Integrand size = 183, antiderivative size = 24 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=e^{3 \left (1+\frac {-5+\log (5)}{e+e^{e^x}}\right )^2}+x \]
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\[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=\int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+\exp \left (\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}\right ) \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {e^3+e^{3 x}+3 e^{2+x}+3 e^{1+2 x}-6 e^{x+\frac {3 \left (-5+e+e^x+\log (5)\right )^2}{\left (e+e^x\right )^2}} x (-5+\log (5)) \left (-5+e+e^x+\log (5)\right )}{\left (e+e^x\right )^3 x} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (\frac {e^3}{\left (e+e^x\right )^3 x}+\frac {e^{3 x}}{\left (e+e^x\right )^3 x}+\frac {3 e^{2+x}}{\left (e+e^x\right )^3 x}+\frac {3 e^{1+2 x}}{\left (e+e^x\right )^3 x}+\frac {6 e^{x+\frac {3 \left (-5+e+e^x+\log (5)\right )^2}{\left (e+e^x\right )^2}} \left (e^x-5 \left (1+\frac {1}{5} (-e-\log (5))\right )\right ) (5-\log (5))}{\left (e+e^x\right )^3}\right ) \, dx,x,e^x\right ) \\ & = 3 \text {Subst}\left (\int \frac {e^{2+x}}{\left (e+e^x\right )^3 x} \, dx,x,e^x\right )+3 \text {Subst}\left (\int \frac {e^{1+2 x}}{\left (e+e^x\right )^3 x} \, dx,x,e^x\right )+e^3 \text {Subst}\left (\int \frac {1}{\left (e+e^x\right )^3 x} \, dx,x,e^x\right )+(6 (5-\log (5))) \text {Subst}\left (\int \frac {e^{x+\frac {3 \left (-5+e+e^x+\log (5)\right )^2}{\left (e+e^x\right )^2}} \left (e^x-5 \left (1+\frac {1}{5} (-e-\log (5))\right )\right )}{\left (e+e^x\right )^3} \, dx,x,e^x\right )+\text {Subst}\left (\int \frac {e^{3 x}}{\left (e+e^x\right )^3 x} \, dx,x,e^x\right ) \\ & = 3 \text {Subst}\left (\int \frac {e^{1+2 x}}{\left (e+e^x\right )^3 x} \, dx,x,e^x\right )+\left (3 e^2\right ) \text {Subst}\left (\int \frac {e^x}{\left (e+e^x\right )^3 x} \, dx,x,e^x\right )+e^3 \text {Subst}\left (\int \frac {1}{\left (e+e^x\right )^3 x} \, dx,x,e^x\right )+(6 (5-\log (5))) \text {Subst}\left (\int \frac {e^{\frac {3 (-5+e+x+\log (5))^2}{(e+x)^2}} (-5+e+x+\log (5))}{(e+x)^3} \, dx,x,e^{e^x}\right )+\text {Subst}\left (\int \left (\frac {1}{x}-\frac {e^3}{\left (e+e^x\right )^3 x}+\frac {3 e^2}{\left (e+e^x\right )^2 x}-\frac {3 e}{\left (e+e^x\right ) x}\right ) \, dx,x,e^x\right ) \\ & = e^{\frac {3 \left (5-e-e^{e^x}-\log (5)\right )^2}{\left (e+e^{e^x}\right )^2}}-\frac {3 e^{2-x}}{2 \left (e+e^{e^x}\right )^2}+x+3 \text {Subst}\left (\int \frac {e^{1+2 x}}{\left (e+e^x\right )^3 x} \, dx,x,e^x\right )-(3 e) \text {Subst}\left (\int \frac {1}{\left (e+e^x\right ) x} \, dx,x,e^x\right )-\frac {1}{2} \left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (e+e^x\right )^2 x^2} \, dx,x,e^x\right )+\left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (e+e^x\right )^2 x} \, dx,x,e^x\right ) \\ \end{align*}
Time = 1.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=e^{\frac {3 \left (-5+e+e^{e^x}+\log (5)\right )^2}{\left (e+e^{e^x}\right )^2}}+x \]
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Timed out.
