Integrand size = 73, antiderivative size = 26 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=5-x+e^{-16-2 \left (1+e^3-x\right )^2+2 x} x \]
[Out]
Time = 1.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {6, 6873, 6874, 2266, 2236, 2272, 2273} \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=x \exp \left (-2 x^2+2 \left (3+2 e^3\right ) x-2 \left (9+2 e^3+e^6\right )\right )-x \]
[In]
[Out]
Rule 6
Rule 2236
Rule 2266
Rule 2272
Rule 2273
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+\left (6+4 e^3\right ) x-4 x^2\right ) \, dx \\ & = \int \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+\left (6+4 e^3\right ) x-4 x^2\right ) \, dx \\ & = \int \left (-1+\exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right )+2 \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) \left (3+2 e^3\right ) x-4 \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) x^2\right ) \, dx \\ & = -x-4 \int \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) x^2 \, dx+\left (2 \left (3+2 e^3\right )\right ) \int \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) x \, dx+\int \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) \, dx \\ & = -\frac {1}{2} \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) \left (3+2 e^3\right )-x+\exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) x+e^{-\frac {27}{2}+2 e^3} \int e^{-\frac {1}{8} \left (2 \left (3+2 e^3\right )-4 x\right )^2} \, dx-\left (2 \left (3+2 e^3\right )\right ) \int \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) x \, dx+\left (3+2 e^3\right )^2 \int \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) \, dx-\int \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) \, dx \\ & = -x+\exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) x-\frac {1}{2} e^{-\frac {27}{2}+2 e^3} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {3+2 e^3-2 x}{\sqrt {2}}\right )-e^{-\frac {27}{2}+2 e^3} \int e^{-\frac {1}{8} \left (2 \left (3+2 e^3\right )-4 x\right )^2} \, dx-\left (3+2 e^3\right )^2 \int \exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) \, dx+\left (e^{-\frac {27}{2}+2 e^3} \left (3+2 e^3\right )^2\right ) \int e^{-\frac {1}{8} \left (2 \left (3+2 e^3\right )-4 x\right )^2} \, dx \\ & = -x+\exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) x-\frac {1}{2} e^{-\frac {27}{2}+2 e^3} \left (3+2 e^3\right )^2 \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {3+2 e^3-2 x}{\sqrt {2}}\right )-\left (e^{-\frac {27}{2}+2 e^3} \left (3+2 e^3\right )^2\right ) \int e^{-\frac {1}{8} \left (2 \left (3+2 e^3\right )-4 x\right )^2} \, dx \\ & = -x+\exp \left (-2 \left (9+2 e^3+e^6\right )+2 \left (3+2 e^3\right ) x-2 x^2\right ) x \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=\left (-1+e^{-2 \left (9+e^6-2 e^3 (-1+x)-3 x+x^2\right )}\right ) x \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19
method | result | size |
risch | \(-x +x \,{\mathrm e}^{4 x \,{\mathrm e}^{3}-2 x^{2}-4 \,{\mathrm e}^{3}-2 \,{\mathrm e}^{6}+6 x -18}\) | \(31\) |
parallelrisch | \(-\left (x \,{\mathrm e}^{2 \,{\mathrm e}^{6}+2 \left (2-2 x \right ) {\mathrm e}^{3}+2 x^{2}-6 x +18}-x \right ) {\mathrm e}^{-2 \,{\mathrm e}^{6}-2 \left (2-2 x \right ) {\mathrm e}^{3}-2 x^{2}+6 x -18}\) | \(55\) |
default | \(-x +\frac {{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{4}+6 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )-4 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \left (-\frac {x \,{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )}{4}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )+4 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} {\mathrm e}^{3} \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )\) | \(361\) |
