Integrand size = 24, antiderivative size = 21 \[ \int e^{-\frac {9+e^2 (5+2 x)}{e^2}} (1-2 x) \, dx=2+\frac {4}{e^2}+e^{-5-\frac {9}{e^2}-2 x} x \]
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Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2218, 2207, 2225} \[ \int e^{-\frac {9+e^2 (5+2 x)}{e^2}} (1-2 x) \, dx=\frac {1}{2} e^{-2 x-\frac {9}{e^2}-5}-\frac {1}{2} e^{-2 x-\frac {9}{e^2}-5} (1-2 x) \]
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Rule 2207
Rule 2218
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \int e^{-5-\frac {9}{e^2}-2 x} (1-2 x) \, dx \\ & = -\frac {1}{2} e^{-5-\frac {9}{e^2}-2 x} (1-2 x)-\int e^{-5-\frac {9}{e^2}-2 x} \, dx \\ & = \frac {1}{2} e^{-5-\frac {9}{e^2}-2 x}-\frac {1}{2} e^{-5-\frac {9}{e^2}-2 x} (1-2 x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int e^{-\frac {9+e^2 (5+2 x)}{e^2}} (1-2 x) \, dx=e^{-5-\frac {9}{e^2}-2 x} x \]
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Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
method | result | size |
risch | \(x \,{\mathrm e}^{-\left (2 \,{\mathrm e}^{2} x +5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}}\) | \(19\) |
norman | \(x \,{\mathrm e}^{-\left (\left (5+2 x \right ) {\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}}\) | \(21\) |
parallelrisch | \(x \,{\mathrm e}^{-\left (\left (5+2 x \right ) {\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}}\) | \(21\) |
gosper | \(x \,{\mathrm e}^{-\left (2 \,{\mathrm e}^{2} x +5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}}\) | \(22\) |
meijerg | \(\frac {{\mathrm e}^{-2 x -9 \,{\mathrm e}^{-2}+2 \,{\mathrm e}^{-5} x} \left (1-{\mathrm e}^{-2 \,{\mathrm e}^{-5} x}\right )}{2}-\frac {{\mathrm e}^{-2 x -9 \,{\mathrm e}^{-2}+2 \,{\mathrm e}^{-5} x +5} \left (1-\frac {\left (2+4 \,{\mathrm e}^{-5} x \right ) {\mathrm e}^{-2 \,{\mathrm e}^{-5} x}}{2}\right )}{2}\) | \(62\) |
derivativedivides | \(-\frac {5 \,{\mathrm e}^{-2 x -\left (5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}}}{2}+\frac {{\mathrm e}^{-2 x -\left (5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}} \left (2 x +\left (5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}\right )}{2}-\frac {9 \,{\mathrm e}^{-2 x -\left (5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}} {\mathrm e}^{-2}}{2}\) | \(81\) |
default | \(-\frac {5 \,{\mathrm e}^{-2 x -\left (5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}}}{2}+\frac {{\mathrm e}^{-2 x -\left (5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}} \left (2 x +\left (5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}\right )}{2}-\frac {9 \,{\mathrm e}^{-2 x -\left (5 \,{\mathrm e}^{2}+9\right ) {\mathrm e}^{-2}} {\mathrm e}^{-2}}{2}\) | \(81\) |
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int e^{-\frac {9+e^2 (5+2 x)}{e^2}} (1-2 x) \, dx=x e^{\left (-{\left ({\left (2 \, x + 5\right )} e^{2} + 9\right )} e^{\left (-2\right )}\right )} \]
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Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int e^{-\frac {9+e^2 (5+2 x)}{e^2}} (1-2 x) \, dx=x e^{- \frac {\left (2 x + 5\right ) e^{2} + 9}{e^{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int e^{-\frac {9+e^2 (5+2 x)}{e^2}} (1-2 x) \, dx=\frac {1}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x - 9 \, e^{\left (-2\right )} - 5\right )} - \frac {1}{2} \, e^{\left (-2 \, x - 9 \, e^{\left (-2\right )} - 5\right )} \]
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int e^{-\frac {9+e^2 (5+2 x)}{e^2}} (1-2 x) \, dx=x e^{\left (-{\left (2 \, x e^{2} + 5 \, e^{2} + 9\right )} e^{\left (-2\right )}\right )} \]
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Time = 0.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int e^{-\frac {9+e^2 (5+2 x)}{e^2}} (1-2 x) \, dx=x\,{\mathrm {e}}^{-9\,{\mathrm {e}}^{-2}}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-5} \]
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