Integrand size = 34, antiderivative size = 20 \begin {dmath*} \int e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (3+e^2 \left (12 x-3 x^2\right )\right ) \, dx=-20+3 e^{\frac {1}{2} e^2 (8-x) x} x \end {dmath*}
Time = 0.40 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \begin {dmath*} \int e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (3+e^2 \left (12 x-3 x^2\right )\right ) \, dx=3 e^{-\frac {1}{2} e^2 (-8+x) x} x \end {dmath*}
Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2726}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (e^2 \left (12 x-3 x^2\right )+3\right ) \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {3 e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (4 x-x^2\right )}{4-x}\) |
3.11.60.3.1 Defintions of rubi rules used
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 0.36 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(3 \,{\mathrm e}^{-\frac {x \left (-8+x \right ) {\mathrm e}^{2}}{2}} x\) | \(13\) |
risch | \(3 \,{\mathrm e}^{-\frac {x \left (-8+x \right ) {\mathrm e}^{2}}{2}} x\) | \(13\) |
norman | \(3 x \,{\mathrm e}^{\frac {\left (-x^{2}+8 x \right ) {\mathrm e}^{2}}{2}}\) | \(18\) |
parallelrisch | \(3 x \,{\mathrm e}^{\frac {\left (-x^{2}+8 x \right ) {\mathrm e}^{2}}{2}}\) | \(18\) |
default | \(\frac {3 \sqrt {\pi }\, {\mathrm e}^{8 \,{\mathrm e}^{2}} {\mathrm e}^{-1} \sqrt {2}\, \operatorname {erf}\left (\frac {{\mathrm e} \sqrt {2}\, x}{2}-2 \,{\mathrm e}^{2} {\mathrm e}^{-1} \sqrt {2}\right )}{2}-3 \,{\mathrm e}^{2} \left (-{\mathrm e}^{-2} x \,{\mathrm e}^{-\frac {x^{2} {\mathrm e}^{2}}{2}+4 \,{\mathrm e}^{2} x}-4 \,{\mathrm e}^{-2} {\mathrm e}^{-\frac {x^{2} {\mathrm e}^{2}}{2}+4 \,{\mathrm e}^{2} x}+8 \sqrt {\pi }\, {\mathrm e}^{8 \,{\mathrm e}^{2}} {\mathrm e}^{-1} \sqrt {2}\, \operatorname {erf}\left (\frac {{\mathrm e} \sqrt {2}\, x}{2}-2 \,{\mathrm e}^{2} {\mathrm e}^{-1} \sqrt {2}\right )+\frac {{\mathrm e}^{-2} \sqrt {\pi }\, {\mathrm e}^{8 \,{\mathrm e}^{2}} {\mathrm e}^{-1} \sqrt {2}\, \operatorname {erf}\left (\frac {{\mathrm e} \sqrt {2}\, x}{2}-2 \,{\mathrm e}^{2} {\mathrm e}^{-1} \sqrt {2}\right )}{2}\right )+12 \,{\mathrm e}^{2} \left (-{\mathrm e}^{-2} {\mathrm e}^{-\frac {x^{2} {\mathrm e}^{2}}{2}+4 \,{\mathrm e}^{2} x}+2 \sqrt {\pi }\, {\mathrm e}^{8 \,{\mathrm e}^{2}} {\mathrm e}^{-1} \sqrt {2}\, \operatorname {erf}\left (\frac {{\mathrm e} \sqrt {2}\, x}{2}-2 \,{\mathrm e}^{2} {\mathrm e}^{-1} \sqrt {2}\right )\right )\) | \(229\) |
parts | \(-\frac {3 \sqrt {\pi }\, {\mathrm e}^{8 \,{\mathrm e}^{2}} {\mathrm e}^{-1} \sqrt {2}\, \operatorname {erf}\left (\frac {{\mathrm e} \sqrt {2}\, x}{2}-2 \,{\mathrm e}^{2} {\mathrm e}^{-1} \sqrt {2}\right ) {\mathrm e}^{2} x^{2}}{2}+6 \sqrt {\pi }\, {\mathrm e}^{8 \,{\mathrm e}^{2}} {\mathrm e}^{-1} \sqrt {2}\, \operatorname {erf}\left (\frac {{\mathrm e} \sqrt {2}\, x}{2}-2 \,{\mathrm e}^{2} {\mathrm e}^{-1} \sqrt {2}\right ) {\mathrm e}^{2} x +\frac {3 \sqrt {\pi }\, {\mathrm e}^{8 \,{\mathrm e}^{2}} {\mathrm e}^{-1} \sqrt {2}\, \operatorname {erf}\left (\frac {{\mathrm e} \sqrt {2}\, x}{2}-2 \,{\mathrm e}^{2} {\mathrm e}^{-1} \sqrt {2}\right )}{2}+\frac {{\mathrm e}^{8 \,{\mathrm e}^{2}} {\mathrm e}^{-5} \sqrt {2}\, \left (-3 \,{\mathrm e}^{2} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (-{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-1}}{2}\right ) x^{2} {\mathrm e}^{4} \sqrt {\pi }+12 \,{\mathrm e}^{2} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (-{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-1}}{2}\right ) x \,{\mathrm e}^{4} \sqrt {\pi }+3 \sqrt {2}\, {\mathrm e}^{2} {\mathrm e}^{-\frac {\left (-{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}\right )^{2} {\mathrm e}^{-2}}{2}} x \,{\mathrm e}^{3}-48 \,{\mathrm e}^{4} {\mathrm e}^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (-{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-1}}{2}\right )-12 \,{\mathrm e}^{2} {\mathrm e}^{3} \sqrt {2}\, {\mathrm e}^{-\frac {\left (-{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}\right )^{2} {\mathrm e}^{-2}}{2}}+48 \,{\mathrm e}^{6} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (-{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-1}}{2}\right )+12 \,{\mathrm e}^{4} {\mathrm e} \sqrt {2}\, {\mathrm e}^{-\frac {\left (-{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}\right )^{2} {\mathrm e}^{-2}}{2}}+3 \left ({\mathrm e}^{2}\right )^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (-{\mathrm e}^{2} x +4 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-1}}{2}\right )\right )}{2}\) | \(405\) |