parts | \(-x +\frac {{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{4}+6 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )-4 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} \left (-\frac {x \,{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )}{4}+\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )+4 \,{\mathrm e}^{-2 \,{\mathrm e}^{6}} {\mathrm e}^{-4 \,{\mathrm e}^{3}} {\mathrm e}^{-18} {\mathrm e}^{3} \left (-\frac {{\mathrm e}^{-2 x^{2}+\left (6+4 \,{\mathrm e}^{3}\right ) x}}{4}+\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {\left (6+4 \,{\mathrm e}^{3}\right )^{2}}{8}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, x -\frac {\left (6+4 \,{\mathrm e}^{3}\right ) \sqrt {2}}{4}\right )}{16}\right )\) | \(361\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=-{\left (x e^{\left (2 \, x^{2} - 4 \, {\left (x - 1\right )} e^{3} - 6 \, x + 2 \, e^{6} + 18\right )} - x\right )} e^{\left (-2 \, x^{2} + 4 \, {\left (x - 1\right )} e^{3} + 6 \, x - 2 \, e^{6} - 18\right )} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=x e^{- 2 x^{2} + 6 x - 2 \cdot \left (2 - 2 x\right ) e^{3} - 2 e^{6} - 18} - x \]
[In]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.37 (sec) , antiderivative size = 397, normalized size of antiderivative = 15.27 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=\frac {1}{4} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} x - \frac {1}{2} \, \sqrt {2} {\left (2 \, e^{3} + 3\right )}\right ) e^{\left (\frac {1}{2} \, {\left (2 \, e^{3} + 3\right )}^{2} - 2 \, e^{6} - 4 \, e^{3} - 18\right )} - \frac {1}{2} i \, \sqrt {2} {\left (\frac {i \, \sqrt {\pi } {\left (2 \, x - 2 \, e^{3} - 3\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}\right ) - 1\right )} {\left (2 \, e^{3} + 3\right )}}{\sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}} - i \, \sqrt {2} e^{\left (-\frac {1}{2} \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )}\right )} e^{\left (\frac {1}{2} \, {\left (2 \, e^{3} + 3\right )}^{2} - 2 \, e^{6} - 4 \, e^{3} - 15\right )} + \frac {1}{4} i \, \sqrt {2} {\left (\frac {i \, \sqrt {\pi } {\left (2 \, x - 2 \, e^{3} - 3\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}\right ) - 1\right )} {\left (2 \, e^{3} + 3\right )}^{2}}{\sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}} - 2 i \, \sqrt {2} {\left (2 \, e^{3} + 3\right )} e^{\left (-\frac {1}{2} \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )} - \frac {2 i \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{2} \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )}{{\left ({\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )}^{\frac {3}{2}}}\right )} e^{\left (\frac {1}{2} \, {\left (2 \, e^{3} + 3\right )}^{2} - 2 \, e^{6} - 4 \, e^{3} - 18\right )} - \frac {3}{4} i \, \sqrt {2} {\left (\frac {i \, \sqrt {\pi } {\left (2 \, x - 2 \, e^{3} - 3\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}\right ) - 1\right )} {\left (2 \, e^{3} + 3\right )}}{\sqrt {{\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}}} - i \, \sqrt {2} e^{\left (-\frac {1}{2} \, {\left (2 \, x - 2 \, e^{3} - 3\right )}^{2}\right )}\right )} e^{\left (\frac {1}{2} \, {\left (2 \, e^{3} + 3\right )}^{2} - 2 \, e^{6} - 4 \, e^{3} - 18\right )} - x \]
[In]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.15 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=\frac {1}{2} \, \sqrt {2} \sqrt {\pi } {\left (2 \, e^{3} + 3\right )} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - 2 \, e^{3} - 3\right )}\right ) e^{\left (2 \, e^{3} - \frac {21}{2}\right )} - \frac {1}{2} \, \sqrt {2} \sqrt {\pi } {\left (2 \, e^{6} + 3 \, e^{3}\right )} \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - 2 \, e^{3} - 3\right )}\right ) e^{\left (2 \, e^{3} - \frac {27}{2}\right )} + {\left (x + e^{3}\right )} e^{\left (-2 \, x^{2} + 4 \, x e^{3} + 6 \, x - 2 \, e^{6} - 4 \, e^{3} - 18\right )} - x - e^{\left (-2 \, x^{2} + 4 \, x e^{3} + 6 \, x - 2 \, e^{6} - 4 \, e^{3} - 15\right )} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int e^{-18-2 e^6-2 e^3 (2-2 x)+6 x-2 x^2} \left (1-e^{18+2 e^6+2 e^3 (2-2 x)-6 x+2 x^2}+6 x+4 e^3 x-4 x^2\right ) \, dx=x\,{\mathrm {e}}^{-4\,{\mathrm {e}}^3}\,{\mathrm {e}}^{-2\,{\mathrm {e}}^6}\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{-18}\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^3}-x \]
[In]
[Out]