Integrand size = 68, antiderivative size = 26 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx=e^{-2 x} \left (1+e+2 x-e^{-e^4} (1+x)\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(26)=52\).
Time = 0.17 (sec) , antiderivative size = 192, normalized size of antiderivative = 7.38, number of steps used = 23, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2227, 2207, 2225} \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx=e^{-2 x-2 e^4} x^2-4 e^{-2 x-e^4} x^2+4 e^{-2 x} x^2+4 e^{1-2 x} x+2 e^{-2 x-2 e^4} x-4 e^{-2 x-e^4} x+4 e^{-2 x} x-2 (1+e) e^{-2 x-e^4} x+2 e^{1-2 x}+e^{-2 x-2 e^4}+(1-e) e^{-2 x-e^4}-2 e^{-2 x-e^4}+2 e^{-2 x}-\left (1-e^2\right ) e^{-2 x}-(1+e) e^{-2 x-e^4} \]
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Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \int \left (-2 e^{-2 e^4-2 x} x-2 e^{-2 e^4-2 x} x^2-2 e^{-2 x} \left (-1+e^2+4 e x+4 x^2\right )+2 e^{-e^4-2 x} \left (-1+e+2 (1+e) x+4 x^2\right )\right ) \, dx \\ & = -\left (2 \int e^{-2 e^4-2 x} x \, dx\right )-2 \int e^{-2 e^4-2 x} x^2 \, dx-2 \int e^{-2 x} \left (-1+e^2+4 e x+4 x^2\right ) \, dx+2 \int e^{-e^4-2 x} \left (-1+e+2 (1+e) x+4 x^2\right ) \, dx \\ & = e^{-2 e^4-2 x} x+e^{-2 e^4-2 x} x^2-2 \int e^{-2 e^4-2 x} x \, dx+2 \int \left ((-1+e) e^{-e^4-2 x}+2 e^{-e^4-2 x} (1+e) x+4 e^{-e^4-2 x} x^2\right ) \, dx-2 \int \left (e^{-2 x} \left (-1+e^2\right )+4 e^{1-2 x} x+4 e^{-2 x} x^2\right ) \, dx-\int e^{-2 e^4-2 x} \, dx \\ & = \frac {1}{2} e^{-2 e^4-2 x}+2 e^{-2 e^4-2 x} x+e^{-2 e^4-2 x} x^2-8 \int e^{1-2 x} x \, dx+8 \int e^{-e^4-2 x} x^2 \, dx-8 \int e^{-2 x} x^2 \, dx-(2 (1-e)) \int e^{-e^4-2 x} \, dx+(4 (1+e)) \int e^{-e^4-2 x} x \, dx+\left (2 \left (1-e^2\right )\right ) \int e^{-2 x} \, dx-\int e^{-2 e^4-2 x} \, dx \\ & = e^{-2 e^4-2 x}+(1-e) e^{-e^4-2 x}-e^{-2 x} \left (1-e^2\right )+4 e^{1-2 x} x+2 e^{-2 e^4-2 x} x-2 e^{-e^4-2 x} (1+e) x+e^{-2 e^4-2 x} x^2-4 e^{-e^4-2 x} x^2+4 e^{-2 x} x^2-4 \int e^{1-2 x} \, dx+8 \int e^{-e^4-2 x} x \, dx-8 \int e^{-2 x} x \, dx+(2 (1+e)) \int e^{-e^4-2 x} \, dx \\ & = 2 e^{1-2 x}+e^{-2 e^4-2 x}+(1-e) e^{-e^4-2 x}-e^{-e^4-2 x} (1+e)-e^{-2 x} \left (1-e^2\right )+4 e^{1-2 x} x+2 e^{-2 e^4-2 x} x-4 e^{-e^4-2 x} x+4 e^{-2 x} x-2 e^{-e^4-2 x} (1+e) x+e^{-2 e^4-2 x} x^2-4 e^{-e^4-2 x} x^2+4 e^{-2 x} x^2+4 \int e^{-e^4-2 x} \, dx-4 \int e^{-2 x} \, dx \\ & = 2 e^{1-2 x}+e^{-2 e^4-2 x}-2 e^{-e^4-2 x}+(1-e) e^{-e^4-2 x}+2 e^{-2 x}-e^{-e^4-2 x} (1+e)-e^{-2 x} \left (1-e^2\right )+4 e^{1-2 x} x+2 e^{-2 e^4-2 x} x-4 e^{-e^4-2 x} x+4 e^{-2 x} x-2 e^{-e^4-2 x} (1+e) x+e^{-2 e^4-2 x} x^2-4 e^{-e^4-2 x} x^2+4 e^{-2 x} x^2 \\ \end{align*}
Time = 5.45 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx=e^{-2 \left (e^4+x\right )} \left (-1+e^{1+e^4}-x+e^{e^4} (1+2 x)\right )^2 \]
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Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27
method | result | size |
gosper | \(\left ({\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+2 x \,{\mathrm e}^{{\mathrm e}^{4}}+{\mathrm e}^{{\mathrm e}^{4}}-x -1\right )^{2} {\mathrm e}^{-2 x} {\mathrm e}^{-2 \,{\mathrm e}^{4}}\) | \(33\) |
