Integrand size = 107, antiderivative size = 30 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=5-e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+x \]
Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=-e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+x \]
Integrate[(3*E^x*x^2 + E^E^E^(E^((2 - x^2)/x)/3)*(3*x^2 + E^(E^(E^((2 - x^ 2)/x)/3) + E^((2 - x^2)/x)/3 + (2 - x^2)/x)*(2 + x^2)))/(3*E^x*x^2),x]
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.
\(\displaystyle \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{\frac {2-x^2}{x}+e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}} \left (x^2+2\right )\right )\right )}{3 x^2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{\frac {2-x^2}{x}+e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}} \left (x^2+2\right )\right )\right )}{x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{3} \int \left (\frac {\exp \left (-2 x+e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}\right ) \left (x^2+2\right )}{x^2}+3 e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}+3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (2 \int \frac {\exp \left (-2 x+e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}\right )}{x^2}dx+\int \exp \left (-2 x+e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}+e^{\frac {1}{3} e^{\frac {2}{x}-x}}+\frac {1}{3} e^{\frac {2}{x}-x}+\frac {2}{x}\right )dx+3 \int e^{e^{e^{\frac {1}{3} e^{\frac {2}{x}-x}}}-x}dx+3 x\right )\) |
Int[(3*E^x*x^2 + E^E^E^(E^((2 - x^2)/x)/3)*(3*x^2 + E^(E^(E^((2 - x^2)/x)/ 3) + E^((2 - x^2)/x)/3 + (2 - x^2)/x)*(2 + x^2)))/(3*E^x*x^2),x]
3.30.26.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 7.68 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
method | result | size |
risch | \(x -{\mathrm e}^{-x +{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{-\frac {x^{2}-2}{x}}}{3}}}}\) | \(25\) |
parallelrisch | \(\frac {\left (3 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{-\frac {x^{2}-2}{x}}}{3}}}}\right ) {\mathrm e}^{-x}}{3}\) | \(31\) |
int(1/3*(((x^2+2)*exp((-x^2+2)/x)*exp(1/3*exp((-x^2+2)/x))*exp(exp(1/3*exp ((-x^2+2)/x)))+3*x^2)*exp(exp(exp(1/3*exp((-x^2+2)/x))))+3*exp(x)*x^2)/exp (x)/x^2,x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.67 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx={\left (x e^{x} - e^{\left (e^{\left (-\frac {3 \, x^{2} - 3 \, x e^{\left (\frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )} - x e^{\left (-\frac {x^{2} - 2}{x}\right )} - 6}{3 \, x} + \frac {x^{2} - 2}{x} - \frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )}\right )}\right )} e^{\left (-x\right )} \]
integrate(1/3*(((x^2+2)*exp((-x^2+2)/x)*exp(1/3*exp((-x^2+2)/x))*exp(exp(1 /3*exp((-x^2+2)/x)))+3*x^2)*exp(exp(exp(1/3*exp((-x^2+2)/x))))+3*exp(x)*x^ 2)/exp(x)/x^2,x, algorithm=\
(x*e^x - e^(e^(-1/3*(3*x^2 - 3*x*e^(1/3*e^(-(x^2 - 2)/x)) - x*e^(-(x^2 - 2 )/x) - 6)/x + (x^2 - 2)/x - 1/3*e^(-(x^2 - 2)/x))))*e^(-x)
Time = 6.39 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=x - e^{- x} e^{e^{e^{\frac {e^{\frac {2 - x^{2}}{x}}}{3}}}} \]
integrate(1/3*(((x**2+2)*exp((-x**2+2)/x)*exp(1/3*exp((-x**2+2)/x))*exp(ex p(1/3*exp((-x**2+2)/x)))+3*x**2)*exp(exp(exp(1/3*exp((-x**2+2)/x))))+3*exp (x)*x**2)/exp(x)/x**2,x)
Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=x - e^{\left (-x + e^{\left (e^{\left (\frac {1}{3} \, e^{\left (-x + \frac {2}{x}\right )}\right )}\right )}\right )} \]
integrate(1/3*(((x^2+2)*exp((-x^2+2)/x)*exp(1/3*exp((-x^2+2)/x))*exp(exp(1 /3*exp((-x^2+2)/x)))+3*x^2)*exp(exp(exp(1/3*exp((-x^2+2)/x))))+3*exp(x)*x^ 2)/exp(x)/x^2,x, algorithm=\
\[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=\int { \frac {{\left (3 \, x^{2} e^{x} + {\left (3 \, x^{2} + {\left (x^{2} + 2\right )} e^{\left (-\frac {x^{2} - 2}{x} + e^{\left (\frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )} + \frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )}\right )} e^{\left (e^{\left (e^{\left (\frac {1}{3} \, e^{\left (-\frac {x^{2} - 2}{x}\right )}\right )}\right )}\right )}\right )} e^{\left (-x\right )}}{3 \, x^{2}} \,d x } \]
integrate(1/3*(((x^2+2)*exp((-x^2+2)/x)*exp(1/3*exp((-x^2+2)/x))*exp(exp(1 /3*exp((-x^2+2)/x)))+3*x^2)*exp(exp(exp(1/3*exp((-x^2+2)/x))))+3*exp(x)*x^ 2)/exp(x)/x^2,x, algorithm=\
integrate(1/3*(3*x^2*e^x + (3*x^2 + (x^2 + 2)*e^(-(x^2 - 2)/x + e^(1/3*e^( -(x^2 - 2)/x)) + 1/3*e^(-(x^2 - 2)/x)))*e^(e^(e^(1/3*e^(-(x^2 - 2)/x)))))* e^(-x)/x^2, x)
Time = 12.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x} \left (3 e^x x^2+e^{e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}}} \left (3 x^2+e^{e^{\frac {1}{3} e^{\frac {2-x^2}{x}}}+\frac {1}{3} e^{\frac {2-x^2}{x}}+\frac {2-x^2}{x}} \left (2+x^2\right )\right )\right )}{3 x^2} \, dx=-{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{2/x}}{3}}}}-x\,{\mathrm {e}}^x\right ) \]