Integrand size = 27, antiderivative size = 13 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{\left (-8-e^3\right ) x^2} \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6, 12, 2257, 2240} \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{-\left (\left (8+e^3\right ) x^2\right )} \]
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Rule 6
Rule 12
Rule 2240
Rule 2257
Rubi steps \begin{align*} \text {integral}& = \int e^{-8 x^2-e^3 x^2} \left (-16-2 e^3\right ) x \, dx \\ & = -\left (\left (2 \left (8+e^3\right )\right ) \int e^{-8 x^2-e^3 x^2} x \, dx\right ) \\ & = -\left (\left (2 \left (8+e^3\right )\right ) \int e^{-\left (\left (8+e^3\right ) x^2\right )} x \, dx\right ) \\ & = e^{-\left (\left (8+e^3\right ) x^2\right )} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{-\left (\left (8+e^3\right ) x^2\right )} \]
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Time = 0.14 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85
method | result | size |
risch | \({\mathrm e}^{-x^{2} \left ({\mathrm e}^{3}+8\right )}\) | \(11\) |
parallelrisch | \({\mathrm e}^{-x^{2} \left ({\mathrm e}^{3}+8\right )}\) | \(11\) |
gosper | \({\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}\) | \(15\) |
derivativedivides | \({\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}\) | \(15\) |
norman | \({\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}\) | \(15\) |
meijerg | \(\frac {\left (-2 \,{\mathrm e}^{3}-16\right ) \left (1-{\mathrm e}^{-x^{2} \left ({\mathrm e}^{3}+8\right )}\right )}{2 \,{\mathrm e}^{3}+16}\) | \(29\) |
default | \(\frac {8 \,{\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}}{{\mathrm e}^{3}+8}+\frac {{\mathrm e}^{3} {\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}}{{\mathrm e}^{3}+8}\) | \(47\) |
parts | \(-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {{\mathrm e}^{3}+8}\, x \right ) x \,{\mathrm e}^{3}}{\sqrt {{\mathrm e}^{3}+8}}-\frac {8 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {{\mathrm e}^{3}+8}\, x \right ) x}{\sqrt {{\mathrm e}^{3}+8}}+x \sqrt {\pi }\, \sqrt {{\mathrm e}^{3}+8}\, \operatorname {erf}\left (\sqrt {{\mathrm e}^{3}+8}\, x \right )+{\mathrm e}^{-x^{2} \left ({\mathrm e}^{3}+8\right )}\) | \(76\) |
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Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{\left (-x^{2} e^{3} - 8 \, x^{2}\right )} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{- x^{2} e^{3} - 8 x^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{\left (-x^{2} e^{3} - 8 \, x^{2}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{\left (-x^{2} e^{3} - 8 \, x^{2}\right )} \]
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Time = 14.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx={\mathrm {e}}^{-x^2\,{\mathrm {e}}^3-8\,x^2} \]
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