Integrand size = 39, antiderivative size = 23 \[ \int \frac {e^{-2 x} \left ((-6+12 x) \log (2)+e^{2 x} \left (e^4+6 x \log (2)+\log (2) \log (5)\right )\right )}{\log (2)} \, dx=x \left (-6 e^{-2 x}+3 x+\frac {e^4}{\log (2)}+\log (5)\right ) \]
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Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 6874, 2207, 2225} \[ \int \frac {e^{-2 x} \left ((-6+12 x) \log (2)+e^{2 x} \left (e^4+6 x \log (2)+\log (2) \log (5)\right )\right )}{\log (2)} \, dx=3 x^2-3 e^{-2 x}+3 e^{-2 x} (1-2 x)+\frac {x \left (e^4+\log (2) \log (5)\right )}{\log (2)} \]
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Rule 12
Rule 2207
Rule 2225
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int e^{-2 x} \left ((-6+12 x) \log (2)+e^{2 x} \left (e^4+6 x \log (2)+\log (2) \log (5)\right )\right ) \, dx}{\log (2)} \\ & = \frac {\int \left (e^4+6 x \log (2)+6 e^{-2 x} (-1+2 x) \log (2)+\log (2) \log (5)\right ) \, dx}{\log (2)} \\ & = 3 x^2+\frac {x \left (e^4+\log (2) \log (5)\right )}{\log (2)}+6 \int e^{-2 x} (-1+2 x) \, dx \\ & = 3 e^{-2 x} (1-2 x)+3 x^2+\frac {x \left (e^4+\log (2) \log (5)\right )}{\log (2)}+6 \int e^{-2 x} \, dx \\ & = -3 e^{-2 x}+3 e^{-2 x} (1-2 x)+3 x^2+\frac {x \left (e^4+\log (2) \log (5)\right )}{\log (2)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-2 x} \left ((-6+12 x) \log (2)+e^{2 x} \left (e^4+6 x \log (2)+\log (2) \log (5)\right )\right )}{\log (2)} \, dx=-6 e^{-2 x} x+3 x^2+\frac {e^4 x}{\log (2)}+x \log (5) \]
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Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13
method | result | size |
risch | \(x \ln \left (5\right )+3 x^{2}+\frac {x \,{\mathrm e}^{4}}{\ln \left (2\right )}-6 \,{\mathrm e}^{-2 x} x\) | \(26\) |
parts | \(x \ln \left (5\right )+3 x^{2}+\frac {x \,{\mathrm e}^{4}}{\ln \left (2\right )}-6 \,{\mathrm e}^{-2 x} x\) | \(28\) |
norman | \(\left (\frac {\left (\ln \left (2\right ) \ln \left (5\right )+{\mathrm e}^{4}\right ) x \,{\mathrm e}^{2 x}}{\ln \left (2\right )}-6 x +3 \,{\mathrm e}^{2 x} x^{2}\right ) {\mathrm e}^{-2 x}\) | \(39\) |
parallelrisch | \(\frac {\left (\ln \left (2\right ) \ln \left (5\right ) x \,{\mathrm e}^{2 x}+3 x^{2} \ln \left (2\right ) {\mathrm e}^{2 x}+{\mathrm e}^{2 x} {\mathrm e}^{4} x -6 x \ln \left (2\right )\right ) {\mathrm e}^{-2 x}}{\ln \left (2\right )}\) | \(47\) |
default | \(\frac {x \ln \left (2\right ) \ln \left (5\right )+3 x^{2} \ln \left (2\right )+3 \ln \left (2\right ) {\mathrm e}^{-2 x}+3 \ln \left (2\right ) \left (-2 \,{\mathrm e}^{-2 x} x -{\mathrm e}^{-2 x}\right )+x \,{\mathrm e}^{4}}{\ln \left (2\right )}\) | \(56\) |
derivativedivides | \(\frac {2 x \ln \left (2\right ) \ln \left (5\right )+6 x^{2} \ln \left (2\right )+6 \ln \left (2\right ) {\mathrm e}^{-2 x}+6 \ln \left (2\right ) \left (-2 \,{\mathrm e}^{-2 x} x -{\mathrm e}^{-2 x}\right )+2 x \,{\mathrm e}^{4}}{2 \ln \left (2\right )}\) | \(59\) |
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Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-2 x} \left ((-6+12 x) \log (2)+e^{2 x} \left (e^4+6 x \log (2)+\log (2) \log (5)\right )\right )}{\log (2)} \, dx=\frac {{\left ({\left (3 \, x^{2} \log \left (2\right ) + x \log \left (5\right ) \log \left (2\right ) + x e^{4}\right )} e^{\left (2 \, x\right )} - 6 \, x \log \left (2\right )\right )} e^{\left (-2 \, x\right )}}{\log \left (2\right )} \]
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Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-2 x} \left ((-6+12 x) \log (2)+e^{2 x} \left (e^4+6 x \log (2)+\log (2) \log (5)\right )\right )}{\log (2)} \, dx=3 x^{2} + \frac {x \left (\log {\left (2 \right )} \log {\left (5 \right )} + e^{4}\right )}{\log {\left (2 \right )}} - 6 x e^{- 2 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (21) = 42\).
Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {e^{-2 x} \left ((-6+12 x) \log (2)+e^{2 x} \left (e^4+6 x \log (2)+\log (2) \log (5)\right )\right )}{\log (2)} \, dx=\frac {3 \, x^{2} \log \left (2\right ) - 3 \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} \log \left (2\right ) + x \log \left (5\right ) \log \left (2\right ) + x e^{4} + 3 \, e^{\left (-2 \, x\right )} \log \left (2\right )}{\log \left (2\right )} \]
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Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {e^{-2 x} \left ((-6+12 x) \log (2)+e^{2 x} \left (e^4+6 x \log (2)+\log (2) \log (5)\right )\right )}{\log (2)} \, dx=\frac {3 \, x^{2} \log \left (2\right ) - 6 \, x e^{\left (-2 \, x\right )} \log \left (2\right ) + x \log \left (5\right ) \log \left (2\right ) + x e^{4}}{\log \left (2\right )} \]
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Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {e^{-2 x} \left ((-6+12 x) \log (2)+e^{2 x} \left (e^4+6 x \log (2)+\log (2) \log (5)\right )\right )}{\log (2)} \, dx=x\,\ln \left (5\right )-6\,x\,{\mathrm {e}}^{-2\,x}+3\,x^2+\frac {x\,{\mathrm {e}}^4}{\ln \left (2\right )} \]
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