Integrand size = 28, antiderivative size = 20 \[ \int \frac {e^{-2 x} \left (1-2 x+\left (3-6 x^2\right ) \log (4)\right )}{3 \log (4)} \, dx=e^{-2 x} \left (x+x^2+\frac {x}{3 \log (4)}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 2227, 2225, 2207} \[ \int \frac {e^{-2 x} \left (1-2 x+\left (3-6 x^2\right ) \log (4)\right )}{3 \log (4)} \, dx=e^{-2 x} x^2+e^{-2 x} x+\frac {e^{-2 x} x}{3 \log (4)} \]
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {\int e^{-2 x} \left (1-2 x+\left (3-6 x^2\right ) \log (4)\right ) \, dx}{3 \log (4)} \\ & = \frac {\int \left (e^{-2 x}-2 e^{-2 x} x-3 e^{-2 x} \left (-1+2 x^2\right ) \log (4)\right ) \, dx}{3 \log (4)} \\ & = \frac {\int e^{-2 x} \, dx}{3 \log (4)}-\frac {2 \int e^{-2 x} x \, dx}{3 \log (4)}-\int e^{-2 x} \left (-1+2 x^2\right ) \, dx \\ & = -\frac {e^{-2 x}}{6 \log (4)}+\frac {e^{-2 x} x}{3 \log (4)}-\frac {\int e^{-2 x} \, dx}{3 \log (4)}-\int \left (-e^{-2 x}+2 e^{-2 x} x^2\right ) \, dx \\ & = \frac {e^{-2 x} x}{3 \log (4)}-2 \int e^{-2 x} x^2 \, dx+\int e^{-2 x} \, dx \\ & = -\frac {1}{2} e^{-2 x}+e^{-2 x} x^2+\frac {e^{-2 x} x}{3 \log (4)}-2 \int e^{-2 x} x \, dx \\ & = -\frac {1}{2} e^{-2 x}+e^{-2 x} x+e^{-2 x} x^2+\frac {e^{-2 x} x}{3 \log (4)}-\int e^{-2 x} \, dx \\ & = e^{-2 x} x+e^{-2 x} x^2+\frac {e^{-2 x} x}{3 \log (4)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-2 x} \left (1-2 x+\left (3-6 x^2\right ) \log (4)\right )}{3 \log (4)} \, dx=\frac {e^{-2 x} x (1+\log (64)+x \log (64))}{\log (64)} \]
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Time = 2.54 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25
method | result | size |
gosper | \({\mathrm e}^{\ln \left (\frac {x \left (6 x \ln \left (2\right )+6 \ln \left (2\right )+1\right )}{6 \ln \left (2\right )}\right )-2 x}\) | \(25\) |
risch | \(\frac {\left (2 \left (3 x^{2}+3 x \right ) \ln \left (2\right )+x \right ) {\mathrm e}^{-2 x}}{6 \ln \left (2\right )}\) | \(26\) |
norman | \({\mathrm e}^{\ln \left (\frac {2 \left (3 x^{2}+3 x \right ) \ln \left (2\right )+x}{6 \ln \left (2\right )}\right )-2 x}\) | \(28\) |
parallelrisch | \({\mathrm e}^{\ln \left (\frac {2 \left (3 x^{2}+3 x \right ) \ln \left (2\right )+x}{6 \ln \left (2\right )}\right )-2 x}\) | \(28\) |
default | \(\frac {x \,{\mathrm e}^{-2 x}-3 \,{\mathrm e}^{-2 x} \ln \left (2\right )+\frac {3 \ln \left (2\right ) \left (4 x^{2} {\mathrm e}^{-2 x}+4 x \,{\mathrm e}^{-2 x}+2 \,{\mathrm e}^{-2 x}\right )}{2}}{6 \ln \left (2\right )}\) | \(49\) |
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-2 x} \left (1-2 x+\left (3-6 x^2\right ) \log (4)\right )}{3 \log (4)} \, dx=e^{\left (-2 \, x + \log \left (\frac {6 \, {\left (x^{2} + x\right )} \log \left (2\right ) + x}{6 \, \log \left (2\right )}\right )\right )} \]
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Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-2 x} \left (1-2 x+\left (3-6 x^2\right ) \log (4)\right )}{3 \log (4)} \, dx=\frac {\left (6 x^{2} \log {\left (2 \right )} + x + 6 x \log {\left (2 \right )}\right ) e^{- 2 x}}{6 \log {\left (2 \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (18) = 36\).
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {e^{-2 x} \left (1-2 x+\left (3-6 x^2\right ) \log (4)\right )}{3 \log (4)} \, dx=\frac {6 \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \log \left (2\right ) + {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} - 6 \, e^{\left (-2 \, x\right )} \log \left (2\right ) - e^{\left (-2 \, x\right )}}{12 \, \log \left (2\right )} \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-2 x} \left (1-2 x+\left (3-6 x^2\right ) \log (4)\right )}{3 \log (4)} \, dx=\frac {6 \, x^{2} e^{\left (-2 \, x\right )} \log \left (2\right ) + 6 \, x e^{\left (-2 \, x\right )} \log \left (2\right ) + x e^{\left (-2 \, x\right )}}{6 \, \log \left (2\right )} \]
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Time = 16.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.90 \[ \int \frac {e^{-2 x} \left (1-2 x+\left (3-6 x^2\right ) \log (4)\right )}{3 \log (4)} \, dx=\frac {{\mathrm {e}}^{-2\,x}\,\left (x-3\,\ln \left (2\right )+\ln \left (8\right )+72\,x^2\,{\ln \left (2\right )}^2+36\,x^3\,{\ln \left (2\right )}^2+3\,x\,\ln \left (2\right )+x\,\ln \left (512\right )+36\,x\,{\ln \left (2\right )}^2+6\,x^2\,\ln \left (2\right )+x^2\,\ln \left (64\right )\right )}{6\,\ln \left (2\right )\,\left (6\,\ln \left (2\right )+6\,x\,\ln \left (2\right )+1\right )} \]
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