Integrand size = 20, antiderivative size = 22 \[ \int \frac {1}{2} \left (-4+e^{e^3-x}+8 e^x\right ) \, dx=-\frac {1}{2} e^{e^3-x}+4 e^x-2 x \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 2225} \[ \int \frac {1}{2} \left (-4+e^{e^3-x}+8 e^x\right ) \, dx=-2 x-\frac {e^{e^3-x}}{2}+4 e^x \]
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Rule 12
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \left (-4+e^{e^3-x}+8 e^x\right ) \, dx \\ & = -2 x+\frac {1}{2} \int e^{e^3-x} \, dx+4 \int e^x \, dx \\ & = -\frac {1}{2} e^{e^3-x}+4 e^x-2 x \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{2} \left (-4+e^{e^3-x}+8 e^x\right ) \, dx=\frac {1}{2} \left (-e^{e^3-x}+8 e^x-4 x\right ) \]
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Time = 0.47 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {{\mathrm e}^{-x +{\mathrm e}^{3}}}{2}-2 x +4 \,{\mathrm e}^{x}\) | \(18\) |
risch | \(-\frac {{\mathrm e}^{-x +{\mathrm e}^{3}}}{2}-2 x +4 \,{\mathrm e}^{x}\) | \(18\) |
parallelrisch | \(-\frac {{\mathrm e}^{-x +{\mathrm e}^{3}}}{2}-2 x +4 \,{\mathrm e}^{x}\) | \(18\) |
parts | \(-\frac {{\mathrm e}^{-x +{\mathrm e}^{3}}}{2}-2 x +4 \,{\mathrm e}^{x}\) | \(18\) |
norman | \(\left (4 \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x -\frac {{\mathrm e}^{{\mathrm e}^{3}}}{2}\right ) {\mathrm e}^{-x}\) | \(23\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {1}{2} \left (-4+e^{e^3-x}+8 e^x\right ) \, dx=-\frac {1}{2} \, {\left (4 \, x e^{\left (-x + e^{3}\right )} + e^{\left (-2 \, x + 2 \, e^{3}\right )} - 8 \, e^{\left (e^{3}\right )}\right )} e^{\left (x - e^{3}\right )} \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1}{2} \left (-4+e^{e^3-x}+8 e^x\right ) \, dx=- 2 x + 4 e^{x} - \frac {e^{- x} e^{e^{3}}}{2} \]
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Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1}{2} \left (-4+e^{e^3-x}+8 e^x\right ) \, dx=-2 \, x + 4 \, e^{x} - \frac {1}{2} \, e^{\left (-x + e^{3}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1}{2} \left (-4+e^{e^3-x}+8 e^x\right ) \, dx=-2 \, x + 4 \, e^{x} - \frac {1}{2} \, e^{\left (-x + e^{3}\right )} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1}{2} \left (-4+e^{e^3-x}+8 e^x\right ) \, dx=4\,{\mathrm {e}}^x-2\,x-\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^3}}{2} \]
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