Integrand size = 99, antiderivative size = 26 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=\left (2+e^2\right )^2 \left (-3+x+e^{2+x-e^{-x} x} x\right ) \]
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\[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=\int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int e^{-e^{-x} x} \left (2+e^2\right )^2 \left (e^{e^{-x} x}+e^2 (-1+x) x+e^{2+x} (1+x)\right ) \, dx \\ & = \left (2+e^2\right )^2 \int e^{-e^{-x} x} \left (e^{e^{-x} x}+e^2 (-1+x) x+e^{2+x} (1+x)\right ) \, dx \\ & = \left (2+e^2\right )^2 \int \left (1+e^{2-e^{-x} x} (-1+x) x+e^{2+x-e^{-x} x} (1+x)\right ) \, dx \\ & = \left (2+e^2\right )^2 x+\left (2+e^2\right )^2 \int e^{2-e^{-x} x} (-1+x) x \, dx+\left (2+e^2\right )^2 \int e^{2+x-e^{-x} x} (1+x) \, dx \\ & = \left (2+e^2\right )^2 x+\left (2+e^2\right )^2 \int \left (e^{2+x-e^{-x} x}+e^{2+x-e^{-x} x} x\right ) \, dx+\left (2+e^2\right )^2 \int \left (-e^{2-e^{-x} x} x+e^{2-e^{-x} x} x^2\right ) \, dx \\ & = \left (2+e^2\right )^2 x+\left (2+e^2\right )^2 \int e^{2+x-e^{-x} x} \, dx-\left (2+e^2\right )^2 \int e^{2-e^{-x} x} x \, dx+\left (2+e^2\right )^2 \int e^{2+x-e^{-x} x} x \, dx+\left (2+e^2\right )^2 \int e^{2-e^{-x} x} x^2 \, dx \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=\left (2+e^2\right )^2 \left (1+e^{2+x-e^{-x} x}\right ) x \]
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Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62
method | result | size |
risch | \(4 \,{\mathrm e}^{2} x +x \,{\mathrm e}^{4}+4 x +x \left ({\mathrm e}^{4}+4 \,{\mathrm e}^{2}+4\right ) {\mathrm e}^{\left ({\mathrm e}^{x} x +2 \,{\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}\) | \(42\) |
norman | \(\left (\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{2}+4\right ) x \,{\mathrm e}^{x}+\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{2}+4\right ) x \,{\mathrm e}^{x} {\mathrm e}^{\left (\left (2+x \right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\) | \(51\) |
parallelrisch | \({\mathrm e}^{4} {\mathrm e}^{\left (\left (2+x \right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}} x +x \,{\mathrm e}^{4}+4 \,{\mathrm e}^{2} {\mathrm e}^{\left (\left (2+x \right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}} x +4 \,{\mathrm e}^{2} x +4 \,{\mathrm e}^{\left (\left (2+x \right ) {\mathrm e}^{x}-x \right ) {\mathrm e}^{-x}} x +4 x\) | \(78\) |
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=x e^{4} + 4 \, x e^{2} + {\left (x e^{4} + 4 \, x e^{2} + 4 \, x\right )} e^{\left ({\left ({\left (x + 2\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )} + 4 \, x \]
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Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=x \left (4 + 4 e^{2} + e^{4}\right ) + \left (4 x + 4 x e^{2} + x e^{4}\right ) e^{\left (- x + \left (x + 2\right ) e^{x}\right ) e^{- x}} \]
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\[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=\int { {\left ({\left (4 \, x^{2} + {\left (x^{2} - x\right )} e^{4} + 4 \, {\left (x^{2} - x\right )} e^{2} + {\left ({\left (x + 1\right )} e^{4} + 4 \, {\left (x + 1\right )} e^{2} + 4 \, x + 4\right )} e^{x} - 4 \, x\right )} e^{\left ({\left ({\left (x + 2\right )} e^{x} - x\right )} e^{\left (-x\right )}\right )} + {\left (e^{4} + 4 \, e^{2} + 4\right )} e^{x}\right )} e^{\left (-x\right )} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=x e^{4} + 4 \, x e^{2} + x e^{\left (-x e^{\left (-x\right )} + x + 6\right )} + 4 \, x e^{\left (-x e^{\left (-x\right )} + x + 4\right )} + 4 \, x e^{\left (-x e^{\left (-x\right )} + x + 2\right )} + 4 \, x \]
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Time = 9.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int e^{-x} \left (e^x \left (4+4 e^2+e^4\right )+e^{e^{-x} \left (-x+e^x (2+x)\right )} \left (-4 x+4 x^2+e^4 \left (-x+x^2\right )+e^2 \left (-4 x+4 x^2\right )+e^x \left (4+4 x+e^4 (1+x)+e^2 (4+4 x)\right )\right )\right ) \, dx=x\,\left ({\mathrm {e}}^{x-x\,{\mathrm {e}}^{-x}+2}+1\right )\,{\left ({\mathrm {e}}^2+2\right )}^2 \]
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