Integrand size = 39, antiderivative size = 23 \[ \int e^{-3-2 x} \left (4 e^{2+2 x}+e \left (1+4 x-4 x^2\right )+e (1-2 x) \log (x)\right ) \, dx=-2+\frac {4 x}{e}+e^{-2-2 x} x (2 x+\log (x)) \]
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Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {6873, 6874, 2225, 2207, 2634, 12, 2209, 2230} \[ \int e^{-3-2 x} \left (4 e^{2+2 x}+e \left (1+4 x-4 x^2\right )+e (1-2 x) \log (x)\right ) \, dx=2 e^{-2 x-2} x^2+\frac {4 x}{e}+e^{-2 x-2} x \log (x) \]
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Rule 12
Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2634
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int e^{-2-2 x} \left (1+4 e^{1+2 x}+4 x-4 x^2+\log (x)-2 x \log (x)\right ) \, dx \\ & = \int \left (\frac {4}{e}+e^{-2-2 x}+4 e^{-2-2 x} x-4 e^{-2-2 x} x^2+e^{-2-2 x} \log (x)-2 e^{-2-2 x} x \log (x)\right ) \, dx \\ & = \frac {4 x}{e}-2 \int e^{-2-2 x} x \log (x) \, dx+4 \int e^{-2-2 x} x \, dx-4 \int e^{-2-2 x} x^2 \, dx+\int e^{-2-2 x} \, dx+\int e^{-2-2 x} \log (x) \, dx \\ & = -\frac {1}{2} e^{-2-2 x}+\frac {4 x}{e}-2 e^{-2-2 x} x+2 e^{-2-2 x} x^2+e^{-2-2 x} x \log (x)+2 \int e^{-2-2 x} \, dx+2 \int \frac {e^{-2-2 x} (-1-2 x)}{4 x} \, dx-4 \int e^{-2-2 x} x \, dx-\int -\frac {e^{-2-2 x}}{2 x} \, dx \\ & = -\frac {3}{2} e^{-2-2 x}+\frac {4 x}{e}+2 e^{-2-2 x} x^2+e^{-2-2 x} x \log (x)+\frac {1}{2} \int \frac {e^{-2-2 x}}{x} \, dx+\frac {1}{2} \int \frac {e^{-2-2 x} (-1-2 x)}{x} \, dx-2 \int e^{-2-2 x} \, dx \\ & = -\frac {1}{2} e^{-2-2 x}+\frac {4 x}{e}+2 e^{-2-2 x} x^2+\frac {\text {Ei}(-2 x)}{2 e^2}+e^{-2-2 x} x \log (x)+\frac {1}{2} \int \left (-2 e^{-2-2 x}-\frac {e^{-2-2 x}}{x}\right ) \, dx \\ & = -\frac {1}{2} e^{-2-2 x}+\frac {4 x}{e}+2 e^{-2-2 x} x^2+\frac {\text {Ei}(-2 x)}{2 e^2}+e^{-2-2 x} x \log (x)-\frac {1}{2} \int \frac {e^{-2-2 x}}{x} \, dx-\int e^{-2-2 x} \, dx \\ & = \frac {4 x}{e}+2 e^{-2-2 x} x^2+e^{-2-2 x} x \log (x) \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int e^{-3-2 x} \left (4 e^{2+2 x}+e \left (1+4 x-4 x^2\right )+e (1-2 x) \log (x)\right ) \, dx=e^{-2 (1+x)} x \left (4 e^{1+2 x}+2 x+\log (x)\right ) \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13
method | result | size |
parts | \(\left (x \ln \left (x \right )+2 x^{2}\right ) {\mathrm e}^{-2-2 x}+4 \,{\mathrm e}^{-1} x\) | \(26\) |
risch | \(2 \,{\mathrm e}^{-2-2 x} x^{2}+x \,{\mathrm e}^{-2-2 x} \ln \left (x \right )+4 \,{\mathrm e}^{-1} x\) | \(28\) |
default | \({\mathrm e}^{-1} \left (4 x +\left (x \,{\mathrm e} \ln \left (x \right )+2 x^{2} {\mathrm e}\right ) {\mathrm e}^{-2-2 x}\right )\) | \(31\) |
norman | \(\left (x \ln \left (x \right )+2 x^{2}+4 x \,{\mathrm e}^{-1} {\mathrm e}^{2+2 x}\right ) {\mathrm e}^{-2-2 x}\) | \(31\) |
parallelrisch | \({\mathrm e}^{-1} \left (2 x^{2} {\mathrm e}+x \,{\mathrm e} \ln \left (x \right )+4 x \,{\mathrm e}^{2+2 x}\right ) {\mathrm e}^{-2-2 x}\) | \(35\) |
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int e^{-3-2 x} \left (4 e^{2+2 x}+e \left (1+4 x-4 x^2\right )+e (1-2 x) \log (x)\right ) \, dx={\left (2 \, x^{2} e + x e \log \left (x\right ) + 4 \, x e^{\left (2 \, x + 2\right )}\right )} e^{\left (-2 \, x - 3\right )} \]
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Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int e^{-3-2 x} \left (4 e^{2+2 x}+e \left (1+4 x-4 x^2\right )+e (1-2 x) \log (x)\right ) \, dx=\frac {4 x}{e} + \left (2 x^{2} + x \log {\left (x \right )}\right ) e^{- 2 x - 2} \]
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\[ \int e^{-3-2 x} \left (4 e^{2+2 x}+e \left (1+4 x-4 x^2\right )+e (1-2 x) \log (x)\right ) \, dx=\int { -{\left ({\left (2 \, x - 1\right )} e \log \left (x\right ) + {\left (4 \, x^{2} - 4 \, x - 1\right )} e - 4 \, e^{\left (2 \, x + 2\right )}\right )} e^{\left (-2 \, x - 3\right )} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int e^{-3-2 x} \left (4 e^{2+2 x}+e \left (1+4 x-4 x^2\right )+e (1-2 x) \log (x)\right ) \, dx={\left (2 \, x^{2} e^{\left (-2 \, x + 1\right )} + x e^{\left (-2 \, x + 1\right )} \log \left (x\right ) + 4 \, x e^{2}\right )} e^{\left (-3\right )} \]
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Time = 12.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int e^{-3-2 x} \left (4 e^{2+2 x}+e \left (1+4 x-4 x^2\right )+e (1-2 x) \log (x)\right ) \, dx=4\,x\,{\mathrm {e}}^{-1}+2\,x^2\,{\mathrm {e}}^{-2\,x-2}+x\,{\mathrm {e}}^{-2\,x-2}\,\ln \left (x\right ) \]
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