Optimal. Leaf size=24 \[ 2+\frac {x}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \]
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Rubi [F] time = 1.73, 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 {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {x^2 \left (-e+9 x^3+x^4\right )}{e}\right ) \log ^2\left (\log \left (\frac {x^2 \left (-e+9 x^3+x^4\right )}{e}\right )\right )} \, dx\\ &=\int \left (\frac {-2 e+45 x^3+6 x^4}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}+\frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}\right ) \, dx\\ &=\int \frac {-2 e+45 x^3+6 x^4}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\\ &=\int \left (-\frac {6}{\log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}+\frac {4 e-9 x^3}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}\right ) \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\\ &=-\left (6 \int \frac {1}{\log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\right )+\int \frac {4 e-9 x^3}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\\ &=-\left (6 \int \frac {1}{\log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\right )+\int \left (\frac {4 e}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}-\frac {9 x^3}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )}\right ) \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\\ &=-\left (6 \int \frac {1}{\log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\right )-9 \int \frac {x^3}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx+(4 e) \int \frac {1}{\left (e-9 x^3-x^4\right ) \log \left (-x^2+\frac {x^5 (9+x)}{e}\right ) \log ^2\left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx+\int \frac {1}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 22, normalized size = 0.92 \begin {gather*} \frac {x}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 25, normalized size = 1.04 \begin {gather*} \frac {x}{\log \left (\log \left ({\left (x^{6} + 9 \, x^{5} - x^{2} e\right )} e^{\left (-1\right )}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.41, size = 24, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (\log \left (x^{6} + 9 \, x^{5} - x^{2} e\right ) - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left ({\mathrm e}-x^{4}-9 x^{3}\right ) \ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right ) \ln \left (\ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right )\right )-2 \,{\mathrm e}+6 x^{4}+45 x^{3}}{\left ({\mathrm e}-x^{4}-9 x^{3}\right ) \ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right ) \ln \left (\ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 25, normalized size = 1.04 \begin {gather*} \frac {x}{\log \left (\log \left (x^{4} + 9 \, x^{3} - e\right ) + 2 \, \log \relax (x) - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 170, normalized size = 7.08 \begin {gather*} \frac {\ln \left (x^4+9\,x^3-\mathrm {e}\right )}{4}+\frac {\ln \relax (x)}{2}+\frac {x}{\ln \left (\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )\right )}-\frac {45\,x^3\,\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )}{4\,\left (6\,x^4+45\,x^3-2\,\mathrm {e}\right )}-\frac {3\,x^4\,\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )}{2\,\left (6\,x^4+45\,x^3-2\,\mathrm {e}\right )}+\frac {\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )\,\mathrm {e}}{2\,\left (6\,x^4+45\,x^3-2\,\mathrm {e}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 22, normalized size = 0.92 \begin {gather*} \frac {x}{\log {\left (\log {\left (\frac {x^{6} + 9 x^{5} - e x^{2}}{e} \right )} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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