Integrand size = 123, antiderivative size = 29 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 e^{-x^2 \left (e^{3-e^{2 x}}+\log (\log (4))\right )^2}}{x} \]
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Leaf count is larger than twice the leaf count of optimal. \(200\) vs. \(2(29)=58\).
Time = 0.97 (sec) , antiderivative size = 200, normalized size of antiderivative = 6.90, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2326} \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 e^{-e^{6-2 e^{2 x}} x^2-x^2 \log ^2(\log (4))} \log ^{-2 e^{3-e^{2 x}} x^2}(4) \left (x^2 \log ^2(\log (4))+e^{6-2 e^{2 x}} \left (x^2-2 e^{2 x} x^3\right )+2 e^{3-e^{2 x}} \left (x^2-e^{2 x} x^3\right ) \log (\log (4))\right )}{x^2 \left (-2 e^{2 x-2 e^{2 x}+6} x^2-2 e^{2 x-e^{2 x}+3} x^2 \log (\log (4))+e^{6-2 e^{2 x}} x+x \log ^2(\log (4))+2 e^{3-e^{2 x}} x \log (\log (4))\right )} \]
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Rule 2326
Rubi steps \begin{align*} \text {integral}& = \frac {4 e^{-e^{6-2 e^{2 x}} x^2-x^2 \log ^2(\log (4))} \log ^{-2 e^{3-e^{2 x}} x^2}(4) \left (e^{6-2 e^{2 x}} \left (x^2-2 e^{2 x} x^3\right )+2 e^{3-e^{2 x}} \left (x^2-e^{2 x} x^3\right ) \log (\log (4))+x^2 \log ^2(\log (4))\right )}{x^2 \left (e^{6-2 e^{2 x}} x-2 e^{6-2 e^{2 x}+2 x} x^2+2 e^{3-e^{2 x}} x \log (\log (4))-2 e^{3-e^{2 x}+2 x} x^2 \log (\log (4))+x \log ^2(\log (4))\right )} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 e^{-x^2 \left (e^{6-2 e^{2 x}}+\log ^2(\log (4))\right )} \log ^{-2 e^{3-e^{2 x}} x^2}(4)}{x} \]
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Time = 3.50 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66
method | result | size |
parallelrisch | \(\frac {4 \,{\mathrm e}^{-x^{2} \left ({\mathrm e}^{3-{\mathrm e}^{2 x}} \ln \left (4 \ln \left (2\right )^{2}\right )+\ln \left (2 \ln \left (2\right )\right )^{2}+{\mathrm e}^{6-2 \,{\mathrm e}^{2 x}}\right )}}{x}\) | \(48\) |
risch | \(\frac {4 \ln \left (2\right )^{-2 x^{2} {\mathrm e}^{3-{\mathrm e}^{2 x}}} 4^{-x^{2} {\mathrm e}^{3-{\mathrm e}^{2 x}}} \ln \left (2\right )^{-2 x^{2} \ln \left (2\right )} {\mathrm e}^{-x^{2} \left (\ln \left (2\right )^{2}+\ln \left (\ln \left (2\right )\right )^{2}+{\mathrm e}^{6-2 \,{\mathrm e}^{2 x}}\right )}}{x}\) | \(79\) |
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Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 \, e^{\left (-2 \, x^{2} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (2 \, \log \left (2\right )\right ) - x^{2} \log \left (2 \, \log \left (2\right )\right )^{2} - x^{2} e^{\left (-2 \, e^{\left (2 \, x\right )} + 6\right )}\right )}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 e^{- 2 x^{2} e^{3 - e^{2 x}} \log {\left (2 \log {\left (2 \right )} \right )} - x^{2} e^{6 - 2 e^{2 x}} - x^{2} \log {\left (2 \log {\left (2 \right )} \right )}^{2}}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (28) = 56\).
Time = 0.40 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.86 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4 \, e^{\left (-2 \, x^{2} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (2\right ) - x^{2} \log \left (2\right )^{2} - 2 \, x^{2} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (\log \left (2\right )\right ) - 2 \, x^{2} \log \left (2\right ) \log \left (\log \left (2\right )\right ) - x^{2} \log \left (\log \left (2\right )\right )^{2} - x^{2} e^{\left (-2 \, e^{\left (2 \, x\right )} + 6\right )}\right )}}{x} \]
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\[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\int { -\frac {4 \, {\left (2 \, x^{2} \log \left (2 \, \log \left (2\right )\right )^{2} - 4 \, {\left (x^{3} e^{\left (2 \, x\right )} - x^{2}\right )} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (2 \, \log \left (2\right )\right ) - 2 \, {\left (2 \, x^{3} e^{\left (2 \, x\right )} - x^{2}\right )} e^{\left (-2 \, e^{\left (2 \, x\right )} + 6\right )} + 1\right )} e^{\left (-2 \, x^{2} e^{\left (-e^{\left (2 \, x\right )} + 3\right )} \log \left (2 \, \log \left (2\right )\right ) - x^{2} \log \left (2 \, \log \left (2\right )\right )^{2} - x^{2} e^{\left (-2 \, e^{\left (2 \, x\right )} + 6\right )}\right )}}{x^{2}} \,d x } \]
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Time = 0.48 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {e^{-e^{6-2 e^{2 x}} x^2-2 e^{3-e^{2 x}} x^2 \log (\log (4))-x^2 \log ^2(\log (4))} \left (-4+e^{6-2 e^{2 x}} \left (-8 x^2+16 e^{2 x} x^3\right )+e^{3-e^{2 x}} \left (-16 x^2+16 e^{2 x} x^3\right ) \log (\log (4))-8 x^2 \log ^2(\log (4))\right )}{x^2} \, dx=\frac {4\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{-2\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^6}\,{\mathrm {e}}^{-x^2\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{-x^2\,{\ln \left (\ln \left (2\right )\right )}^2}}{2^{2\,x^2\,\ln \left (\ln \left (2\right )\right )}\,2^{2\,x^2\,{\mathrm {e}}^{-{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^3}\,x\,{\ln \left (2\right )}^{2\,x^2\,{\mathrm {e}}^{-{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^3}} \]
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