Integrand size = 140, antiderivative size = 33 \[ \int \frac {e^{\frac {2 \left (-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x\right )}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (288-96 e^5 x+16 x^2\right )+e^{\frac {-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (4608-1536 e^5 x+256 x^2\right )}{x^4} \, dx=\left (16+e^{5+\frac {8 \left (-e^{\frac {\left (3-e^5 x\right )^2}{x^2}}+x\right )}{x}}\right )^2 \] Output:
(16+exp(5+8*(x-exp((-x*exp(5)+3)^2/x^2))/x))^2
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {e^{\frac {2 \left (-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x\right )}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (288-96 e^5 x+16 x^2\right )+e^{\frac {-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (4608-1536 e^5 x+256 x^2\right )}{x^4} \, dx=e^{13-\frac {16 e^{\frac {\left (-3+e^5 x\right )^2}{x^2}}}{x}} \left (e^{13}+32 e^{\frac {8 e^{\frac {\left (-3+e^5 x\right )^2}{x^2}}}{x}}\right ) \] Input:
Integrate[(E^((2*(-8*E^((9 - 6*E^5*x + E^10*x^2)/x^2) + 13*x))/x + (9 - 6* E^5*x + E^10*x^2)/x^2)*(288 - 96*E^5*x + 16*x^2) + E^((-8*E^((9 - 6*E^5*x + E^10*x^2)/x^2) + 13*x)/x + (9 - 6*E^5*x + E^10*x^2)/x^2)*(4608 - 1536*E^ 5*x + 256*x^2))/x^4,x]
Output:
E^(13 - (16*E^((-3 + E^5*x)^2/x^2))/x)*(E^13 + 32*E^((8*E^((-3 + E^5*x)^2/ 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.
| \(\displaystyle \int \frac {\left (16 x^2-96 e^5 x+288\right ) \exp \left (\frac {2 \left (13 x-8 e^{\frac {e^{10} x^2-6 e^5 x+9}{x^2}}\right )}{x}+\frac {e^{10} x^2-6 e^5 x+9}{x^2}\right )+\left (256 x^2-1536 e^5 x+4608\right ) \exp \left (\frac {13 x-8 e^{\frac {e^{10} x^2-6 e^5 x+9}{x^2}}}{x}+\frac {e^{10} x^2-6 e^5 x+9}{x^2}\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {16 \exp \left (-\frac {16 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}+\frac {9}{x^2}-\frac {6 e^5}{x}+26 \left (1+\frac {e^{10}}{26}\right )\right )}{x^2}+\frac {256 \exp \left (-\frac {8 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}+\frac {9}{x^2}-\frac {6 e^5}{x}+13 \left (1+\frac {e^{10}}{13}\right )\right )}{x^2}+\frac {288 \exp \left (-\frac {16 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}+\frac {9}{x^2}-\frac {6 e^5}{x}+26 \left (1+\frac {e^{10}}{26}\right )\right )}{x^4}+\frac {4608 \exp \left (-\frac {8 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}+\frac {9}{x^2}-\frac {6 e^5}{x}+13 \left (1+\frac {e^{10}}{13}\right )\right )}{x^4}-\frac {96 \exp \left (-\frac {16 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}+\frac {9}{x^2}-\frac {6 e^5}{x}+31 \left (1+\frac {e^{10}}{31}\right )\right )}{x^3}-\frac {1536 \exp \left (-\frac {8 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}+\frac {9}{x^2}-\frac {6 e^5}{x}+18 \left (1+\frac {e^{10}}{18}\right )\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \int \frac {\exp \left (26 \left (1+\frac {e^{10}}{26}\right )-\frac {16 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}-\frac {6 e^5}{x}+\frac {9}{x^2}\right )}{x^2}dx+256 \int \frac {\exp \left (13 \left (1+\frac {e^{10}}{13}\right )-\frac {8 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}-\frac {6 e^5}{x}+\frac {9}{x^2}\right )}{x^2}dx+288 \int \frac {\exp \left (26 \left (1+\frac {e^{10}}{26}\right )-\frac {16 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}-\frac {6 e^5}{x}+\frac {9}{x^2}\right )}{x^4}dx+4608 \int \frac {\exp \left (13 \left (1+\frac {e^{10}}{13}\right )-\frac {8 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}-\frac {6 e^5}{x}+\frac {9}{x^2}\right )}{x^4}dx-96 \int \frac {\exp \left (31 \left (1+\frac {e^{10}}{31}\right )-\frac {16 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}-\frac {6 e^5}{x}+\frac {9}{x^2}\right )}{x^3}dx-1536 \int \frac {\exp \left (18 \left (1+\frac {e^{10}}{18}\right )-\frac {8 e^{\frac {\left (e^5 x-3\right )^2}{x^2}}}{x}-\frac {6 e^5}{x}+\frac {9}{x^2}\right )}{x^3}dx\) |
Input:
Int[(E^((2*(-8*E^((9 - 6*E^5*x + E^10*x^2)/x^2) + 13*x))/x + (9 - 6*E^5*x + E^10*x^2)/x^2)*(288 - 96*E^5*x + 16*x^2) + E^((-8*E^((9 - 6*E^5*x + E^10 *x^2)/x^2) + 13*x)/x + (9 - 6*E^5*x + E^10*x^2)/x^2)*(4608 - 1536*E^5*x + 256*x^2))/x^4,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(30)=60\).
