Integrand size = 49, antiderivative size = 26 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 \left (2 x-\left (-1+e^{2 x-e^x x}-x\right ) x\right ) \]
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\[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=\int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (3 e^{3 x-e^x x} x (1+x)-3 e^{2 x-e^x x} (1+2 x)+3 (3+2 x)\right ) \, dx \\ & = \frac {3}{4} (3+2 x)^2+3 \int e^{3 x-e^x x} x (1+x) \, dx-3 \int e^{2 x-e^x x} (1+2 x) \, dx \\ & = \frac {3}{4} (3+2 x)^2-3 \int \left (e^{2 x-e^x x}+2 e^{2 x-e^x x} x\right ) \, dx+3 \int \left (e^{3 x-e^x x} x+e^{3 x-e^x x} x^2\right ) \, dx \\ & = \frac {3}{4} (3+2 x)^2-3 \int e^{2 x-e^x x} \, dx+3 \int e^{3 x-e^x x} x \, dx+3 \int e^{3 x-e^x x} x^2 \, dx-6 \int e^{2 x-e^x x} x \, dx \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 x \left (3-e^{-\left (\left (-2+e^x\right ) x\right )}+x\right ) \]
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Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81
method | result | size |
risch | \(3 x^{2}+9 x -3 x \,{\mathrm e}^{-x \left ({\mathrm e}^{x}-2\right )}\) | \(21\) |
norman | \(\left (-3 x \,{\mathrm e}^{2 x}+3 x^{2} {\mathrm e}^{{\mathrm e}^{x} x}+9 \,{\mathrm e}^{{\mathrm e}^{x} x} x \right ) {\mathrm e}^{-{\mathrm e}^{x} x}\) | \(35\) |
parallelrisch | \(-\left (-3 x^{2} {\mathrm e}^{{\mathrm e}^{x} x}+3 x \,{\mathrm e}^{2 x}-9 \,{\mathrm e}^{{\mathrm e}^{x} x} x \right ) {\mathrm e}^{-{\mathrm e}^{x} x}\) | \(36\) |
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 \, {\left ({\left (x^{2} + 3 \, x\right )} e^{\left (x e^{x}\right )} - x e^{\left (2 \, x\right )}\right )} e^{\left (-x e^{x}\right )} \]
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Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 x^{2} - 3 x e^{2 x} e^{- x e^{x}} + 9 x \]
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3 \, x^{2} - 3 \, x e^{\left (-x e^{x} + 2 \, x\right )} + 9 \, x \]
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\[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=\int { 3 \, {\left ({\left (2 \, x + 3\right )} e^{\left (x e^{x}\right )} + {\left (x^{2} + x\right )} e^{\left (3 \, x\right )} - {\left (2 \, x + 1\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-x e^{x}\right )} \,d x } \]
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Time = 12.65 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int e^{-e^x x} \left (e^{2 x} (-3-6 x)+e^{e^x x} (9+6 x)+e^{3 x} \left (3 x+3 x^2\right )\right ) \, dx=3\,x\,\left (x-{\mathrm {e}}^{2\,x-x\,{\mathrm {e}}^x}+3\right ) \]
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