Integrand size = 7, antiderivative size = 16 \[ \int e^{-x} x \, dx=-e^{-x}-e^{-x} x \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2207, 2225} \[ \int e^{-x} x \, dx=-e^{-x} x-e^{-x} \]
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Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = -e^{-x} x+\int e^{-x} \, dx \\ & = -e^{-x}-e^{-x} x \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int e^{-x} x \, dx=e^{-x} (-1-x) \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(-\left (1+x \right ) {\mathrm e}^{-x}\) | \(10\) |
norman | \(\left (-1-x \right ) {\mathrm e}^{-x}\) | \(11\) |
risch | \(\left (-1-x \right ) {\mathrm e}^{-x}\) | \(11\) |
parallelrisch | \(\left (-1-x \right ) {\mathrm e}^{-x}\) | \(11\) |
meijerg | \(1-\frac {\left (2 x +2\right ) {\mathrm e}^{-x}}{2}\) | \(14\) |
default | \(-{\mathrm e}^{-x}-x \,{\mathrm e}^{-x}\) | \(15\) |
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none
Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int e^{-x} x \, dx=-{\left (x + 1\right )} e^{\left (-x\right )} \]
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Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.44 \[ \int e^{-x} x \, dx=\left (- x - 1\right ) e^{- x} \]
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none
Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int e^{-x} x \, dx=-{\left (x + 1\right )} e^{\left (-x\right )} \]
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none
Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int e^{-x} x \, dx=-{\left (x + 1\right )} e^{\left (-x\right )} \]
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Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int e^{-x} x \, dx=-{\mathrm {e}}^{-x}\,\left (x+1\right ) \]
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