Integrand size = 143, antiderivative size = 30 \[ \int \frac {e^{\frac {-5 x-x^2}{3-4 x+e^{e x+x^2} x-x^2}} \left (-15-6 x-x^2+e^{e x+x^2} \left (-x^2+10 x^3+2 x^4+e \left (5 x^2+x^3\right )\right )\right )}{9-24 x+10 x^2+e^{2 e x+2 x^2} x^2+8 x^3+x^4+e^{e x+x^2} \left (6 x-8 x^2-2 x^3\right )} \, dx=e^{\frac {5+x}{3-e^{x (e+x)}-\frac {3-x}{x}+x}} \]
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\frac {-5 x-x^2}{3-4 x+e^{e x+x^2} x-x^2}} \left (-15-6 x-x^2+e^{e x+x^2} \left (-x^2+10 x^3+2 x^4+e \left (5 x^2+x^3\right )\right )\right )}{9-24 x+10 x^2+e^{2 e x+2 x^2} x^2+8 x^3+x^4+e^{e x+x^2} \left (6 x-8 x^2-2 x^3\right )} \, dx=e^{\frac {x (5+x)}{-3-\left (-4+e^{x (e+x)}\right ) x+x^2}} \]
Integrate[(E^((-5*x - x^2)/(3 - 4*x + E^(E*x + x^2)*x - x^2))*(-15 - 6*x - x^2 + E^(E*x + x^2)*(-x^2 + 10*x^3 + 2*x^4 + E*(5*x^2 + x^3))))/(9 - 24*x + 10*x^2 + E^(2*E*x + 2*x^2)*x^2 + 8*x^3 + x^4 + E^(E*x + x^2)*(6*x - 8*x ^2 - 2*x^3)),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.
\(\displaystyle \int \frac {\left (-x^2+e^{x^2+e x} \left (2 x^4+10 x^3-x^2+e \left (x^3+5 x^2\right )\right )-6 x-15\right ) \exp \left (\frac {-x^2-5 x}{-x^2+e^{x^2+e x} x-4 x+3}\right )}{x^4+8 x^3+e^{2 x^2+2 e x} x^2+10 x^2+e^{x^2+e x} \left (-2 x^3-8 x^2+6 x\right )-24 x+9} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-x^2+e^{x^2+e x} \left (2 x^4+10 x^3-x^2+e \left (x^3+5 x^2\right )\right )-6 x-15\right ) \exp \left (\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}\right )}{\left (-x^2+e^{x (x+e)} x-4 x+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {(1-5 e) x^2 \exp \left (x^2+\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}+e x\right )}{\left (-x^2+e^{x (x+e)} x-4 x+3\right )^2}-\frac {x^2 \exp \left (\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}\right )}{\left (x^2-e^{x (x+e)} x+4 x-3\right )^2}-\frac {6 x \exp \left (\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}\right )}{\left (x^2-e^{x (x+e)} x+4 x-3\right )^2}-\frac {15 \exp \left (\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}\right )}{\left (x^2-e^{x (x+e)} x+4 x-3\right )^2}+\frac {2 x^4 \exp \left (x^2+\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}+e x\right )}{\left (-x^2+e^{x (x+e)} x-4 x+3\right )^2}+\frac {10 \left (1+\frac {e}{10}\right ) x^3 \exp \left (x^2+\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}+e x\right )}{\left (-x^2+e^{x (x+e)} x-4 x+3\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\left ((1-5 e) \int \frac {\exp \left (x^2+\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}+e x\right ) x^2}{\left (-x^2+e^{x (x+e)} x-4 x+3\right )^2}dx\right )-15 \int \frac {\exp \left (\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}\right )}{\left (x^2-e^{x (x+e)} x+4 x-3\right )^2}dx-6 \int \frac {\exp \left (\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}\right ) x}{\left (x^2-e^{x (x+e)} x+4 x-3\right )^2}dx-\int \frac {\exp \left (\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}\right ) x^2}{\left (x^2-e^{x (x+e)} x+4 x-3\right )^2}dx+2 \int \frac {\exp \left (x^2+\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}+e x\right ) x^4}{\left (-x^2+e^{x (x+e)} x-4 x+3\right )^2}dx+(10+e) \int \frac {\exp \left (x^2+\frac {(-x-5) x}{-x^2+e^{x^2+e x} x-4 x+3}+e x\right ) x^3}{\left (-x^2+e^{x (x+e)} x-4 x+3\right )^2}dx\) |
Int[(E^((-5*x - x^2)/(3 - 4*x + E^(E*x + x^2)*x - x^2))*(-15 - 6*x - x^2 + E^(E*x + x^2)*(-x^2 + 10*x^3 + 2*x^4 + E*(5*x^2 + x^3))))/(9 - 24*x + 10* x^2 + E^(2*E*x + 2*x^2)*x^2 + 8*x^3 + x^4 + E^(E*x + x^2)*(6*x - 8*x^2 - 2 *x^3)),x]
3.13.76.3.1 Defintions of rubi rules used
Time = 13.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90
method | result | size |
risch | \({\mathrm e}^{\frac {\left (5+x \right ) x}{-x \,{\mathrm e}^{x \left (x +{\mathrm e}\right )}+x^{2}+4 x -3}}\) | \(27\) |
parallelrisch | \({\mathrm e}^{\frac {-x^{2}-5 x}{x \,{\mathrm e}^{x \left (x +{\mathrm e}\right )}-x^{2}-4 x +3}}\) | \(33\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {-x^{2}-5 x}{x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}-x^{2}-4 x +3}}+4 x \,{\mathrm e}^{\frac {-x^{2}-5 x}{x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}-x^{2}-4 x +3}}-x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}} {\mathrm e}^{\frac {-x^{2}-5 x}{x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}-x^{2}-4 x +3}}-3 \,{\mathrm e}^{\frac {-x^{2}-5 x}{x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}-x^{2}-4 x +3}}}{-x \,{\mathrm e}^{x \,{\mathrm e}+x^{2}}+x^{2}+4 x -3}\) | \(182\) |
int((((x^3+5*x^2)*exp(1)+2*x^4+10*x^3-x^2)*exp(x*exp(1)+x^2)-x^2-6*x-15)*e xp((-x^2-5*x)/(x*exp(x*exp(1)+x^2)-x^2-4*x+3))/(x^2*exp(x*exp(1)+x^2)^2+(- 2*x^3-8*x^2+6*x)*exp(x*exp(1)+x^2)+x^4+8*x^3+10*x^2-24*x+9),x,method=_RETU RNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {-5 x-x^2}{3-4 x+e^{e x+x^2} x-x^2}} \left (-15-6 x-x^2+e^{e x+x^2} \left (-x^2+10 x^3+2 x^4+e \left (5 x^2+x^3\right )\right )\right )}{9-24 x+10 x^2+e^{2 e x+2 x^2} x^2+8 x^3+x^4+e^{e x+x^2} \left (6 x-8 x^2-2 x^3\right )} \, dx=e^{\left (\frac {x^{2} + 5 \, x}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3}\right )} \]
integrate((((x^3+5*x^2)*exp(1)+2*x^4+10*x^3-x^2)*exp(x*exp(1)+x^2)-x^2-6*x -15)*exp((-x^2-5*x)/(x*exp(x*exp(1)+x^2)-x^2-4*x+3))/(x^2*exp(x*exp(1)+x^2 )^2+(-2*x^3-8*x^2+6*x)*exp(x*exp(1)+x^2)+x^4+8*x^3+10*x^2-24*x+9),x, algor ithm=\
