Integrand size = 21, antiderivative size = 25 \[ \int \left (e^x (8+4 x)+e^x (-2-2 x) \log (x)\right ) \, dx=2 e^x \left (5-x \left (\frac {2 \left (x-x^2\right )}{x^2}+\log (x)\right )\right ) \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2207, 2225, 2634, 12} \[ \int \left (e^x (8+4 x)+e^x (-2-2 x) \log (x)\right ) \, dx=4 e^x (x+2)-2 e^x+2 e^x \log (x)-2 e^x (x+1) \log (x) \]
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Rule 12
Rule 2207
Rule 2225
Rule 2634
Rubi steps \begin{align*} \text {integral}& = \int e^x (8+4 x) \, dx+\int e^x (-2-2 x) \log (x) \, dx \\ & = 4 e^x (2+x)+2 e^x \log (x)-2 e^x (1+x) \log (x)-4 \int e^x \, dx-\int -2 e^x \, dx \\ & = -4 e^x+4 e^x (2+x)+2 e^x \log (x)-2 e^x (1+x) \log (x)+2 \int e^x \, dx \\ & = -2 e^x+4 e^x (2+x)+2 e^x \log (x)-2 e^x (1+x) \log (x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int \left (e^x (8+4 x)+e^x (-2-2 x) \log (x)\right ) \, dx=-2 e^x (-3-2 x+x \log (x)) \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72
method | result | size |
default | \(4 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}-2 x \,{\mathrm e}^{x} \ln \left (x \right )\) | \(18\) |
norman | \(4 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}-2 x \,{\mathrm e}^{x} \ln \left (x \right )\) | \(18\) |
risch | \(4 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}-2 x \,{\mathrm e}^{x} \ln \left (x \right )\) | \(18\) |
parallelrisch | \(4 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{x}-2 x \,{\mathrm e}^{x} \ln \left (x \right )\) | \(18\) |
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none
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \left (e^x (8+4 x)+e^x (-2-2 x) \log (x)\right ) \, dx=-2 \, x e^{x} \log \left (x\right ) + 2 \, {\left (2 \, x + 3\right )} e^{x} \]
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Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int \left (e^x (8+4 x)+e^x (-2-2 x) \log (x)\right ) \, dx=\left (- 2 x \log {\left (x \right )} + 4 x + 6\right ) e^{x} \]
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none
Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (e^x (8+4 x)+e^x (-2-2 x) \log (x)\right ) \, dx=-2 \, x e^{x} \log \left (x\right ) + 4 \, {\left (x - 1\right )} e^{x} + 10 \, e^{x} \]
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none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \left (e^x (8+4 x)+e^x (-2-2 x) \log (x)\right ) \, dx=-2 \, x e^{x} \log \left (x\right ) + 4 \, {\left (x + 1\right )} e^{x} + 2 \, e^{x} \]
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Time = 8.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.56 \[ \int \left (e^x (8+4 x)+e^x (-2-2 x) \log (x)\right ) \, dx=2\,{\mathrm {e}}^x\,\left (2\,x-x\,\ln \left (x\right )+3\right ) \]
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