Integrand size = 37, antiderivative size = 22 \[ \int e^{-1-e^x} \left (e^{1+e^x}-18 e^x+80 e^{2 x}-32 e^{3 x}\right ) \, dx=2 e^{-1-e^x} \left (-1+4 e^x\right )^2+x \]
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Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {2320, 14, 2227, 2225, 2207} \[ \int e^{-1-e^x} \left (e^{1+e^x}-18 e^x+80 e^{2 x}-32 e^{3 x}\right ) \, dx=x+2 e^{-e^x-1}-16 e^{x-e^x-1}+32 e^{2 x-e^x-1} \]
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Rule 14
Rule 2207
Rule 2225
Rule 2227
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1-2 e^{-1-x} x \left (9-40 x+16 x^2\right )}{x} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{x}-2 e^{-1-x} (-9+4 x) (-1+4 x)\right ) \, dx,x,e^x\right ) \\ & = x-2 \text {Subst}\left (\int e^{-1-x} (-9+4 x) (-1+4 x) \, dx,x,e^x\right ) \\ & = x-2 \text {Subst}\left (\int \left (9 e^{-1-x}-40 e^{-1-x} x+16 e^{-1-x} x^2\right ) \, dx,x,e^x\right ) \\ & = x-18 \text {Subst}\left (\int e^{-1-x} \, dx,x,e^x\right )-32 \text {Subst}\left (\int e^{-1-x} x^2 \, dx,x,e^x\right )+80 \text {Subst}\left (\int e^{-1-x} x \, dx,x,e^x\right ) \\ & = 18 e^{-1-e^x}-80 e^{-1-e^x+x}+32 e^{-1-e^x+2 x}+x-64 \text {Subst}\left (\int e^{-1-x} x \, dx,x,e^x\right )+80 \text {Subst}\left (\int e^{-1-x} \, dx,x,e^x\right ) \\ & = -62 e^{-1-e^x}-16 e^{-1-e^x+x}+32 e^{-1-e^x+2 x}+x-64 \text {Subst}\left (\int e^{-1-x} \, dx,x,e^x\right ) \\ & = 2 e^{-1-e^x}-16 e^{-1-e^x+x}+32 e^{-1-e^x+2 x}+x \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int e^{-1-e^x} \left (e^{1+e^x}-18 e^x+80 e^{2 x}-32 e^{3 x}\right ) \, dx=e^{-e^x} \left (\frac {2}{e}-16 e^{-1+x}+32 e^{-1+2 x}\right )+x \]
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Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
risch | \(x +\left (32 \,{\mathrm e}^{2 x}-16 \,{\mathrm e}^{x}+2\right ) {\mathrm e}^{-{\mathrm e}^{x}-1}\) | \(23\) |
parallelrisch | \({\mathrm e}^{-4} \left ({\mathrm e}^{4} x \,{\mathrm e}^{{\mathrm e}^{x}-3}+2+32 \,{\mathrm e}^{2 x}-16 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}\) | \(34\) |
norman | \(\left (x \,{\mathrm e}^{{\mathrm e}^{x}-3}+2 \,{\mathrm e}^{-4}+32 \,{\mathrm e}^{-4} {\mathrm e}^{2 x}-16 \,{\mathrm e}^{x} {\mathrm e}^{-4}\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}\) | \(41\) |
derivativedivides | \({\mathrm e}^{-4} \left ({\mathrm e}^{4} \ln \left ({\mathrm e}^{x}\right )+402 \,{\mathrm e}^{-{\mathrm e}^{x}+3}+208 \left ({\mathrm e}^{x}-5\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}+32 \left (\left ({\mathrm e}^{x}-3\right )^{2}+11-{\mathrm e}^{x}\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}\right )\) | \(56\) |
default | \({\mathrm e}^{-4} \left ({\mathrm e}^{4} \ln \left ({\mathrm e}^{x}\right )+402 \,{\mathrm e}^{-{\mathrm e}^{x}+3}+208 \left ({\mathrm e}^{x}-5\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}+32 \left (\left ({\mathrm e}^{x}-3\right )^{2}+11-{\mathrm e}^{x}\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}\right )\) | \(56\) |
parts | \(x +18 \,{\mathrm e}^{-4} {\mathrm e}^{-{\mathrm e}^{x}+3}+80 \,{\mathrm e}^{-4} \left (-\left ({\mathrm e}^{x}-3\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}-4 \,{\mathrm e}^{-{\mathrm e}^{x}+3}\right )-32 \,{\mathrm e}^{-4} \left (-{\mathrm e}^{-{\mathrm e}^{x}+3} \left ({\mathrm e}^{x}-3\right )^{2}-8 \left ({\mathrm e}^{x}-3\right ) {\mathrm e}^{-{\mathrm e}^{x}+3}-17 \,{\mathrm e}^{-{\mathrm e}^{x}+3}\right )\) | \(89\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int e^{-1-e^x} \left (e^{1+e^x}-18 e^x+80 e^{2 x}-32 e^{3 x}\right ) \, dx={\left (x e^{\left (e^{x} + 1\right )} + 32 \, e^{\left (2 \, x\right )} - 16 \, e^{x} + 2\right )} e^{\left (-e^{x} - 1\right )} \]
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Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int e^{-1-e^x} \left (e^{1+e^x}-18 e^x+80 e^{2 x}-32 e^{3 x}\right ) \, dx=x + \frac {\left (32 e^{2 x} - 16 e^{x} + 2\right ) e^{3 - e^{x}}}{e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (19) = 38\).
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.95 \[ \int e^{-1-e^x} \left (e^{1+e^x}-18 e^x+80 e^{2 x}-32 e^{3 x}\right ) \, dx=32 \, {\left (e^{\left (2 \, x\right )} + 2 \, e^{x} + 2\right )} e^{\left (-e^{x} - 1\right )} - 80 \, {\left (e^{x} + 1\right )} e^{\left (-e^{x} - 1\right )} + x + 18 \, e^{\left (-e^{x} - 1\right )} \]
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int e^{-1-e^x} \left (e^{1+e^x}-18 e^x+80 e^{2 x}-32 e^{3 x}\right ) \, dx={\left (x e + 32 \, e^{\left (2 \, x - e^{x}\right )} - 16 \, e^{\left (x - e^{x}\right )} + 2 \, e^{\left (-e^{x}\right )}\right )} e^{\left (-1\right )} \]
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Time = 8.37 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int e^{-1-e^x} \left (e^{1+e^x}-18 e^x+80 e^{2 x}-32 e^{3 x}\right ) \, dx=x+2\,{\mathrm {e}}^{-{\mathrm {e}}^x-1}-16\,{\mathrm {e}}^{x-{\mathrm {e}}^x-1}+32\,{\mathrm {e}}^{2\,x-{\mathrm {e}}^x-1} \]
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