Integrand size = 39, antiderivative size = 17 \[ \int e^{2-4 e-8 x+4 x^2+(-8+8 x) \log (4)+4 \log ^2(4)} (-8+8 x+8 \log (4)) \, dx=e^{-2-4 e+(2-2 (x+\log (4)))^2} \]
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Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2276, 2268} \[ \int e^{2-4 e-8 x+4 x^2+(-8+8 x) \log (4)+4 \log ^2(4)} (-8+8 x+8 \log (4)) \, dx=\exp \left (4 x^2-8 x (1-\log (4))+2 \left (1-2 e+2 \log ^2(4)-4 \log (4)\right )\right ) \]
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Rule 2268
Rule 2276
Rubi steps \begin{align*} \text {integral}& = \int \exp \left (4 x^2-8 x (1-\log (4))+2 \left (1-2 e-4 \log (4)+2 \log ^2(4)\right )\right ) (8 x-8 (1-\log (4))) \, dx \\ & = \exp \left (4 x^2-8 x (1-\log (4))+2 \left (1-2 e-4 \log (4)+2 \log ^2(4)\right )\right ) \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int e^{2-4 e-8 x+4 x^2+(-8+8 x) \log (4)+4 \log ^2(4)} (-8+8 x+8 \log (4)) \, dx=65536^{-1+x} e^{2-4 e-8 x+4 x^2-16 \log ^2(2)+8 \log ^2(4)} \]
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76
method | result | size |
risch | \(2^{16 x -16} {\mathrm e}^{16 \ln \left (2\right )^{2}+2-4 \,{\mathrm e}+4 x^{2}-8 x}\) | \(30\) |
gosper | \({\mathrm e}^{16 \ln \left (2\right )^{2}+16 x \ln \left (2\right )+4 x^{2}-16 \ln \left (2\right )-4 \,{\mathrm e}-8 x +2}\) | \(31\) |
derivativedivides | \({\mathrm e}^{16 \ln \left (2\right )^{2}+2 \left (8 x -8\right ) \ln \left (2\right )-4 \,{\mathrm e}+4 x^{2}-8 x +2}\) | \(31\) |
norman | \({\mathrm e}^{16 \ln \left (2\right )^{2}+2 \left (8 x -8\right ) \ln \left (2\right )-4 \,{\mathrm e}+4 x^{2}-8 x +2}\) | \(31\) |
parallelrisch | \({\mathrm e}^{16 \ln \left (2\right )^{2}+2 \left (8 x -8\right ) \ln \left (2\right )-4 \,{\mathrm e}+4 x^{2}-8 x +2}\) | \(31\) |
default | \({\mathrm e}^{4 x^{2}+\left (16 \ln \left (2\right )-8\right ) x +16 \ln \left (2\right )^{2}-16 \ln \left (2\right )-4 \,{\mathrm e}+2}+\frac {i \left (16 \ln \left (2\right )-8\right ) \sqrt {\pi }\, {\mathrm e}^{16 \ln \left (2\right )^{2}-16 \ln \left (2\right )-4 \,{\mathrm e}+2-\frac {\left (16 \ln \left (2\right )-8\right )^{2}}{16}} \operatorname {erf}\left (2 i x +\frac {i \left (16 \ln \left (2\right )-8\right )}{4}\right )}{4}-4 i \ln \left (2\right ) \sqrt {\pi }\, {\mathrm e}^{16 \ln \left (2\right )^{2}-16 \ln \left (2\right )-4 \,{\mathrm e}+2-\frac {\left (16 \ln \left (2\right )-8\right )^{2}}{16}} \operatorname {erf}\left (2 i x +\frac {i \left (16 \ln \left (2\right )-8\right )}{4}\right )+2 i \sqrt {\pi }\, {\mathrm e}^{16 \ln \left (2\right )^{2}-16 \ln \left (2\right )-4 \,{\mathrm e}+2-\frac {\left (16 \ln \left (2\right )-8\right )^{2}}{16}} \operatorname {erf}\left (2 i x +\frac {i \left (16 \ln \left (2\right )-8\right )}{4}\right )\) | \(184\) |
