Integrand size = 53, antiderivative size = 27 \[ \int \frac {e^5 (5-2 x)-e^5 \log (4)+e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right )}{e^5} \, dx=\left (5-e^{\frac {e^e-2 x^2}{e^5}}-x\right ) (x+\log (4)) \]
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Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 2326} \[ \int \frac {e^5 (5-2 x)-e^5 \log (4)+e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right )}{e^5} \, dx=-\frac {e^{\frac {e^e-2 x^2}{e^5}} \left (x^2+x \log (4)\right )}{x}-\frac {1}{4} (5-2 x)^2-x \log (4) \]
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Rule 12
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (e^5 (5-2 x)-e^5 \log (4)+e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right )\right ) \, dx}{e^5} \\ & = -\frac {1}{4} (5-2 x)^2-x \log (4)+\frac {\int e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right ) \, dx}{e^5} \\ & = -\frac {1}{4} (5-2 x)^2-x \log (4)-\frac {e^{\frac {e^e-2 x^2}{e^5}} \left (x^2+x \log (4)\right )}{x} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^5 (5-2 x)-e^5 \log (4)+e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right )}{e^5} \, dx=-x (-5+x+\log (4))-\frac {1}{4} e^{\frac {e^e-2 x^2}{e^5}} (4 x+\log (256)) \]
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Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63
method | result | size |
risch | \(-2 x \ln \left (2\right )-x^{2}+5 x +\left (-2 \,{\mathrm e}^{5} \ln \left (2\right )-x \,{\mathrm e}^{5}\right ) {\mathrm e}^{-2 x^{2} {\mathrm e}^{-5}+{\mathrm e}^{{\mathrm e}} {\mathrm e}^{-5}-5}\) | \(44\) |
norman | \(\left (-2 \ln \left (2\right )+5\right ) x -x^{2}-x \,{\mathrm e}^{\left ({\mathrm e}^{{\mathrm e}}-2 x^{2}\right ) {\mathrm e}^{-5}}-2 \ln \left (2\right ) {\mathrm e}^{\left ({\mathrm e}^{{\mathrm e}}-2 x^{2}\right ) {\mathrm e}^{-5}}\) | \(52\) |
parts | \(-x^{2}+5 x -{\mathrm e}^{{\mathrm e}^{{\mathrm e}} {\mathrm e}^{-5}} x \,{\mathrm e}^{-2 x^{2} {\mathrm e}^{-5}}-2 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}} {\mathrm e}^{-5}} \ln \left (2\right ) {\mathrm e}^{-2 x^{2} {\mathrm e}^{-5}}-2 x \ln \left (2\right )\) | \(61\) |
parallelrisch | \({\mathrm e}^{-5} \left (-x^{2} {\mathrm e}^{5}-{\mathrm e}^{\left ({\mathrm e}^{{\mathrm e}}-2 x^{2}\right ) {\mathrm e}^{-5}} {\mathrm e}^{5} x +5 x \,{\mathrm e}^{5}-2 \,{\mathrm e}^{\left ({\mathrm e}^{{\mathrm e}}-2 x^{2}\right ) {\mathrm e}^{-5}} {\mathrm e}^{5} \ln \left (2\right )-2 x \,{\mathrm e}^{5} \ln \left (2\right )\right )\) | \(67\) |
default | \({\mathrm e}^{-5} \left ({\mathrm e}^{5} \left (-x^{2}+5 x \right )-{\mathrm e}^{{\mathrm e}^{{\mathrm e}} {\mathrm e}^{-5}} {\mathrm e}^{5} x \,{\mathrm e}^{-2 x^{2} {\mathrm e}^{-5}}-2 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}} {\mathrm e}^{-5}} \ln \left (2\right ) {\mathrm e}^{-2 x^{2} {\mathrm e}^{-5}} {\mathrm e}^{5}-2 x \,{\mathrm e}^{5} \ln \left (2\right )\right )\) | \(76\) |
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Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^5 (5-2 x)-e^5 \log (4)+e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right )}{e^5} \, dx=-x^{2} - {\left (x + 2 \, \log \left (2\right )\right )} e^{\left (-{\left (2 \, x^{2} - e^{e}\right )} e^{\left (-5\right )}\right )} - 2 \, x \log \left (2\right ) + 5 \, x \]
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Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^5 (5-2 x)-e^5 \log (4)+e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right )}{e^5} \, dx=- x^{2} + x \left (5 - 2 \log {\left (2 \right )}\right ) + \left (- x - 2 \log {\left (2 \right )}\right ) e^{\frac {- 2 x^{2} + e^{e}}{e^{5}}} \]
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Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {e^5 (5-2 x)-e^5 \log (4)+e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right )}{e^5} \, dx=-{\left (2 \, x e^{5} \log \left (2\right ) + {\left (x^{2} - 5 \, x\right )} e^{5} + {\left (x e^{\left (e^{\left (e - 5\right )} + 5\right )} + 2 \, e^{\left (e^{\left (e - 5\right )} + 5\right )} \log \left (2\right )\right )} e^{\left (-2 \, x^{2} e^{\left (-5\right )}\right )}\right )} e^{\left (-5\right )} \]
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Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int \frac {e^5 (5-2 x)-e^5 \log (4)+e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right )}{e^5} \, dx=-{\left (2 \, x e^{5} \log \left (2\right ) + {\left (x^{2} - 5 \, x\right )} e^{5} + {\left (x e^{5} + 2 \, e^{5} \log \left (2\right )\right )} e^{\left (-{\left (2 \, x^{2} - e^{e}\right )} e^{\left (-5\right )}\right )}\right )} e^{\left (-5\right )} \]
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Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {e^5 (5-2 x)-e^5 \log (4)+e^{\frac {e^e-2 x^2}{e^5}} \left (-e^5+4 x^2+4 x \log (4)\right )}{e^5} \, dx=5\,x-2\,x\,\ln \left (2\right )-2\,{\mathrm {e}}^{{\mathrm {e}}^{-5}\,{\mathrm {e}}^{\mathrm {e}}-2\,x^2\,{\mathrm {e}}^{-5}}\,\ln \left (2\right )-x\,{\mathrm {e}}^{{\mathrm {e}}^{-5}\,{\mathrm {e}}^{\mathrm {e}}-2\,x^2\,{\mathrm {e}}^{-5}}-x^2 \]
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