Optimal. Leaf size=26 \[ e^{\frac {4}{3 e^{2/3}}-2 x+\frac {-2+\log \left (\log \left (x^2\right )\right )}{x}} \]
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Rubi [F] time = 2.46, 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 {\exp \left (\frac {4 x+e^{2/3} \left (-6-6 x^2\right )+3 e^{2/3} \log \left (\log \left (x^2\right )\right )}{3 e^{2/3} x}\right ) \left (2+\left (2-2 x^2\right ) \log \left (x^2\right )-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )\right )}{x^2 \log \left (x^2\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 {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \log ^{-1+\frac {1}{x}}\left (x^2\right ) \left (2-\log \left (x^2\right ) \left (-2+2 x^2+\log \left (\log \left (x^2\right )\right )\right )\right )}{x^2} \, dx\\ &=\int \left (-\frac {2 e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \log ^{-1+\frac {1}{x}}\left (x^2\right ) \left (-1-\log \left (x^2\right )+x^2 \log \left (x^2\right )\right )}{x^2}-\frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \log \left (\log \left (x^2\right )\right )}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \log ^{-1+\frac {1}{x}}\left (x^2\right ) \left (-1-\log \left (x^2\right )+x^2 \log \left (x^2\right )\right )}{x^2} \, dx\right )-\int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \log \left (\log \left (x^2\right )\right )}{x^2} \, dx\\ &=-\left (2 \int \left (-\frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \log ^{-1+\frac {1}{x}}\left (x^2\right )}{x^2}+\frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \left (-1+x^2\right ) \sqrt [x]{\log \left (x^2\right )}}{x^2}\right ) \, dx\right )-\int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \log \left (\log \left (x^2\right )\right )}{x^2} \, dx\\ &=2 \int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \log ^{-1+\frac {1}{x}}\left (x^2\right )}{x^2} \, dx-2 \int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \left (-1+x^2\right ) \sqrt [x]{\log \left (x^2\right )}}{x^2} \, dx-\int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \log \left (\log \left (x^2\right )\right )}{x^2} \, dx\\ &=2 \int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \log ^{-1+\frac {1}{x}}\left (x^2\right )}{x^2} \, dx-2 \int \left (e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )}-\frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )}}{x^2}\right ) \, dx-\int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \log \left (\log \left (x^2\right )\right )}{x^2} \, dx\\ &=2 \int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \log ^{-1+\frac {1}{x}}\left (x^2\right )}{x^2} \, dx-2 \int e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \, dx+2 \int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )}}{x^2} \, dx-\int \frac {e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \log \left (\log \left (x^2\right )\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.39, size = 29, normalized size = 1.12 \begin {gather*} e^{\frac {4}{3 e^{2/3}}-\frac {2}{x}-2 x} \sqrt [x]{\log \left (x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 30, normalized size = 1.15 \begin {gather*} e^{\left (-\frac {{\left (6 \, {\left (x^{2} + 1\right )} e^{\frac {2}{3}} - 3 \, e^{\frac {2}{3}} \log \left (\log \left (x^{2}\right )\right ) - 4 \, x\right )} e^{\left (-\frac {2}{3}\right )}}{3 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 23, normalized size = 0.88 \begin {gather*} e^{\left (-2 \, x + \frac {\log \left (\log \left (x^{2}\right )\right )}{x} - \frac {2}{x} + \frac {4}{3} \, e^{\left (-\frac {2}{3}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (-\ln \left (x^{2}\right ) \ln \left (\ln \left (x^{2}\right )\right )+\left (-2 x^{2}+2\right ) \ln \left (x^{2}\right )+2\right ) {\mathrm e}^{\frac {\left (3 \,{\mathrm e}^{\frac {2}{3}} \ln \left (\ln \left (x^{2}\right )\right )+\left (-6 x^{2}-6\right ) {\mathrm e}^{\frac {2}{3}}+4 x \right ) {\mathrm e}^{-\frac {2}{3}}}{3 x}}}{x^{2} \ln \left (x^{2}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 27, normalized size = 1.04 \begin {gather*} e^{\left (-2 \, x + \frac {\log \relax (2)}{x} + \frac {\log \left (\log \relax (x)\right )}{x} - \frac {2}{x} + \frac {4}{3} \, e^{\left (-\frac {2}{3}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 24, normalized size = 0.92 \begin {gather*} {\ln \left (x^2\right )}^{1/x}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{-\frac {2}{3}}}{3}}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-\frac {2}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 39, normalized size = 1.50 \begin {gather*} e^{\frac {\frac {4 x}{3} + \frac {\left (- 6 x^{2} - 6\right ) e^{\frac {2}{3}}}{3} + e^{\frac {2}{3}} \log {\left (\log {\left (x^{2} \right )} \right )}}{x e^{\frac {2}{3}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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