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \begin {dmath*} \int e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (3+e^2 \left (12 x-3 x^2\right )\right ) \, dx=3 \, x e^{\left (-\frac {1}{2} \, {\left (x^{2} - 8 \, x\right )} e^{2}\right )} \end {dmath*}
Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \begin {dmath*} \int e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (3+e^2 \left (12 x-3 x^2\right )\right ) \, dx=3 x e^{\left (- \frac {x^{2}}{2} + 4 x\right ) e^{2}} \end {dmath*}
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.35 (sec) , antiderivative size = 311, normalized size of antiderivative = 15.55 \begin {dmath*} \int e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (3+e^2 \left (12 x-3 x^2\right )\right ) \, dx=\frac {3}{2} \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} \, \sqrt {2} x e - 2 \, \sqrt {2} e\right ) e^{\left (8 \, e^{2} - 1\right )} + \frac {24 \, \sqrt {\frac {1}{2}} {\left (\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\pi } {\left (x e^{2} - 4 \, e^{2}\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (x e^{2} - 4 \, e^{2}\right )}^{2}} e^{\left (-1\right )}\right ) - 1\right )} e^{3}}{\sqrt {{\left (x e^{2} - 4 \, e^{2}\right )}^{2}} \left (-e^{2}\right )^{\frac {3}{2}}} - \frac {\sqrt {\frac {1}{2}} e^{\left (-\frac {1}{2} \, {\left (x e^{2} - 4 \, e^{2}\right )}^{2} e^{\left (-2\right )} + 2\right )}}{\left (-e^{2}\right )^{\frac {3}{2}}}\right )} e^{\left (8 \, e^{2} + 2\right )}}{\sqrt {-e^{2}}} - \frac {6 \, \sqrt {\frac {1}{2}} {\left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} {\left (x e^{2} - 4 \, e^{2}\right )}^{3} e^{3} \Gamma \left (\frac {3}{2}, \frac {1}{2} \, {\left (x e^{2} - 4 \, e^{2}\right )}^{2} e^{\left (-2\right )}\right )}{{\left ({\left (x e^{2} - 4 \, e^{2}\right )}^{2}\right )}^{\frac {3}{2}} \left (-e^{2}\right )^{\frac {5}{2}}} - \frac {8 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\pi } {\left (x e^{2} - 4 \, e^{2}\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {{\left (x e^{2} - 4 \, e^{2}\right )}^{2}} e^{\left (-1\right )}\right ) - 1\right )} e^{5}}{\sqrt {{\left (x e^{2} - 4 \, e^{2}\right )}^{2}} \left (-e^{2}\right )^{\frac {5}{2}}} + \frac {8 \, \sqrt {\frac {1}{2}} e^{\left (-\frac {1}{2} \, {\left (x e^{2} - 4 \, e^{2}\right )}^{2} e^{\left (-2\right )} + 4\right )}}{\left (-e^{2}\right )^{\frac {5}{2}}}\right )} e^{\left (8 \, e^{2} + 2\right )}}{\sqrt {-e^{2}}} \end {dmath*}
3/2*sqrt(2)*sqrt(pi)*erf(1/2*sqrt(2)*x*e - 2*sqrt(2)*e)*e^(8*e^2 - 1) + 24 *sqrt(1/2)*(2*sqrt(2)*sqrt(1/2)*sqrt(pi)*(x*e^2 - 4*e^2)*(erf(sqrt(1/2)*sq rt((x*e^2 - 4*e^2)^2)*e^(-1)) - 1)*e^3/(sqrt((x*e^2 - 4*e^2)^2)*(-e^2)^(3/ 2)) - sqrt(1/2)*e^(-1/2*(x*e^2 - 4*e^2)^2*e^(-2) + 2)/(-e^2)^(3/2))*e^(8*e ^2 + 2)/sqrt(-e^2) - 6*sqrt(1/2)*(sqrt(2)*sqrt(1/2)*(x*e^2 - 4*e^2)^3*e^3* gamma(3/2, 1/2*(x*e^2 - 4*e^2)^2*e^(-2))/(((x*e^2 - 4*e^2)^2)^(3/2)*(-e^2) ^(5/2)) - 8*sqrt(2)*sqrt(1/2)*sqrt(pi)*(x*e^2 - 4*e^2)*(erf(sqrt(1/2)*sqrt ((x*e^2 - 4*e^2)^2)*e^(-1)) - 1)*e^5/(sqrt((x*e^2 - 4*e^2)^2)*(-e^2)^(5/2) ) + 8*sqrt(1/2)*e^(-1/2*(x*e^2 - 4*e^2)^2*e^(-2) + 4)/(-e^2)^(5/2))*e^(8*e ^2 + 2)/sqrt(-e^2)
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \begin {dmath*} \int e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (3+e^2 \left (12 x-3 x^2\right )\right ) \, dx=3 \, x e^{\left (-\frac {1}{2} \, x^{2} e^{2} + 4 \, x e^{2}\right )} \end {dmath*}
Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \begin {dmath*} \int e^{\frac {1}{2} e^2 \left (8 x-x^2\right )} \left (3+e^2 \left (12 x-3 x^2\right )\right ) \, dx=3\,x\,{\mathrm {e}}^{-\frac {x^2\,{\mathrm {e}}^2}{2}}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^2} \end {dmath*}