Integrand size = 117, antiderivative size = 24 \[ \int \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx=\left (-3+e^{e^x}+x^2 \left (-e^x+4 x \log (4)\right )\right )^2 \]
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\[ \int \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx=\int \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -24 x^3 \log (4)+16 x^6 \log ^2(4)+2 \int e^{2 e^x+x} \, dx+\int e^{2 x} \left (4 x^3+2 x^4\right ) \, dx+\int e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right ) \, dx+\int e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right ) \, dx \\ & = -24 x^3 \log (4)+16 x^6 \log ^2(4)+2 \text {Subst}\left (\int e^{2 x} \, dx,x,e^x\right )+\int e^{2 x} x^3 (4+2 x) \, dx+\int \left (12 e^x x+6 e^x x^2-8 e^x x^4 (5+x) \log (4)\right ) \, dx+\int \left (-2 e^{e^x+2 x} x^2+24 e^{e^x} x^2 \log (4)+2 e^{e^x+x} \left (-3-2 x-x^2+4 x^3 \log (4)\right )\right ) \, dx \\ & = e^{2 e^x}-24 x^3 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+2 x} x^2 \, dx+2 \int e^{e^x+x} \left (-3-2 x-x^2+4 x^3 \log (4)\right ) \, dx+6 \int e^x x^2 \, dx+12 \int e^x x \, dx-(8 \log (4)) \int e^x x^4 (5+x) \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx+\int \left (4 e^{2 x} x^3+2 e^{2 x} x^4\right ) \, dx \\ & = e^{2 e^x}+12 e^x x+6 e^x x^2-24 x^3 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+2 x} x^2 \, dx+2 \int e^{2 x} x^4 \, dx+2 \int \left (-3 e^{e^x+x}-2 e^{e^x+x} x-e^{e^x+x} x^2+4 e^{e^x+x} x^3 \log (4)\right ) \, dx+4 \int e^{2 x} x^3 \, dx-12 \int e^x \, dx-12 \int e^x x \, dx-(8 \log (4)) \int \left (5 e^x x^4+e^x x^5\right ) \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx \\ & = e^{2 e^x}-12 e^x+6 e^x x^2+2 e^{2 x} x^3+e^{2 x} x^4-24 x^3 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx-4 \int e^{2 x} x^3 \, dx-6 \int e^{e^x+x} \, dx-6 \int e^{2 x} x^2 \, dx+12 \int e^x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx-(8 \log (4)) \int e^x x^5 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx-(40 \log (4)) \int e^x x^4 \, dx \\ & = e^{2 e^x}+6 e^x x^2-3 e^{2 x} x^2+e^{2 x} x^4-24 x^3 \log (4)-40 e^x x^4 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx+6 \int e^{2 x} x \, dx+6 \int e^{2 x} x^2 \, dx-6 \text {Subst}\left (\int e^x \, dx,x,e^x\right )+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx+(40 \log (4)) \int e^x x^4 \, dx+(160 \log (4)) \int e^x x^3 \, dx \\ & = -6 e^{e^x}+e^{2 e^x}+3 e^{2 x} x+6 e^x x^2+e^{2 x} x^4-24 x^3 \log (4)+160 e^x x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-3 \int e^{2 x} \, dx-4 \int e^{e^x+x} x \, dx-6 \int e^{2 x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx-(160 \log (4)) \int e^x x^3 \, dx-(480 \log (4)) \int e^x x^2 \, dx \\ & = -6 e^{e^x}+e^{2 e^x}-\frac {3 e^{2 x}}{2}+6 e^x x^2+e^{2 x} x^4-480 e^x x^2 \log (4)-24 x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx+3 \int e^{2 x} \, dx-4 \int e^{e^x+x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx+(480 \log (4)) \int e^x x^2 \, dx+(960 \log (4)) \int e^x x \, dx \\ & = -6 e^{e^x}+e^{2 e^x}+6 e^x x^2+e^{2 x} x^4+960 e^x x \log (4)-24 x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx-(960 \log (4)) \int e^x \, dx-(960 \log (4)) \int e^x x \, dx \\ & = -6 e^{e^x}+e^{2 e^x}+6 e^x x^2+e^{2 x} x^4-960 e^x \log (4)-24 x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx+(960 \log (4)) \int e^x \, dx \\ & = -6 e^{e^x}+e^{2 e^x}+6 e^x x^2+e^{2 x} x^4-24 x^3 \log (4)-8 e^x x^5 \log (4)+16 x^6 \log ^2(4)-2 \int e^{e^x+x} x^2 \, dx-2 \int e^{e^x+2 x} x^2 \, dx-4 \int e^{e^x+x} x \, dx+(8 \log (4)) \int e^{e^x+x} x^3 \, dx+(24 \log (4)) \int e^{e^x} x^2 \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(24)=48\).
