Integrand size = 21, antiderivative size = 19 \[ \int \frac {1}{2} \left (-3 e^{3 x}+e^x (2+x+\log (2))\right ) \, dx=\frac {1}{2} e^x \left (1-e^{2 x}+x+\log (2)\right ) \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2225, 2207} \[ \int \frac {1}{2} \left (-3 e^{3 x}+e^x (2+x+\log (2))\right ) \, dx=-\frac {e^x}{2}-\frac {e^{3 x}}{2}+\frac {1}{2} e^x (x+2+\log (2)) \]
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Rule 12
Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \left (-3 e^{3 x}+e^x (2+x+\log (2))\right ) \, dx \\ & = \frac {1}{2} \int e^x (2+x+\log (2)) \, dx-\frac {3}{2} \int e^{3 x} \, dx \\ & = -\frac {e^{3 x}}{2}+\frac {1}{2} e^x (2+x+\log (2))-\frac {\int e^x \, dx}{2} \\ & = -\frac {e^x}{2}-\frac {e^{3 x}}{2}+\frac {1}{2} e^x (2+x+\log (2)) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {1}{2} \left (-3 e^{3 x}+e^x (2+x+\log (2))\right ) \, dx=\frac {1}{2} \left (-e^{3 x}+e^x (1+x+\log (2))\right ) \]
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Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {{\mathrm e}^{3 x}}{2}+\frac {\left (\ln \left (2\right )+x +1\right ) {\mathrm e}^{x}}{2}\) | \(17\) |
norman | \(\left (\frac {\ln \left (2\right )}{2}+\frac {1}{2}\right ) {\mathrm e}^{x}-\frac {{\mathrm e}^{3 x}}{2}+\frac {{\mathrm e}^{x} x}{2}\) | \(22\) |
default | \(-\frac {{\mathrm e}^{3 x}}{2}+\frac {{\mathrm e}^{x} \ln \left (2\right )}{2}+\frac {{\mathrm e}^{x} x}{2}+\frac {{\mathrm e}^{x}}{2}\) | \(23\) |
parallelrisch | \(\frac {{\mathrm e}^{x} x}{2}-\frac {{\mathrm e}^{x} {\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{x} \ln \left (2\right )}{2}+\frac {{\mathrm e}^{x}}{2}\) | \(25\) |
meijerg | \(1-\frac {{\mathrm e}^{3 x}}{2}-\frac {\left (2-2 x \right ) {\mathrm e}^{x}}{4}-\left (1+\frac {\ln \left (2\right )}{2}\right ) \left (1-{\mathrm e}^{x}\right )\) | \(32\) |
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Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {1}{2} \left (-3 e^{3 x}+e^x (2+x+\log (2))\right ) \, dx=\frac {1}{2} \, {\left (x + \log \left (2\right ) + 1\right )} e^{x} - \frac {1}{2} \, e^{\left (3 \, x\right )} \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {1}{2} \left (-3 e^{3 x}+e^x (2+x+\log (2))\right ) \, dx=\frac {\left (2 x + 2 \log {\left (2 \right )} + 2\right ) e^{x}}{4} - \frac {e^{3 x}}{2} \]
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Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {1}{2} \left (-3 e^{3 x}+e^x (2+x+\log (2))\right ) \, dx=\frac {1}{2} \, {\left (x - 1\right )} e^{x} + \frac {1}{2} \, e^{x} \log \left (2\right ) - \frac {1}{2} \, e^{\left (3 \, x\right )} + e^{x} \]
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {1}{2} \left (-3 e^{3 x}+e^x (2+x+\log (2))\right ) \, dx=\frac {1}{2} \, {\left (x + \log \left (2\right ) + 1\right )} e^{x} - \frac {1}{2} \, e^{\left (3 \, x\right )} \]
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Time = 8.68 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{2} \left (-3 e^{3 x}+e^x (2+x+\log (2))\right ) \, dx=\frac {{\mathrm {e}}^x\,\left (x-{\mathrm {e}}^{2\,x}+\ln \left (2\right )+1\right )}{2} \]
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