Integrand size = 146, antiderivative size = 32 \[ \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \]
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\[ \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=\int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (2 x^4+e^{-e^{15/x}} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \left (1+e^{e^{15/x}}+x\right )-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )}{x^3} \, dx \\ & = \int \left (2 e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x+\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x+e^{e^{15/x}} x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )}{x^3}\right ) \, dx \\ & = 2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x+e^{e^{15/x}} x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )}{x^3} \, dx \\ & = 2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx+\int \left (\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (1+e^{e^{15/x}}+x\right )}{x^2}-\frac {15 e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \log \left (\frac {e^{-x}}{x}\right )}{x^3}\right ) \, dx \\ & = 2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx-15 \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \log \left (\frac {e^{-x}}{x}\right )}{x^3} \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (1+e^{e^{15/x}}+x\right )}{x^2} \, dx \\ & = 2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx+15 \int \frac {(-1-x) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx}{x} \, dx-\left (15 \log \left (\frac {e^{-x}}{x}\right )\right ) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx+\int \left (\frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2}+\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} (1+x)}{x^2}\right ) \, dx \\ & = 2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx+15 \int \left (-\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx-\frac {\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx}{x}\right ) \, dx-\left (15 \log \left (\frac {e^{-x}}{x}\right )\right ) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx+\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2} \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} (1+x)}{x^2} \, dx \\ & = 2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx-15 \int \left (\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx\right ) \, dx-15 \int \frac {\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx}{x} \, dx-\left (15 \log \left (\frac {e^{-x}}{x}\right )\right ) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx+\int \left (\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2}+\frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}\right ) \, dx+\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2} \, dx \\ & = 2 \int e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} x \, dx-15 \int \left (\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx\right ) \, dx-15 \int \frac {\int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx}{x} \, dx-\left (15 \log \left (\frac {e^{-x}}{x}\right )\right ) \int \frac {e^{-e^{15/x}+\frac {15}{x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3} \, dx+\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2} \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2} \, dx+\int \frac {e^{-e^{15/x}-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x} \, dx \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.54 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.06
\[{\mathrm e}^{\frac {-{\mathrm e}^{-\frac {\left (i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )+2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-{\mathrm e}^{\frac {15}{x}}}}{2}}+x^{3}}{x}}\]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=e^{\left (\frac {x^{3} - e^{\left (e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )}\right )}}{x}\right )} \]
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Timed out. \[ \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=\text {Timed out} \]
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Time = 0.43 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=e^{\left (x^{2} - \frac {e^{\left (-x e^{\left (-e^{\frac {15}{x}}\right )} - e^{\left (-e^{\frac {15}{x}}\right )} \log \left (x\right )\right )}}{x}\right )} \]
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\[ \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=\int { \frac {{\left (2 \, x^{4} \log \left (\frac {e^{\left (-x\right )}}{x}\right ) + {\left ({\left (x^{2} - 15 \, e^{\frac {15}{x}} \log \left (\frac {e^{\left (-x\right )}}{x}\right ) + x\right )} e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )} + x \log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )} e^{\left (e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )}\right )}\right )} e^{\left (\frac {x^{3} - e^{\left (e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )}\right )}}{x}\right )}}{x^{3} \log \left (\frac {e^{\left (-x\right )}}{x}\right )} \,d x } \]
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Time = 15.53 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx={\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-x\,{\mathrm {e}}^{-{\mathrm {e}}^{15/x}}}\,{\left (\frac {1}{x}\right )}^{{\mathrm {e}}^{-{\mathrm {e}}^{15/x}}}}{x}} \]
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