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.38 \[ \int \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx=-6 e^{e^x}+e^{2 e^x}-2 e^{e^x+x} x^2+e^{2 x} x^4-24 x^3 \log (4)+8 e^{e^x} x^3 \log (4)+16 x^6 \log ^2(4)-2 e^x x \left (-3 x+4 x^4 \log (4)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(21)=42\).
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.79
method | result | size |
risch | \({\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (16 x^{3} \ln \left (2\right )-2 \,{\mathrm e}^{x} x^{2}-6\right ) {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2 x} x^{4}+\left (-16 x^{5} \ln \left (2\right )+6 x^{2}\right ) {\mathrm e}^{x}+64 x^{6} \ln \left (2\right )^{2}-48 x^{3} \ln \left (2\right )\) | \(67\) |
default | \(64 x^{6} \ln \left (2\right )^{2}-16 \ln \left (2\right ) {\mathrm e}^{x} x^{5}+16 x^{3} \ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2 x} x^{4}-48 x^{3} \ln \left (2\right )-2 x^{2} {\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}+6 \,{\mathrm e}^{x} x^{2}-6 \,{\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2 \,{\mathrm e}^{x}}\) | \(72\) |
parallelrisch | \(64 x^{6} \ln \left (2\right )^{2}-16 \ln \left (2\right ) {\mathrm e}^{x} x^{5}+16 x^{3} \ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2 x} x^{4}-48 x^{3} \ln \left (2\right )-2 x^{2} {\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}+6 \,{\mathrm e}^{x} x^{2}-6 \,{\mathrm e}^{{\mathrm e}^{x}}+{\mathrm e}^{2 \,{\mathrm e}^{x}}\) | \(72\) |
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.83 \[ \int \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx=64 \, x^{6} \log \left (2\right )^{2} + x^{4} e^{\left (2 \, x\right )} - 48 \, x^{3} \log \left (2\right ) - 2 \, {\left (8 \, x^{5} \log \left (2\right ) - 3 \, x^{2}\right )} e^{x} + 2 \, {\left (8 \, x^{3} \log \left (2\right ) - x^{2} e^{x} - 3\right )} e^{\left (e^{x}\right )} + e^{\left (2 \, e^{x}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (20) = 40\).
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.04 \[ \int \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx=64 x^{6} \log {\left (2 \right )}^{2} + x^{4} e^{2 x} - 48 x^{3} \log {\left (2 \right )} + \left (- 16 x^{5} \log {\left (2 \right )} + 6 x^{2}\right ) e^{x} + \left (16 x^{3} \log {\left (2 \right )} - 2 x^{2} e^{x} - 6\right ) e^{e^{x}} + e^{2 e^{x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.83 \[ \int \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx=64 \, x^{6} \log \left (2\right )^{2} + x^{4} e^{\left (2 \, x\right )} - 48 \, x^{3} \log \left (2\right ) - 2 \, {\left (8 \, x^{5} \log \left (2\right ) - 3 \, x^{2}\right )} e^{x} + 2 \, {\left (8 \, x^{3} \log \left (2\right ) - x^{2} e^{x} - 3\right )} e^{\left (e^{x}\right )} + e^{\left (2 \, e^{x}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.54 \[ \int \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx=64 \, x^{6} \log \left (2\right )^{2} + x^{4} e^{\left (2 \, x\right )} - 48 \, x^{3} \log \left (2\right ) + 2 \, {\left (8 \, x^{3} e^{\left (x + e^{x}\right )} \log \left (2\right ) - x^{2} e^{\left (2 \, x + e^{x}\right )} - 3 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} - 2 \, {\left (8 \, x^{5} \log \left (2\right ) - 3 \, x^{2}\right )} e^{x} + e^{\left (2 \, e^{x}\right )} \]
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Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.96 \[ \int \left (2 e^{2 e^x+x}+e^{2 x} \left (4 x^3+2 x^4\right )-72 x^2 \log (4)+96 x^5 \log ^2(4)+e^x \left (12 x+6 x^2+\left (-40 x^4-8 x^5\right ) \log (4)\right )+e^{e^x} \left (-2 e^{2 x} x^2+24 x^2 \log (4)+e^x \left (-6-4 x-2 x^2+8 x^3 \log (4)\right )\right )\right ) \, dx={\mathrm {e}}^{2\,{\mathrm {e}}^x}-6\,{\mathrm {e}}^{{\mathrm {e}}^x}+64\,x^6\,{\ln \left (2\right )}^2+6\,x^2\,{\mathrm {e}}^x-2\,x^2\,{\mathrm {e}}^{x+{\mathrm {e}}^x}+x^4\,{\mathrm {e}}^{2\,x}-48\,x^3\,\ln \left (2\right )-16\,x^5\,{\mathrm {e}}^x\,\ln \left (2\right )+16\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\ln \left (2\right ) \]
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