Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{2} \left (2+e^{x/2}+2 e^4 \log \left (\frac {9}{4}\right )\right ) \, dx=4+e^{x/2}+x+e^4 \left (2+x \log \left (\frac {9}{4}\right )\right ) \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 2225} \[ \int \frac {1}{2} \left (2+e^{x/2}+2 e^4 \log \left (\frac {9}{4}\right )\right ) \, dx=e^{x/2}+x \left (1+e^4 \log \left (\frac {9}{4}\right )\right ) \]
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Rule 12
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \left (2+e^{x/2}+2 e^4 \log \left (\frac {9}{4}\right )\right ) \, dx \\ & = x \left (1+e^4 \log \left (\frac {9}{4}\right )\right )+\frac {1}{2} \int e^{x/2} \, dx \\ & = e^{x/2}+x \left (1+e^4 \log \left (\frac {9}{4}\right )\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{2} \left (2+e^{x/2}+2 e^4 \log \left (\frac {9}{4}\right )\right ) \, dx=e^{x/2}+x+e^4 x \log \left (\frac {9}{4}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64
method | result | size |
default | \(x -{\mathrm e}^{4} \ln \left (\frac {4}{9}\right ) x +{\mathrm e}^{\frac {x}{2}}\) | \(14\) |
parts | \(x -{\mathrm e}^{4} \ln \left (\frac {4}{9}\right ) x +{\mathrm e}^{\frac {x}{2}}\) | \(14\) |
parallelrisch | \({\mathrm e}^{\frac {x}{2}}+\left (-{\mathrm e}^{4} \ln \left (\frac {4}{9}\right )+1\right ) x\) | \(16\) |
derivativedivides | \({\mathrm e}^{\frac {x}{2}}+\left (-2 \,{\mathrm e}^{4} \ln \left (\frac {4}{9}\right )+2\right ) \ln \left ({\mathrm e}^{\frac {x}{2}}\right )\) | \(20\) |
risch | \(2 \,{\mathrm e}^{4} x \ln \left (3\right )-2 x \,{\mathrm e}^{4} \ln \left (2\right )+{\mathrm e}^{\frac {x}{2}}+x\) | \(21\) |
norman | \(\left (2 \,{\mathrm e}^{4} \ln \left (3\right )-2 \,{\mathrm e}^{4} \ln \left (2\right )+1\right ) x +{\mathrm e}^{\frac {x}{2}}\) | \(22\) |
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Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {1}{2} \left (2+e^{x/2}+2 e^4 \log \left (\frac {9}{4}\right )\right ) \, dx=-x e^{4} \log \left (\frac {4}{9}\right ) + x + e^{\left (\frac {1}{2} \, x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{2} \left (2+e^{x/2}+2 e^4 \log \left (\frac {9}{4}\right )\right ) \, dx=x \left (- 2 e^{4} \log {\left (2 \right )} + 1 + 2 e^{4} \log {\left (3 \right )}\right ) + e^{\frac {x}{2}} \]
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Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {1}{2} \left (2+e^{x/2}+2 e^4 \log \left (\frac {9}{4}\right )\right ) \, dx=-x e^{4} \log \left (\frac {4}{9}\right ) + x + e^{\left (\frac {1}{2} \, x\right )} \]
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Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {1}{2} \left (2+e^{x/2}+2 e^4 \log \left (\frac {9}{4}\right )\right ) \, dx=-x e^{4} \log \left (\frac {4}{9}\right ) + x + e^{\left (\frac {1}{2} \, x\right )} \]
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Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {1}{2} \left (2+e^{x/2}+2 e^4 \log \left (\frac {9}{4}\right )\right ) \, dx={\mathrm {e}}^{x/2}-x\,\left ({\mathrm {e}}^4\,\ln \left (\frac {4}{9}\right )-1\right ) \]
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