Integrand size = 30, antiderivative size = 30 \[ \int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx=i \pi -x \left (x+e^{-4 e^x} x\right )+\log \left (-2+e^{e^{4 e}}\right ) \]
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\[ \int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx=\int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int 2 e^{-4 e^x} x \left (-1-e^{4 e^x}+2 e^x x\right ) \, dx \\ & = 2 \int e^{-4 e^x} x \left (-1-e^{4 e^x}+2 e^x x\right ) \, dx \\ & = 2 \int \left (-e^{-4 e^x} \left (1+e^{4 e^x}\right ) x+2 e^{-4 e^x+x} x^2\right ) \, dx \\ & = -\left (2 \int e^{-4 e^x} \left (1+e^{4 e^x}\right ) x \, dx\right )+4 \int e^{-4 e^x+x} x^2 \, dx \\ & = -\left (2 \int \left (1+e^{-4 e^x}\right ) x \, dx\right )+4 \int e^{-4 e^x+x} x^2 \, dx \\ & = -\left (2 \int \left (x+e^{-4 e^x} x\right ) \, dx\right )+4 \int e^{-4 e^x+x} x^2 \, dx \\ & = -x^2-2 \int e^{-4 e^x} x \, dx+4 \int e^{-4 e^x+x} x^2 \, dx \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.47 \[ \int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx=-\left (\left (1+e^{-4 e^x}\right ) x^2\right ) \]
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Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-x^{2}-x^{2} {\mathrm e}^{-4 \,{\mathrm e}^{x}}\) | \(17\) |
norman | \(\left (-x^{2}-x^{2} {\mathrm e}^{4 \,{\mathrm e}^{x}}\right ) {\mathrm e}^{-4 \,{\mathrm e}^{x}}\) | \(25\) |
parallelrisch | \(\left (-x^{2}-x^{2} {\mathrm e}^{4 \,{\mathrm e}^{x}}\right ) {\mathrm e}^{-4 \,{\mathrm e}^{x}}\) | \(25\) |
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Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx=-{\left (x^{2} e^{\left (4 \, e^{x}\right )} + x^{2}\right )} e^{\left (-4 \, e^{x}\right )} \]
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Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.47 \[ \int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx=- x^{2} - x^{2} e^{- 4 e^{x}} \]
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Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.53 \[ \int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx=-x^{2} e^{\left (-4 \, e^{x}\right )} - x^{2} \]
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx=-{\left (x^{2} e^{\left (x - 4 \, e^{x}\right )} + x^{2} e^{x}\right )} e^{\left (-x\right )} \]
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Time = 11.73 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.40 \[ \int e^{-4 e^x} \left (-2 x-2 e^{4 e^x} x+4 e^x x^2\right ) \, dx=-x^2\,\left ({\mathrm {e}}^{-4\,{\mathrm {e}}^x}+1\right ) \]
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