Time = 1.55 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91
| method | result | size |
| risch | \({\mathrm e}^{\frac {-16 \,{\mathrm e}^{\frac {x^{2} {\mathrm e}^{10}-6 x \,{\mathrm e}^{5}+9}{x^{2}}}+26 x}{x}}+32 \,{\mathrm e}^{\frac {-8 \,{\mathrm e}^{\frac {x^{2} {\mathrm e}^{10}-6 x \,{\mathrm e}^{5}+9}{x^{2}}}+13 x}{x}}\) | \(63\) |
| parts | \({\mathrm e}^{\frac {-16 \,{\mathrm e}^{\frac {x^{2} {\mathrm e}^{10}-6 x \,{\mathrm e}^{5}+9}{x^{2}}}+26 x}{x}}+32 \,{\mathrm e}^{\frac {-8 \,{\mathrm e}^{\frac {x^{2} {\mathrm e}^{10}-6 x \,{\mathrm e}^{5}+9}{x^{2}}}+13 x}{x}}\) | \(68\) |
| parallelrisch | \({\mathrm e}^{-\frac {2 \left (8 \,{\mathrm e}^{\frac {x^{2} {\mathrm e}^{10}-6 x \,{\mathrm e}^{5}+9}{x^{2}}}-13 x \right )}{x}}+32 \,{\mathrm e}^{-\frac {8 \,{\mathrm e}^{\frac {x^{2} {\mathrm e}^{10}-6 x \,{\mathrm e}^{5}+9}{x^{2}}}-13 x}{x}}\) | \(70\) |
| norman | \(\frac {x^{3} {\mathrm e}^{\frac {-16 \,{\mathrm e}^{\frac {x^{2} {\mathrm e}^{10}-6 x \,{\mathrm e}^{5}+9}{x^{2}}}+26 x}{x}}+32 x^{3} {\mathrm e}^{\frac {-8 \,{\mathrm e}^{\frac {x^{2} {\mathrm e}^{10}-6 x \,{\mathrm e}^{5}+9}{x^{2}}}+13 x}{x}}}{x^{3}}\) | \(79\) |
Input:
int(((-96*x*exp(5)+16*x^2+288)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)*exp((- 8*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)+13*x)/x)^2+(-1536*x*exp(5)+256*x^2+ 4608)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)*exp((-8*exp((x^2*exp(5)^2-6*x*e xp(5)+9)/x^2)+13*x)/x))/x^4,x,method=_RETURNVERBOSE)
Output:
exp(2*(-8*exp((x^2*exp(10)-6*x*exp(5)+9)/x^2)+13*x)/x)+32*exp((-8*exp((x^2 *exp(10)-6*x*exp(5)+9)/x^2)+13*x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (29) = 58\).
Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 3.94 \[ \int \frac {e^{\frac {2 \left (-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x\right )}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (288-96 e^5 x+16 x^2\right )+e^{\frac {-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (4608-1536 e^5 x+256 x^2\right )}{x^4} \, dx={\left (e^{\left (\frac {2 \, {\left (x^{2} e^{10} + 13 \, x^{2} - 6 \, x e^{5} - 8 \, x e^{\left (\frac {x^{2} e^{10} - 6 \, x e^{5} + 9}{x^{2}}\right )} + 9\right )}}{x^{2}}\right )} + 32 \, e^{\left (\frac {x^{2} e^{10} + 13 \, x^{2} - 6 \, x e^{5} - 8 \, x e^{\left (\frac {x^{2} e^{10} - 6 \, x e^{5} + 9}{x^{2}}\right )} + 9}{x^{2}} + \frac {x^{2} e^{10} - 6 \, x e^{5} + 9}{x^{2}}\right )}\right )} e^{\left (-\frac {2 \, {\left (x^{2} e^{10} - 6 \, x e^{5} + 9\right )}}{x^{2}}\right )} \] Input:
integrate(((-96*x*exp(5)+16*x^2+288)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)* exp((-8*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)+13*x)/x)^2+(-1536*x*exp(5)+25 6*x^2+4608)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)*exp((-8*exp((x^2*exp(5)^2 -6*x*exp(5)+9)/x^2)+13*x)/x))/x^4,x, algorithm="fricas")
Output:
(e^(2*(x^2*e^10 + 13*x^2 - 6*x*e^5 - 8*x*e^((x^2*e^10 - 6*x*e^5 + 9)/x^2) + 9)/x^2) + 32*e^((x^2*e^10 + 13*x^2 - 6*x*e^5 - 8*x*e^((x^2*e^10 - 6*x*e^ 5 + 9)/x^2) + 9)/x^2 + (x^2*e^10 - 6*x*e^5 + 9)/x^2))*e^(-2*(x^2*e^10 - 6* x*e^5 + 9)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {e^{\frac {2 \left (-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x\right )}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (288-96 e^5 x+16 x^2\right )+e^{\frac {-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (4608-1536 e^5 x+256 x^2\right )}{x^4} \, dx=e^{\frac {2 \cdot \left (13 x - 8 e^{\frac {x^{2} e^{10} - 6 x e^{5} + 9}{x^{2}}}\right )}{x}} + 32 e^{\frac {13 x - 8 e^{\frac {x^{2} e^{10} - 6 x e^{5} + 9}{x^{2}}}}{x}} \] Input:
integrate(((-96*x*exp(5)+16*x**2+288)*exp((x**2*exp(5)**2-6*x*exp(5)+9)/x* *2)*exp((-8*exp((x**2*exp(5)**2-6*x*exp(5)+9)/x**2)+13*x)/x)**2+(-1536*x*e xp(5)+256*x**2+4608)*exp((x**2*exp(5)**2-6*x*exp(5)+9)/x**2)*exp((-8*exp(( x**2*exp(5)**2-6*x*exp(5)+9)/x**2)+13*x)/x))/x**4,x)
Output:
exp(2*(13*x - 8*exp((x**2*exp(10) - 6*x*exp(5) + 9)/x**2))/x) + 32*exp((13 *x - 8*exp((x**2*exp(10) - 6*x*exp(5) + 9)/x**2))/x)
Time = 0.33 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\frac {2 \left (-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x\right )}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (288-96 e^5 x+16 x^2\right )+e^{\frac {-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (4608-1536 e^5 x+256 x^2\right )}{x^4} \, dx={\left (e^{26} + 32 \, e^{\left (\frac {8 \, e^{\left (-\frac {6 \, e^{5}}{x} + \frac {9}{x^{2}} + e^{10}\right )}}{x} + 13\right )}\right )} e^{\left (-\frac {16 \, e^{\left (-\frac {6 \, e^{5}}{x} + \frac {9}{x^{2}} + e^{10}\right )}}{x}\right )} \] Input:
integrate(((-96*x*exp(5)+16*x^2+288)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)* exp((-8*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)+13*x)/x)^2+(-1536*x*exp(5)+25 6*x^2+4608)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)*exp((-8*exp((x^2*exp(5)^2 -6*x*exp(5)+9)/x^2)+13*x)/x))/x^4,x, algorithm="maxima")
Output:
(e^26 + 32*e^(8*e^(-6*e^5/x + 9/x^2 + e^10)/x + 13))*e^(-16*e^(-6*e^5/x + 9/x^2 + e^10)/x)