Time = 0.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {-5 x-x^2}{3-4 x+e^{e x+x^2} x-x^2}} \left (-15-6 x-x^2+e^{e x+x^2} \left (-x^2+10 x^3+2 x^4+e \left (5 x^2+x^3\right )\right )\right )}{9-24 x+10 x^2+e^{2 e x+2 x^2} x^2+8 x^3+x^4+e^{e x+x^2} \left (6 x-8 x^2-2 x^3\right )} \, dx=e^{\frac {- x^{2} - 5 x}{- x^{2} + x e^{x^{2} + e x} - 4 x + 3}} \]
integrate((((x**3+5*x**2)*exp(1)+2*x**4+10*x**3-x**2)*exp(x*exp(1)+x**2)-x **2-6*x-15)*exp((-x**2-5*x)/(x*exp(x*exp(1)+x**2)-x**2-4*x+3))/(x**2*exp(x *exp(1)+x**2)**2+(-2*x**3-8*x**2+6*x)*exp(x*exp(1)+x**2)+x**4+8*x**3+10*x* *2-24*x+9),x)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
Time = 0.34 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.80 \[ \int \frac {e^{\frac {-5 x-x^2}{3-4 x+e^{e x+x^2} x-x^2}} \left (-15-6 x-x^2+e^{e x+x^2} \left (-x^2+10 x^3+2 x^4+e \left (5 x^2+x^3\right )\right )\right )}{9-24 x+10 x^2+e^{2 e x+2 x^2} x^2+8 x^3+x^4+e^{e x+x^2} \left (6 x-8 x^2-2 x^3\right )} \, dx=e^{\left (\frac {x e^{\left (x^{2} + x e\right )}}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3} + \frac {x}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3} + \frac {3}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3} + 1\right )} \]
integrate((((x^3+5*x^2)*exp(1)+2*x^4+10*x^3-x^2)*exp(x*exp(1)+x^2)-x^2-6*x -15)*exp((-x^2-5*x)/(x*exp(x*exp(1)+x^2)-x^2-4*x+3))/(x^2*exp(x*exp(1)+x^2 )^2+(-2*x^3-8*x^2+6*x)*exp(x*exp(1)+x^2)+x^4+8*x^3+10*x^2-24*x+9),x, algor ithm=\
e^(x*e^(x^2 + x*e)/(x^2 - x*e^(x^2 + x*e) + 4*x - 3) + x/(x^2 - x*e^(x^2 + x*e) + 4*x - 3) + 3/(x^2 - x*e^(x^2 + x*e) + 4*x - 3) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.59 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {e^{\frac {-5 x-x^2}{3-4 x+e^{e x+x^2} x-x^2}} \left (-15-6 x-x^2+e^{e x+x^2} \left (-x^2+10 x^3+2 x^4+e \left (5 x^2+x^3\right )\right )\right )}{9-24 x+10 x^2+e^{2 e x+2 x^2} x^2+8 x^3+x^4+e^{e x+x^2} \left (6 x-8 x^2-2 x^3\right )} \, dx=e^{\left (\frac {x^{2}}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3} + \frac {5 \, x}{x^{2} - x e^{\left (x^{2} + x e\right )} + 4 \, x - 3}\right )} \]
integrate((((x^3+5*x^2)*exp(1)+2*x^4+10*x^3-x^2)*exp(x*exp(1)+x^2)-x^2-6*x -15)*exp((-x^2-5*x)/(x*exp(x*exp(1)+x^2)-x^2-4*x+3))/(x^2*exp(x*exp(1)+x^2 )^2+(-2*x^3-8*x^2+6*x)*exp(x*exp(1)+x^2)+x^4+8*x^3+10*x^2-24*x+9),x, algor ithm=\
Time = 13.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {-5 x-x^2}{3-4 x+e^{e x+x^2} x-x^2}} \left (-15-6 x-x^2+e^{e x+x^2} \left (-x^2+10 x^3+2 x^4+e \left (5 x^2+x^3\right )\right )\right )}{9-24 x+10 x^2+e^{2 e x+2 x^2} x^2+8 x^3+x^4+e^{e x+x^2} \left (6 x-8 x^2-2 x^3\right )} \, dx={\mathrm {e}}^{\frac {x^2+5\,x}{4\,x+x^2-x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x\,\mathrm {e}}-3}} \]