risch | \(\left ({\mathrm e}^{2 \,{\mathrm e}^{4}+2}+4 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x -2 x \,{\mathrm e}^{{\mathrm e}^{4}+1}-4 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+{\mathrm e}^{2 \,{\mathrm e}^{4}}-2 \,{\mathrm e}^{{\mathrm e}^{4}+1}-6 x \,{\mathrm e}^{{\mathrm e}^{4}}+x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{4}}+2 x +1\right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}-2 x}\) | \(102\) |
parallelrisch | \({\mathrm e}^{-2 \,{\mathrm e}^{4}} \left ({\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e} x +4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+1+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}-2 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{4}} x +4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x -4 x^{2} {\mathrm e}^{{\mathrm e}^{4}}-2 \,{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+{\mathrm e}^{2 \,{\mathrm e}^{4}}-6 x \,{\mathrm e}^{{\mathrm e}^{4}}+x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{4}}+2 x \right ) {\mathrm e}^{-2 x}\) | \(105\) |
norman | \(\left (\left ({\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}^{2}-2 \,{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+{\mathrm e}^{2 \,{\mathrm e}^{4}}-2 \,{\mathrm e}^{{\mathrm e}^{4}}+1+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}\right ) {\mathrm e}^{-{\mathrm e}^{4}}+\left (4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}-4 \,{\mathrm e}^{{\mathrm e}^{4}}+1\right ) {\mathrm e}^{-{\mathrm e}^{4}} x^{2}+2 \left (2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e}-{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}-3 \,{\mathrm e}^{{\mathrm e}^{4}}+1+2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}\right ) {\mathrm e}^{-{\mathrm e}^{4}} x \right ) {\mathrm e}^{-2 x} {\mathrm e}^{-{\mathrm e}^{4}}\) | \(117\) |
meijerg | \(-{\mathrm e}^{2} \left (1-{\mathrm e}^{-2 x}\right )+{\mathrm e}^{1-{\mathrm e}^{4}} \left (1-{\mathrm e}^{-2 x}\right )+1-{\mathrm e}^{-2 x}-{\mathrm e}^{-{\mathrm e}^{4}} \left (1-{\mathrm e}^{-2 x}\right )+\frac {\left (-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{{\mathrm e}^{4}+1}+4 \,{\mathrm e}^{{\mathrm e}^{4}}-2\right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (1-\frac {\left (4 x +2\right ) {\mathrm e}^{-2 x}}{2}\right )}{4}+\frac {\left (-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}+8 \,{\mathrm e}^{{\mathrm e}^{4}}-2\right ) {\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (2-\frac {\left (12 x^{2}+12 x +6\right ) {\mathrm e}^{-2 x}}{3}\right )}{8}\) | \(134\) |
default | \({\mathrm e}^{-2 \,{\mathrm e}^{4}} \left (2 \,{\mathrm e}^{-2 x} x +{\mathrm e}^{-2 x}+{\mathrm e}^{-2 x} x^{2}+{\mathrm e}^{-2 x} {\mathrm e}^{{\mathrm e}^{4}}-{\mathrm e}^{-2 x} {\mathrm e}^{2 \,{\mathrm e}^{4}}+4 \,{\mathrm e}^{{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )+8 \,{\mathrm e}^{{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-2 x} x^{2}}{2}-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (-\frac {{\mathrm e}^{-2 x} x^{2}}{2}-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )+{\mathrm e}^{-2 x} {\mathrm e}^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}-{\mathrm e}^{-2 x} {\mathrm e}^{{\mathrm e}^{4}} {\mathrm e}+4 \,{\mathrm e}^{{\mathrm e}^{4}} {\mathrm e} \left (-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )-8 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} {\mathrm e} \left (-\frac {{\mathrm e}^{-2 x} x}{2}-\frac {{\mathrm e}^{-2 x}}{4}\right )\right )\) | \(192\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx={\left (4 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} e + 4 \, x + e^{2} + 1\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (2 \, x^{2} + {\left (x + 1\right )} e + 3 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} + {\left (x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.04 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx=\frac {\left (- 4 x^{2} e^{e^{4}} + x^{2} + 4 x^{2} e^{2 e^{4}} - 6 x e^{e^{4}} - 2 e x e^{e^{4}} + 2 x + 4 x e^{2 e^{4}} + 4 e x e^{2 e^{4}} - 2 e e^{e^{4}} - 2 e^{e^{4}} + 1 + e^{2 e^{4}} + 2 e e^{2 e^{4}} + e^{2} e^{2 e^{4}}\right ) e^{- 2 x}}{e^{2 e^{4}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (25) = 50\).
Time = 0.19 (sec) , antiderivative size = 157, normalized size of antiderivative = 6.04 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx=2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} + 2 \, {\left (2 \, x e + e\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} - {\left (2 \, x e + e\right )} e^{\left (-2 \, x - e^{4}\right )} - {\left (2 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} + \frac {1}{2} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} + \frac {1}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} - e^{\left (-2 \, x\right )} - e^{\left (-2 \, x - e^{4} + 1\right )} + e^{\left (-2 \, x - e^{4}\right )} + e^{\left (-2 \, x + 2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.42 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx={\left (4 \, x^{2} + 4 \, x + 1\right )} e^{\left (-2 \, x\right )} - 2 \, {\left (x + 1\right )} e^{\left (-2 \, x - e^{4} + 1\right )} - 2 \, {\left (2 \, x^{2} + 3 \, x + 1\right )} e^{\left (-2 \, x - e^{4}\right )} + {\left (x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x - 2 \, e^{4}\right )} + 2 \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x + 1\right )} + e^{\left (-2 \, x + 2\right )} \]
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Time = 12.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.19 \[ \int e^{-2 e^4-2 x} \left (-2 x-2 x^2+e^{2 e^4} \left (2-2 e^2-8 e x-8 x^2\right )+e^{e^4} \left (-2+4 x+8 x^2+e (2+4 x)\right )\right ) \, dx={\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^4}\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^4}-2\,{\mathrm {e}}^{{\mathrm {e}}^4+1}+2\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+1}+{\mathrm {e}}^{2\,{\mathrm {e}}^4+2}-2\,{\mathrm {e}}^{{\mathrm {e}}^4}+1\right )+x^2\,{\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^4}\,{\left (2\,{\mathrm {e}}^{{\mathrm {e}}^4}-1\right )}^2+x\,{\mathrm {e}}^{-2\,x-2\,{\mathrm {e}}^4}\,\left (4\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}-2\,{\mathrm {e}}^{{\mathrm {e}}^4+1}+4\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+1}-6\,{\mathrm {e}}^{{\mathrm {e}}^4}+2\right ) \]
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