\[ \int \frac {e^{\frac {2 \left (-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x\right )}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (288-96 e^5 x+16 x^2\right )+e^{\frac {-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (4608-1536 e^5 x+256 x^2\right )}{x^4} \, dx=\int { \frac {16 \, {\left ({\left (x^{2} - 6 \, x e^{5} + 18\right )} e^{\left (\frac {2 \, {\left (13 \, x - 8 \, e^{\left (\frac {x^{2} e^{10} - 6 \, x e^{5} + 9}{x^{2}}\right )}\right )}}{x} + \frac {x^{2} e^{10} - 6 \, x e^{5} + 9}{x^{2}}\right )} + 16 \, {\left (x^{2} - 6 \, x e^{5} + 18\right )} e^{\left (\frac {13 \, x - 8 \, e^{\left (\frac {x^{2} e^{10} - 6 \, x e^{5} + 9}{x^{2}}\right )}}{x} + \frac {x^{2} e^{10} - 6 \, x e^{5} + 9}{x^{2}}\right )}\right )}}{x^{4}} \,d x } \] Input:
integrate(((-96*x*exp(5)+16*x^2+288)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)* exp((-8*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)+13*x)/x)^2+(-1536*x*exp(5)+25 6*x^2+4608)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)*exp((-8*exp((x^2*exp(5)^2 -6*x*exp(5)+9)/x^2)+13*x)/x))/x^4,x, algorithm="giac")
Output:
integrate(16*((x^2 - 6*x*e^5 + 18)*e^(2*(13*x - 8*e^((x^2*e^10 - 6*x*e^5 + 9)/x^2))/x + (x^2*e^10 - 6*x*e^5 + 9)/x^2) + 16*(x^2 - 6*x*e^5 + 18)*e^(( 13*x - 8*e^((x^2*e^10 - 6*x*e^5 + 9)/x^2))/x + (x^2*e^10 - 6*x*e^5 + 9)/x^ 2))/x^4, x)
Time = 3.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64 \[ \int \frac {e^{\frac {2 \left (-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x\right )}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (288-96 e^5 x+16 x^2\right )+e^{\frac {-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (4608-1536 e^5 x+256 x^2\right )}{x^4} \, dx=32\,{\mathrm {e}}^{-\frac {8\,{\mathrm {e}}^{-\frac {6\,{\mathrm {e}}^5}{x}}\,{\mathrm {e}}^{\frac {9}{x^2}}\,{\mathrm {e}}^{{\mathrm {e}}^{10}}}{x}}\,{\mathrm {e}}^{13}+{\mathrm {e}}^{-\frac {16\,{\mathrm {e}}^{-\frac {6\,{\mathrm {e}}^5}{x}}\,{\mathrm {e}}^{\frac {9}{x^2}}\,{\mathrm {e}}^{{\mathrm {e}}^{10}}}{x}}\,{\mathrm {e}}^{26} \] Input:
int((exp((2*(13*x - 8*exp((x^2*exp(10) - 6*x*exp(5) + 9)/x^2)))/x)*exp((x^ 2*exp(10) - 6*x*exp(5) + 9)/x^2)*(16*x^2 - 96*x*exp(5) + 288) + exp((13*x - 8*exp((x^2*exp(10) - 6*x*exp(5) + 9)/x^2))/x)*exp((x^2*exp(10) - 6*x*exp (5) + 9)/x^2)*(256*x^2 - 1536*x*exp(5) + 4608))/x^4,x)
Output:
32*exp(-(8*exp(-(6*exp(5))/x)*exp(9/x^2)*exp(exp(10)))/x)*exp(13) + exp(-( 16*exp(-(6*exp(5))/x)*exp(9/x^2)*exp(exp(10)))/x)*exp(26)
Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {e^{\frac {2 \left (-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x\right )}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (288-96 e^5 x+16 x^2\right )+e^{\frac {-8 e^{\frac {9-6 e^5 x+e^{10} x^2}{x^2}}+13 x}{x}+\frac {9-6 e^5 x+e^{10} x^2}{x^2}} \left (4608-1536 e^5 x+256 x^2\right )}{x^4} \, dx=\frac {e^{13} \left (32 e^{\frac {8 e^{\frac {e^{10} x^{2}+9}{x^{2}}}}{e^{\frac {6 e^{5}}{x}} x}}+e^{13}\right )}{e^{\frac {16 e^{\frac {e^{10} x^{2}+9}{x^{2}}}}{e^{\frac {6 e^{5}}{x}} x}}} \] Input:
int(((-96*x*exp(5)+16*x^2+288)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)*exp((- 8*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)+13*x)/x)^2+(-1536*x*exp(5)+256*x^2+ 4608)*exp((x^2*exp(5)^2-6*x*exp(5)+9)/x^2)*exp((-8*exp((x^2*exp(5)^2-6*x*e xp(5)+9)/x^2)+13*x)/x))/x^4,x)
Output:
(e**13*(32*e**((8*e**((e**10*x**2 + 9)/x**2))/(e**((6*e**5)/x)*x)) + e**13 ))/e**((16*e**((e**10*x**2 + 9)/x**2))/(e**((6*e**5)/x)*x))