parts | \(-4 i \ln \left (2\right ) \sqrt {\pi }\, {\mathrm e}^{16 \ln \left (2\right )^{2}-16 \ln \left (2\right )-4 \,{\mathrm e}+2-\frac {\left (16 \ln \left (2\right )-8\right )^{2}}{16}} \operatorname {erf}\left (2 i x +\frac {i \left (16 \ln \left (2\right )-8\right )}{4}\right )-2 i \sqrt {\pi }\, {\mathrm e}^{16 \ln \left (2\right )^{2}-16 \ln \left (2\right )-4 \,{\mathrm e}+2-\frac {\left (16 \ln \left (2\right )-8\right )^{2}}{16}} \operatorname {erf}\left (2 i x +\frac {i \left (16 \ln \left (2\right )-8\right )}{4}\right ) x +2 i \sqrt {\pi }\, {\mathrm e}^{16 \ln \left (2\right )^{2}-16 \ln \left (2\right )-4 \,{\mathrm e}+2-\frac {\left (16 \ln \left (2\right )-8\right )^{2}}{16}} \operatorname {erf}\left (2 i x +\frac {i \left (16 \ln \left (2\right )-8\right )}{4}\right )+\sqrt {\pi }\, {\mathrm e}^{16 \ln \left (2\right )^{2}-16 \ln \left (2\right )-4 \,{\mathrm e}+2-\frac {\left (16 \ln \left (2\right )-8\right )^{2}}{16}} \left (\operatorname {erf}\left (2 i x +\frac {i \left (16 \ln \left (2\right )-8\right )}{4}\right ) \left (2 i x +\frac {i \left (16 \ln \left (2\right )-8\right )}{4}\right )+\frac {{\mathrm e}^{-\left (2 i x +\frac {i \left (16 \ln \left (2\right )-8\right )}{4}\right )^{2}}}{\sqrt {\pi }}\right )\) | \(234\) |
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Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.65 \[ \int e^{2-4 e-8 x+4 x^2+(-8+8 x) \log (4)+4 \log ^2(4)} (-8+8 x+8 \log (4)) \, dx=e^{\left (4 \, x^{2} + 16 \, {\left (x - 1\right )} \log \left (2\right ) + 16 \, \log \left (2\right )^{2} - 8 \, x - 4 \, e + 2\right )} \]
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Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int e^{2-4 e-8 x+4 x^2+(-8+8 x) \log (4)+4 \log ^2(4)} (-8+8 x+8 \log (4)) \, dx=e^{4 x^{2} - 8 x + \left (16 x - 16\right ) \log {\left (2 \right )} - 4 e + 2 + 16 \log {\left (2 \right )}^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.65 \[ \int e^{2-4 e-8 x+4 x^2+(-8+8 x) \log (4)+4 \log ^2(4)} (-8+8 x+8 \log (4)) \, dx=e^{\left (4 \, x^{2} + 16 \, {\left (x - 1\right )} \log \left (2\right ) + 16 \, \log \left (2\right )^{2} - 8 \, x - 4 \, e + 2\right )} \]
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Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76 \[ \int e^{2-4 e-8 x+4 x^2+(-8+8 x) \log (4)+4 \log ^2(4)} (-8+8 x+8 \log (4)) \, dx=e^{\left (4 \, x^{2} + 16 \, x \log \left (2\right ) + 16 \, \log \left (2\right )^{2} - 8 \, x - 4 \, e - 16 \, \log \left (2\right ) + 2\right )} \]
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Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int e^{2-4 e-8 x+4 x^2+(-8+8 x) \log (4)+4 \log ^2(4)} (-8+8 x+8 \log (4)) \, dx=\frac {2^{16\,x}\,{\mathrm {e}}^{-4\,\mathrm {e}}\,{\mathrm {e}}^{-8\,x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{16\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{4\,x^2}}{65536} \]
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