Optimal. Leaf size=30 \[ e^{2-2 x \left (36-\frac {3}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}\right )} x^2 \]
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Rubi [F] time = 9.35, 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 {2 \left (4 x-36 x^2+(3-108 x) \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}\right ) \left (18 x^2-18 x^3+\left (2 x^3-72 x^4\right ) \log \left (\frac {e^x}{x}\right )+\left (30 x^2-432 x^3\right ) \log \left (\frac {e^x}{x}\right ) \log \left (\log \left (\frac {e^x}{x}\right )\right )+\left (18 x-648 x^2\right ) \log \left (\frac {e^x}{x}\right ) \log ^2\left (\log \left (\frac {e^x}{x}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{x}\right )+6 x \log \left (\frac {e^x}{x}\right ) \log \left (\log \left (\frac {e^x}{x}\right )\right )+9 \log \left (\frac {e^x}{x}\right ) \log ^2\left (\log \left (\frac {e^x}{x}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x \log ^{-1+\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right ) \left (-9 (-1+x) x-\log \left (\frac {e^x}{x}\right ) \left (x^2 (-1+36 x)+3 x (-5+72 x) \log \left (\log \left (\frac {e^x}{x}\right )\right )+9 (-1+36 x) \log ^2\left (\log \left (\frac {e^x}{x}\right )\right )\right )\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2} \, dx\\ &=2 \int \frac {e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x \log ^{-1+\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right ) \left (-9 (-1+x) x-\log \left (\frac {e^x}{x}\right ) \left (x^2 (-1+36 x)+3 x (-5+72 x) \log \left (\log \left (\frac {e^x}{x}\right )\right )+9 (-1+36 x) \log ^2\left (\log \left (\frac {e^x}{x}\right )\right )\right )\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2} \, dx\\ &=2 \int \left (-e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x (-1+36 x) \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )-\frac {3 e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{-1+\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right ) \left (-3+3 x+x \log \left (\frac {e^x}{x}\right )\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2}+\frac {3 e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}\right ) \, dx\\ &=-\left (2 \int e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x (-1+36 x) \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right ) \, dx\right )-6 \int \frac {e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{-1+\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right ) \left (-3+3 x+x \log \left (\frac {e^x}{x}\right )\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2} \, dx+6 \int \frac {e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )} \, dx\\ &=-\left (2 \int \left (-e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )+36 e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )\right ) \, dx\right )+6 \int \frac {e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )} \, dx-6 \int \left (\frac {e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^3 \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2}-\frac {3 e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{-1+\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2}+\frac {3 e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^3 \log ^{-1+\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2}\right ) \, dx\\ &=2 \int e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right ) \, dx-6 \int \frac {e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^3 \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2} \, dx+6 \int \frac {e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )} \, dx+18 \int \frac {e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{-1+\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2} \, dx-18 \int \frac {e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^3 \log ^{-1+\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right )}{\left (x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )\right )^2} \, dx-72 \int e^{\frac {8 (1-9 x) x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \log ^{\frac {6-216 x}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}}\left (\frac {e^x}{x}\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.30, size = 46, normalized size = 1.53 \begin {gather*} e^{\frac {8 (1-9 x) x+(6-216 x) \log \left (\log \left (\frac {e^x}{x}\right )\right )}{x+3 \log \left (\log \left (\frac {e^x}{x}\right )\right )}} x^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 45, normalized size = 1.50 \begin {gather*} x^{2} e^{\left (-\frac {2 \, {\left (36 \, x^{2} + 3 \, {\left (36 \, x - 1\right )} \log \left (\log \left (\frac {e^{x}}{x}\right )\right ) - 4 \, x\right )}}{x + 3 \, \log \left (\log \left (\frac {e^{x}}{x}\right )\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 71.17, size = 48, normalized size = 1.60 \begin {gather*} x^{2} e^{\left (-\frac {2 \, {\left (36 \, x^{2} + 108 \, x \log \left (x - \log \relax (x)\right ) - 4 \, x - 3 \, \log \left (x - \log \relax (x)\right )\right )}}{x + 3 \, \log \left (x - \log \relax (x)\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 205, normalized size = 6.83
method | result | size |
risch | \(x^{2} {\mathrm e}^{-\frac {2 \left (108 \ln \left (-\ln \relax (x )+\ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )}{2}\right ) x +36 x^{2}-3 \ln \left (-\ln \relax (x )+\ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )}{2}\right )-4 x \right )}{3 \ln \left (-\ln \relax (x )+\ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )}{2}\right )+x}}\) | \(205\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {2\,\left (\ln \left (\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\right )\,\left (108\,x-3\right )-4\,x+36\,x^2\right )}{x+3\,\ln \left (\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\right )}}\,\left (\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\left (2\,x^3-72\,x^4\right )+18\,x^2-18\,x^3+\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\ln \left (\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\right )\,\left (30\,x^2-432\,x^3\right )+\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,{\ln \left (\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\right )}^2\,\left (18\,x-648\,x^2\right )\right )}{9\,\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,{\ln \left (\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\right )}^2+x^2\,\ln \left (\frac {{\mathrm {e}}^x}{x}\right )+6\,x\,\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\,\ln \left (\ln \left (\frac {{\mathrm {e}}^x}{x}\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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