-x^2\right )\right ) \]
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Rubi [F] time = 0.74, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {3+2 x^3+\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )+\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right ) \log \left (\log \left (\frac {3-x^3}{x}\right )\right )}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3+2 x^3-3 \log \left (\frac {3-x^3}{x}\right )+x^3 \log \left (\frac {3-x^3}{x}\right )}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )}+\log \left (\log \left (\frac {3-x^3}{x}\right )\right )\right ) \, dx\\ &=\int \frac {3+2 x^3-3 \log \left (\frac {3-x^3}{x}\right )+x^3 \log \left (\frac {3-x^3}{x}\right )}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=\int \left (1+\frac {3+2 x^3}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )}\right ) \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+\int \frac {3+2 x^3}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+\int \left (\frac {2}{\log \left (\frac {3-x^3}{x}\right )}+\frac {9}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )}\right ) \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+2 \int \frac {1}{\log \left (\frac {3-x^3}{x}\right )} \, dx+9 \int \frac {1}{\left (-3+x^3\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+2 \int \frac {1}{\log \left (\frac {3-x^3}{x}\right )} \, dx+9 \int \left (-\frac {1}{3\ 3^{2/3} \left (\sqrt [3]{3}-x\right ) \log \left (\frac {3-x^3}{x}\right )}-\frac {1}{3\ 3^{2/3} \left (\sqrt [3]{3}+\sqrt [3]{-1} x\right ) \log \left (\frac {3-x^3}{x}\right )}-\frac {1}{3\ 3^{2/3} \left (\sqrt [3]{3}-(-1)^{2/3} x\right ) \log \left (\frac {3-x^3}{x}\right )}\right ) \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ &=x+2 \int \frac {1}{\log \left (\frac {3-x^3}{x}\right )} \, dx-\sqrt [3]{3} \int \frac {1}{\left (\sqrt [3]{3}-x\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx-\sqrt [3]{3} \int \frac {1}{\left (\sqrt [3]{3}+\sqrt [3]{-1} x\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx-\sqrt [3]{3} \int \frac {1}{\left (\sqrt [3]{3}-(-1)^{2/3} x\right ) \log \left (\frac {3-x^3}{x}\right )} \, dx+\int \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.21, size = 17, normalized size = 1.00 \begin {gather*} x+x \log \left (\log \left (\frac {3-x^3}{x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 16, normalized size = 0.94 \begin {gather*} x \log \left (\log \left (-\frac {x^{3} - 3}{x}\right )\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 16, normalized size = 0.94 \begin {gather*} x \log \left (\log \left (-\frac {x^{3} - 3}{x}\right )\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-3\right ) \ln \left (\frac {-x^{3}+3}{x}\right ) \ln \left (\ln \left (\frac {-x^{3}+3}{x}\right )\right )+\left (x^{3}-3\right ) \ln \left (\frac {-x^{3}+3}{x}\right )+2 x^{3}+3}{\left (x^{3}-3\right ) \ln \left (\frac {-x^{3}+3}{x}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 18, normalized size = 1.06 \begin {gather*} x \log \left (\log \left (-x^{3} + 3\right ) - \log \relax (x)\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 16, normalized size = 0.94 \begin {gather*} x\,\left (\ln \left (\ln \left (-\frac {x^3-3}{x}\right )\right )+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 12, normalized size = 0.71 \begin {gather*} x \log {\left (\log {\left (\frac {3 - x^{3}}{x} \right )} \right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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