Integrand size = 20, antiderivative size = 22 \[ \int \frac {1}{12} e^{-x} \left (48-12 e+e^x-48 x\right ) \, dx=\frac {1}{3} \left (\frac {x}{4}+3 e^{-x} (e+4 x)\right ) \]
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Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 6873, 6874, 2225, 2207} \[ \int \frac {1}{12} e^{-x} \left (48-12 e+e^x-48 x\right ) \, dx=4 e^{-x} x+\frac {x}{12}-(4-e) e^{-x}+4 e^{-x} \]
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Rule 12
Rule 2207
Rule 2225
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} \int e^{-x} \left (48-12 e+e^x-48 x\right ) \, dx \\ & = \frac {1}{12} \int e^{-x} \left (48 \left (1-\frac {e}{4}\right )+e^x-48 x\right ) \, dx \\ & = \frac {1}{12} \int \left (1-12 (-4+e) e^{-x}-48 e^{-x} x\right ) \, dx \\ & = \frac {x}{12}-4 \int e^{-x} x \, dx+(4-e) \int e^{-x} \, dx \\ & = -\left ((4-e) e^{-x}\right )+\frac {x}{12}+4 e^{-x} x-4 \int e^{-x} \, dx \\ & = 4 e^{-x}-(4-e) e^{-x}+\frac {x}{12}+4 e^{-x} x \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{12} e^{-x} \left (48-12 e+e^x-48 x\right ) \, dx=\frac {1}{12} \left (x+12 e^{-x} (e+4 x)\right ) \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77
method | result | size |
norman | \(\left (4 x +\frac {{\mathrm e}^{x} x}{12}+{\mathrm e}\right ) {\mathrm e}^{-x}\) | \(17\) |
risch | \(\frac {x}{12}+{\mathrm e}^{1-x}+4 x \,{\mathrm e}^{-x}\) | \(18\) |
default | \(\frac {x}{12}+{\mathrm e}^{-x} {\mathrm e}+4 x \,{\mathrm e}^{-x}\) | \(19\) |
parallelrisch | \(\frac {\left ({\mathrm e}^{x} x +12 \,{\mathrm e}+48 x \right ) {\mathrm e}^{-x}}{12}\) | \(19\) |
parts | \(\frac {x}{12}+{\mathrm e}^{-x} {\mathrm e}+4 x \,{\mathrm e}^{-x}\) | \(19\) |
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Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{12} e^{-x} \left (48-12 e+e^x-48 x\right ) \, dx=\frac {1}{12} \, {\left (x e^{x} + 48 \, x + 12 \, e\right )} e^{\left (-x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.55 \[ \int \frac {1}{12} e^{-x} \left (48-12 e+e^x-48 x\right ) \, dx=\frac {x}{12} + \left (4 x + e\right ) e^{- x} \]
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Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1}{12} e^{-x} \left (48-12 e+e^x-48 x\right ) \, dx=4 \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {1}{12} \, x - 4 \, e^{\left (-x\right )} + e^{\left (-x + 1\right )} \]
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1}{12} e^{-x} \left (48-12 e+e^x-48 x\right ) \, dx=4 \, x e^{\left (-x\right )} + \frac {1}{12} \, x + e^{\left (-x + 1\right )} \]
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Time = 11.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{12} e^{-x} \left (48-12 e+e^x-48 x\right ) \, dx=\frac {x}{12}+{\mathrm {e}}^{-x}\,\mathrm {e}+4\,x\,{\mathrm {e}}^{-x} \]
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