\[\int \frac {\left (\left (-6 \ln \left (5\right )+30\right ) {\mathrm e}^{x} {\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (-6 \ln \left (5\right )^{2}+\left (-6 \,{\mathrm e}+60\right ) \ln \left (5\right )+30 \,{\mathrm e}-150\right ) {\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{\frac {3 \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (6 \ln \left (5\right )+6 \,{\mathrm e}-30\right ) {\mathrm e}^{{\mathrm e}^{x}}+3 \ln \left (5\right )^{2}+\left (6 \,{\mathrm e}-30\right ) \ln \left (5\right )+3 \,{\mathrm e}^{2}-30 \,{\mathrm e}+75}{{\mathrm e}^{2 \,{\mathrm e}^{x}}+2 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2}}}+{\mathrm e}^{3 \,{\mathrm e}^{x}}+3 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+3 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{3}}{{\mathrm e}^{3 \,{\mathrm e}^{x}}+3 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+3 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{3}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.46 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=x + e^{\left (\frac {3 \, {\left ({\left (2 \, {\left (e - 5\right )} \log \left (5\right ) + \log \left (5\right )^{2} + e^{2} - 10 \, e + 25\right )} e^{\left (2 \, x\right )} + 2 \, {\left (e + \log \left (5\right ) - 5\right )} e^{\left (2 \, x + e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x}\right )}\right )}}{e^{\left (2 \, x + 2 \, e^{x}\right )} + 2 \, e^{\left (2 \, x + e^{x} + 1\right )} + e^{\left (2 \, x + 2\right )}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (20) = 40\).
Time = 0.56 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.17 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=x + e^{\frac {3 e^{2 e^{x}} + \left (-30 + 6 \log {\left (5 \right )} + 6 e\right ) e^{e^{x}} - 30 e + \left (-30 + 6 e\right ) \log {\left (5 \right )} + 3 \log {\left (5 \right )}^{2} + 3 e^{2} + 75}{e^{2 e^{x}} + 2 e e^{e^{x}} + e^{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (22) = 44\).
Time = 0.42 (sec) , antiderivative size = 296, normalized size of antiderivative = 12.33 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx={\left (x e^{\left (\frac {30 \, e}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {30 \, e^{\left (e^{x}\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {30 \, \log \left (5\right )}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}}\right )} + e^{\left (\frac {6 \, e \log \left (5\right )}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {6 \, e^{\left (e^{x}\right )} \log \left (5\right )}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {3 \, \log \left (5\right )^{2}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {3 \, e^{2}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {3 \, e^{\left (2 \, e^{x}\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {6 \, e^{\left (e^{x} + 1\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} + \frac {75}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}}\right )}\right )} e^{\left (-\frac {30 \, e}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} - \frac {30 \, e^{\left (e^{x}\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}} - \frac {30 \, \log \left (5\right )}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}}\right )} \]
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\[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=\int { -\frac {6 \, {\left ({\left (\log \left (5\right ) - 5\right )} e^{\left (x + 2 \, e^{x}\right )} + {\left ({\left (e - 10\right )} \log \left (5\right ) + \log \left (5\right )^{2} - 5 \, e + 25\right )} e^{\left (x + e^{x}\right )}\right )} e^{\left (\frac {3 \, {\left (2 \, {\left (e + \log \left (5\right ) - 5\right )} e^{\left (e^{x}\right )} + 2 \, {\left (e - 5\right )} \log \left (5\right ) + \log \left (5\right )^{2} + e^{2} - 10 \, e + e^{\left (2 \, e^{x}\right )} + 25\right )}}{e^{2} + e^{\left (2 \, e^{x}\right )} + 2 \, e^{\left (e^{x} + 1\right )}}\right )} - e^{3} - e^{\left (3 \, e^{x}\right )} - 3 \, e^{\left (2 \, e^{x} + 1\right )} - 3 \, e^{\left (e^{x} + 2\right )}}{e^{3} + e^{\left (3 \, e^{x}\right )} + 3 \, e^{\left (2 \, e^{x} + 1\right )} + 3 \, e^{\left (e^{x} + 2\right )}} \,d x } \]
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Time = 12.08 (sec) , antiderivative size = 234, normalized size of antiderivative = 9.75 \[ \int \frac {e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}+e^{\frac {75-30 e+3 e^2+3 e^{2 e^x}+(-30+6 e) \log (5)+3 \log ^2(5)+e^{e^x} (-30+6 e+6 \log (5))}{e^2+e^{2 e^x}+2 e^{1+e^x}}} \left (e^{2 e^x+x} (30-6 \log (5))+e^{e^x+x} \left (-150+30 e+(60-6 e) \log (5)-6 \log ^2(5)\right )\right )}{e^3+e^{3 e^x}+3 e^{2+e^x}+3 e^{1+2 e^x}} \, dx=x+\frac {5^{\frac {6\,{\mathrm {e}}^{{\mathrm {e}}^x}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,5^{\frac {6\,\mathrm {e}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {6\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (5\right )}^2}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {75}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{-\frac {30\,{\mathrm {e}}^{{\mathrm {e}}^x}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^2}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}\,{\mathrm {e}}^{-\frac {30\,\mathrm {e}}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}}{5^{\frac {30}{{\mathrm {e}}^2+{\mathrm {e}}^{2\,{\mathrm {e}}^x}+2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\mathrm {e}}}} \]
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