Integrand size = 38, antiderivative size = 22 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=5 e^{(2+e) x} \left (e^{76 x^2}-\frac {2 x}{3}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {6, 12, 6873, 6874, 2225, 2207, 2268} \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=5 e^{76 x^2+(2+e) x}-\frac {10}{3} e^{(2+e) x} x \]
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Rule 6
Rule 12
Rule 2207
Rule 2225
Rule 2268
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{3} e^{2 x+e x} \left (-10+(-20-10 e) x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx \\ & = \frac {1}{3} \int e^{2 x+e x} \left (-10+(-20-10 e) x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx \\ & = \frac {1}{3} \int e^{(2+e) x} \left (-10+(-20-10 e) x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx \\ & = \frac {1}{3} \int \left (-10 e^{(2+e) x}-10 e^{(2+e) x} (2+e) x+15 e^{(2+e) x+76 x^2} (2+e+152 x)\right ) \, dx \\ & = -\left (\frac {10}{3} \int e^{(2+e) x} \, dx\right )+5 \int e^{(2+e) x+76 x^2} (2+e+152 x) \, dx-\frac {1}{3} (10 (2+e)) \int e^{(2+e) x} x \, dx \\ & = 5 e^{(2+e) x+76 x^2}-\frac {10 e^{(2+e) x}}{3 (2+e)}-\frac {10}{3} e^{(2+e) x} x+\frac {10}{3} \int e^{(2+e) x} \, dx \\ & = 5 e^{(2+e) x+76 x^2}-\frac {10}{3} e^{(2+e) x} x \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=\frac {5}{3} e^{(2+e) x} \left (3 e^{76 x^2}-2 x\right ) \]
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Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {\left (-10 x +15 \,{\mathrm e}^{76 x^{2}}\right ) {\mathrm e}^{x \left ({\mathrm e}+2\right )}}{3}\) | \(22\) |
parallelrisch | \(-\frac {10 x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}}{3}+5 \,{\mathrm e}^{76 x^{2}} {\mathrm e}^{x \left ({\mathrm e}+2\right )}\) | \(27\) |
norman | \(-\frac {10 x \,{\mathrm e}^{x \,{\mathrm e}+2 x}}{3}+5 \,{\mathrm e}^{76 x^{2}} {\mathrm e}^{x \,{\mathrm e}+2 x}\) | \(31\) |
default | \(-\frac {10 \,{\mathrm e}^{x \,{\mathrm e}+2 x}}{3 \left ({\mathrm e}+2\right )}-\frac {20 \left (\left ({\mathrm e}+2\right ) x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}-{\mathrm e}^{x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )^{2}}-\frac {5 i \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{38}-\frac {10 \,{\mathrm e} \left (\left ({\mathrm e}+2\right ) x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}-{\mathrm e}^{x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )^{2}}+5 \,{\mathrm e}^{76 x^{2}+x \left ({\mathrm e}+2\right )}+\frac {5 i \left ({\mathrm e}+2\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}-\frac {5 i {\mathrm e} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}\) | \(215\) |
parts | \(\frac {-\frac {20 \left (\left ({\mathrm e}+2\right ) x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}-{\mathrm e}^{x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )}-\frac {10 \,{\mathrm e} \left (\left ({\mathrm e}+2\right ) x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}-{\mathrm e}^{x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )}-\frac {10 \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}}{3}}{{\mathrm e}+2}-\frac {5 i {\mathrm e} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}-\frac {5 i \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{38}+5 \,{\mathrm e}^{76 x^{2}+x \left ({\mathrm e}+2\right )}+\frac {5 i \left ({\mathrm e}+2\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}\) | \(216\) |
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=-\frac {5}{3} \, {\left (2 \, x - 3 \, e^{\left (76 \, x^{2}\right )}\right )} e^{\left (x e + 2 \, x\right )} \]
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Time = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=\frac {\left (- 10 x + 15 e^{76 x^{2}}\right ) e^{2 x + e x}}{3} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.34 (sec) , antiderivative size = 239, normalized size of antiderivative = 10.86 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=-\frac {5}{38} i \, \sqrt {19} \sqrt {\pi } \operatorname {erf}\left (2 i \, \sqrt {19} x + \frac {1}{76} i \, \sqrt {19} {\left (e + 2\right )}\right ) e^{\left (-\frac {1}{304} \, {\left (e + 2\right )}^{2}\right )} - \frac {5}{76} i \, \sqrt {19} \sqrt {\pi } \operatorname {erf}\left (2 i \, \sqrt {19} x + \frac {1}{76} i \, \sqrt {19} {\left (e + 2\right )}\right ) e^{\left (-\frac {1}{304} \, {\left (e + 2\right )}^{2} + 1\right )} - \frac {5}{76} \, \sqrt {19} {\left (\frac {\sqrt {19} \sqrt {\frac {1}{19}} \sqrt {\pi } {\left (152 \, x + e + 2\right )} {\left (\operatorname {erf}\left (\frac {1}{4} \, \sqrt {\frac {1}{19}} \sqrt {-{\left (152 \, x + e + 2\right )}^{2}}\right ) - 1\right )} {\left (e + 2\right )}}{\sqrt {-{\left (152 \, x + e + 2\right )}^{2}}} - 4 \, \sqrt {19} e^{\left (\frac {1}{304} \, {\left (152 \, x + e + 2\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{304} \, {\left (e + 2\right )}^{2}\right )} - \frac {10 \, {\left (x {\left (e^{2} + 2 \, e\right )} - e\right )} e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} - \frac {20 \, {\left (x {\left (e + 2\right )} - 1\right )} e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} - \frac {10 \, e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.64 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=-\frac {10 \, {\left (x e + 2 \, x - 1\right )} e^{\left (x e + 2 \, x + 1\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} - \frac {10 \, {\left (2 \, x e + 4 \, x + e\right )} e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} + 5 \, e^{\left (76 \, x^{2} + x e + 2 \, x\right )} \]
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Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=-{\mathrm {e}}^{2\,x+x\,\mathrm {e}}\,\left (\frac {10\,x}{3}-5\,{\mathrm {e}}^{76\,x^2}\right ) \]
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