Integrand size = 36, antiderivative size = 20 \[ \int e^{-5+x^2} \left (-4 x+e^5 \left (1-12 x-2 e x+2 e^3 x+2 x^2\right )\right ) \, dx=e^{x^2} \left (-6-\frac {2}{e^5}-e+e^3+x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2258, 2235, 2240, 2243} \[ \int e^{-5+x^2} \left (-4 x+e^5 \left (1-12 x-2 e x+2 e^3 x+2 x^2\right )\right ) \, dx=e^{x^2} x-\left (2+6 e^5+e^6-e^8\right ) e^{x^2-5} \]
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Rule 2235
Rule 2240
Rule 2243
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \int \left (e^{x^2}+2 e^{-5+x^2} \left (-2-6 e^5-e^6+e^8\right ) x+2 e^{x^2} x^2\right ) \, dx \\ & = 2 \int e^{x^2} x^2 \, dx-\left (2 \left (2+6 e^5+e^6-e^8\right )\right ) \int e^{-5+x^2} x \, dx+\int e^{x^2} \, dx \\ & = -e^{-5+x^2} \left (2+6 e^5+e^6-e^8\right )+e^{x^2} x+\frac {1}{2} \sqrt {\pi } \text {erfi}(x)-\int e^{x^2} \, dx \\ & = -e^{-5+x^2} \left (2+6 e^5+e^6-e^8\right )+e^{x^2} x \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int e^{-5+x^2} \left (-4 x+e^5 \left (1-12 x-2 e x+2 e^3 x+2 x^2\right )\right ) \, dx=e^{-5+x^2} \left (-2-e^6+e^8+e^5 (-6+x)\right ) \]
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Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\left ({\mathrm e}^{8}-{\mathrm e}^{6}+x \,{\mathrm e}^{5}-6 \,{\mathrm e}^{5}-2\right ) {\mathrm e}^{x^{2}-5}\) | \(24\) |
gosper | \({\mathrm e}^{x^{2}} \left (x \,{\mathrm e}^{5}+{\mathrm e}^{3} {\mathrm e}^{5}-{\mathrm e} \,{\mathrm e}^{5}-6 \,{\mathrm e}^{5}-2\right ) {\mathrm e}^{-5}\) | \(31\) |
norman | \({\mathrm e}^{x^{2}} x +\left ({\mathrm e}^{3} {\mathrm e}^{5}-{\mathrm e} \,{\mathrm e}^{5}-6 \,{\mathrm e}^{5}-2\right ) {\mathrm e}^{-5} {\mathrm e}^{x^{2}}\) | \(34\) |
parallelrisch | \({\mathrm e}^{-5} \left ({\mathrm e}^{3} {\mathrm e}^{5} {\mathrm e}^{x^{2}}-{\mathrm e} \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}+{\mathrm e}^{x^{2}} x \,{\mathrm e}^{5}-6 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}-2 \,{\mathrm e}^{x^{2}}\right )\) | \(48\) |
meijerg | \(\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{2}-\frac {\left (2 \,{\mathrm e}^{8}-2 \,{\mathrm e}^{6}-12 \,{\mathrm e}^{5}-4\right ) {\mathrm e}^{-5} \left (1-{\mathrm e}^{x^{2}}\right )}{2}+i \left (-i {\mathrm e}^{x^{2}} x +\frac {i \sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{2}\right )\) | \(55\) |
default | \({\mathrm e}^{-5} \left (\frac {{\mathrm e}^{5} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{2}-2 \,{\mathrm e}^{x^{2}}+2 \,{\mathrm e}^{5} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{4}\right )-6 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}-{\mathrm e} \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}+{\mathrm e}^{3} {\mathrm e}^{5} {\mathrm e}^{x^{2}}\right )\) | \(68\) |
parts | \(\sqrt {\pi }\, \operatorname {erfi}\left (x \right ) {\mathrm e}^{3} x -\sqrt {\pi }\, \operatorname {erfi}\left (x \right ) {\mathrm e} x +\sqrt {\pi }\, \operatorname {erfi}\left (x \right ) x^{2}-6 x \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{2}-2 \,{\mathrm e}^{-5} \sqrt {\pi }\, \operatorname {erfi}\left (x \right ) x -\frac {\left (2 \sqrt {\pi }\, {\mathrm e}^{5} {\mathrm e}^{3} x \,\operatorname {erfi}\left (x \right )-2 \sqrt {\pi }\, {\mathrm e}^{5} {\mathrm e} x \,\operatorname {erfi}\left (x \right )+2 x^{2} \operatorname {erfi}\left (x \right ) \sqrt {\pi }\, {\mathrm e}^{5}-12 \sqrt {\pi }\, {\mathrm e}^{5} x \,\operatorname {erfi}\left (x \right )+{\mathrm e}^{5} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )-4 x \sqrt {\pi }\, \operatorname {erfi}\left (x \right )-2 \,{\mathrm e}^{3} {\mathrm e}^{5} {\mathrm e}^{x^{2}}+2 \,{\mathrm e} \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}-2 \,{\mathrm e}^{x^{2}} x \,{\mathrm e}^{5}+12 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}}+4 \,{\mathrm e}^{x^{2}}\right ) {\mathrm e}^{-5}}{2}\) | \(169\) |
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int e^{-5+x^2} \left (-4 x+e^5 \left (1-12 x-2 e x+2 e^3 x+2 x^2\right )\right ) \, dx={\left ({\left (x - 6\right )} e^{5} + e^{8} - e^{6} - 2\right )} e^{\left (x^{2} - 5\right )} \]
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Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int e^{-5+x^2} \left (-4 x+e^5 \left (1-12 x-2 e x+2 e^3 x+2 x^2\right )\right ) \, dx=\frac {\left (x e^{5} - 6 e^{5} - e^{6} - 2 + e^{8}\right ) e^{x^{2}}}{e^{5}} \]
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Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75 \[ \int e^{-5+x^2} \left (-4 x+e^5 \left (1-12 x-2 e x+2 e^3 x+2 x^2\right )\right ) \, dx=x e^{\left (x^{2}\right )} + e^{\left (x^{2} + 3\right )} - e^{\left (x^{2} + 1\right )} - 2 \, e^{\left (x^{2} - 5\right )} - 6 \, e^{\left (x^{2}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int e^{-5+x^2} \left (-4 x+e^5 \left (1-12 x-2 e x+2 e^3 x+2 x^2\right )\right ) \, dx={\left (x - 6\right )} e^{\left (x^{2}\right )} + e^{\left (x^{2} + 3\right )} - e^{\left (x^{2} + 1\right )} - 2 \, e^{\left (x^{2} - 5\right )} \]
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Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int e^{-5+x^2} \left (-4 x+e^5 \left (1-12 x-2 e x+2 e^3 x+2 x^2\right )\right ) \, dx=-{\mathrm {e}}^{x^2-5}\,\left (6\,{\mathrm {e}}^5+{\mathrm {e}}^6-{\mathrm {e}}^8-x\,{\mathrm {e}}^5+2\right ) \]
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