Integrand size = 9, antiderivative size = 32 \[ \int e^{-2 x} x^2 \, dx=-\frac {1}{4} e^{-2 x}-\frac {1}{2} e^{-2 x} x-\frac {1}{2} e^{-2 x} x^2 \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2207, 2225} \[ \int e^{-2 x} x^2 \, dx=-\frac {1}{2} e^{-2 x} x^2-\frac {1}{2} e^{-2 x} x-\frac {e^{-2 x}}{4} \]
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Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} e^{-2 x} x^2+\int e^{-2 x} x \, dx \\ & = -\frac {1}{2} e^{-2 x} x-\frac {1}{2} e^{-2 x} x^2+\frac {1}{2} \int e^{-2 x} \, dx \\ & = -\frac {1}{4} e^{-2 x}-\frac {1}{2} e^{-2 x} x-\frac {1}{2} e^{-2 x} x^2 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int e^{-2 x} x^2 \, dx=-\frac {1}{4} e^{-2 x} \left (1+2 x+2 x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.50
method | result | size |
risch | \(\left (-\frac {1}{2} x^{2}-\frac {1}{2} x -\frac {1}{4}\right ) {\mathrm e}^{-2 x}\) | \(16\) |
norman | \(\left (-\frac {1}{2} x^{2}-\frac {1}{2} x -\frac {1}{4}\right ) {\mathrm e}^{-2 x}\) | \(18\) |
gosper | \(-\frac {\left (2 x^{2}+2 x +1\right ) {\mathrm e}^{-2 x}}{4}\) | \(19\) |
meijerg | \(\frac {1}{4}-\frac {\left (12 x^{2}+12 x +6\right ) {\mathrm e}^{-2 x}}{24}\) | \(19\) |
parallelrisch | \(\frac {\left (-2 x^{2}-2 x -1\right ) {\mathrm e}^{-2 x}}{4}\) | \(19\) |
derivativedivides | \(-\frac {{\mathrm e}^{-2 x}}{4}-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {x^{2} {\mathrm e}^{-2 x}}{2}\) | \(30\) |
default | \(-\frac {{\mathrm e}^{-2 x}}{4}-\frac {x \,{\mathrm e}^{-2 x}}{2}-\frac {x^{2} {\mathrm e}^{-2 x}}{2}\) | \(30\) |
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Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.50 \[ \int e^{-2 x} x^2 \, dx=-\frac {1}{4} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.53 \[ \int e^{-2 x} x^2 \, dx=\frac {\left (- 2 x^{2} - 2 x - 1\right ) e^{- 2 x}}{4} \]
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Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.50 \[ \int e^{-2 x} x^2 \, dx=-\frac {1}{4} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \]
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Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.50 \[ \int e^{-2 x} x^2 \, dx=-\frac {1}{4} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \]
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Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.50 \[ \int e^{-2 x} x^2 \, dx=-\frac {{\mathrm {e}}^{-2\,x}\,\left (4\,x^2+4\,x+2\right )}{8} \]
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