Integrand size = 29, antiderivative size = 10 \[ \int \frac {e^x (-1-x)}{4+8 e^x x+4 e^{2 x} x^2} \, dx=\frac {1}{4+4 e^x x} \]
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Time = 0.09 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.30, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6820, 12, 6818} \[ \int \frac {e^x (-1-x)}{4+8 e^x x+4 e^{2 x} x^2} \, dx=\frac {1}{4 \left (e^x x+1\right )} \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x (-1-x)}{4 \left (1+e^x x\right )^2} \, dx \\ & = \frac {1}{4} \int \frac {e^x (-1-x)}{\left (1+e^x x\right )^2} \, dx \\ & = \frac {1}{4 \left (1+e^x x\right )} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.30 \[ \int \frac {e^x (-1-x)}{4+8 e^x x+4 e^{2 x} x^2} \, dx=\frac {1}{4 \left (1+e^x x\right )} \]
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Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
norman | \(\frac {1}{4 \,{\mathrm e}^{x} x +4}\) | \(11\) |
risch | \(\frac {1}{4 \,{\mathrm e}^{x} x +4}\) | \(11\) |
parallelrisch | \(\frac {1}{4 \,{\mathrm e}^{x} x +4}\) | \(11\) |
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (-1-x)}{4+8 e^x x+4 e^{2 x} x^2} \, dx=\frac {1}{4 \, {\left (x e^{x} + 1\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {e^x (-1-x)}{4+8 e^x x+4 e^{2 x} x^2} \, dx=\frac {1}{4 x e^{x} + 4} \]
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none
Time = 0.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (-1-x)}{4+8 e^x x+4 e^{2 x} x^2} \, dx=\frac {1}{4 \, {\left (x e^{x} + 1\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^x (-1-x)}{4+8 e^x x+4 e^{2 x} x^2} \, dx=\frac {1}{4 \, {\left (x e^{x} + 1\right )}} \]
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Time = 0.13 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90 \[ \int \frac {e^x (-1-x)}{4+8 e^x x+4 e^{2 x} x^2} \, dx=\frac {1}{4\,x\,{\mathrm {e}}^x+4} \]
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