Integrand size = 275, antiderivative size = 32 \[ \int \frac {e^{2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}} \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{405 x^2+5 e^2 x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx=\frac {e^{2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}}}{e^5-x} \] Output:
exp(2/exp(1/5*exp(4)/((3+x)^2+exp(1))))/(exp(5)-x)
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^{2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}} \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{405 x^2+5 e^2 x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx=-\frac {e^{2 e^{-\frac {e^4}{5 \left (e+(3+x)^2\right )}}}}{-e^5+x} \] Input:
Integrate[(E^(2/E^(E^4/(45 + 5*E + 30*x + 5*x^2)) - E^4/(45 + 5*E + 30*x + 5*x^2))*(E^9*(12 + 4*x) + E^4*(-12*x - 4*x^2) + E^(E^4/(45 + 5*E + 30*x + 5*x^2))*(405 + 5*E^2 + 540*x + 270*x^2 + 60*x^3 + 5*x^4 + E*(90 + 60*x + 10*x^2))))/(405*x^2 + 5*E^2*x^2 + 540*x^3 + 270*x^4 + 60*x^5 + 5*x^6 + E*( 90*x^2 + 60*x^3 + 10*x^4) + E^10*(405 + 5*E^2 + 540*x + 270*x^2 + 60*x^3 + 5*x^4 + E*(90 + 60*x + 10*x^2)) + E^5*(-810*x - 10*E^2*x - 1080*x^2 - 540 *x^3 - 120*x^4 - 10*x^5 + E*(-180*x - 120*x^2 - 20*x^3))),x]
Output:
-(E^(2/E^(E^4/(5*(E + (3 + x)^2))))/(-E^5 + 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.
| \(\displaystyle \int \frac {\left (e^4 \left (-4 x^2-12 x\right )+e^{\frac {e^4}{5 x^2+30 x+5 e+45}} \left (5 x^4+60 x^3+270 x^2+e \left (10 x^2+60 x+90\right )+540 x+5 e^2+405\right )+e^9 (4 x+12)\right ) \exp \left (2 e^{-\frac {e^4}{5 x^2+30 x+5 e+45}}-\frac {e^4}{5 x^2+30 x+5 e+45}\right )}{5 x^6+60 x^5+270 x^4+540 x^3+5 e^2 x^2+405 x^2+e \left (10 x^4+60 x^3+90 x^2\right )+e^{10} \left (5 x^4+60 x^3+270 x^2+e \left (10 x^2+60 x+90\right )+540 x+5 e^2+405\right )+e^5 \left (-10 x^5-120 x^4-540 x^3-1080 x^2+e \left (-20 x^3-120 x^2-180 x\right )-10 e^2 x-810 x\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (e^4 \left (-4 x^2-12 x\right )+e^{\frac {e^4}{5 x^2+30 x+5 e+45}} \left (5 x^4+60 x^3+270 x^2+e \left (10 x^2+60 x+90\right )+540 x+5 e^2+405\right )+e^9 (4 x+12)\right ) \exp \left (2 e^{-\frac {e^4}{5 x^2+30 x+5 e+45}}-\frac {e^4}{5 x^2+30 x+5 e+45}\right )}{5 x^6+60 x^5+270 x^4+540 x^3+\left (405+5 e^2\right ) x^2+e \left (10 x^4+60 x^3+90 x^2\right )+e^{10} \left (5 x^4+60 x^3+270 x^2+e \left (10 x^2+60 x+90\right )+540 x+5 e^2+405\right )+e^5 \left (-10 x^5-120 x^4-540 x^3-1080 x^2+e \left (-20 x^3-120 x^2-180 x\right )-10 e^2 x-810 x\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {4 \left (3+e^5\right ) \left (e^4 \left (-4 x^2-12 x\right )+e^{\frac {e^4}{5 x^2+30 x+5 e+45}} \left (5 x^4+60 x^3+270 x^2+e \left (10 x^2+60 x+90\right )+540 x+5 e^2+405\right )+e^9 (4 x+12)\right ) \exp \left (2 e^{-\frac {e^4}{5 x^2+30 x+5 e+45}}-\frac {e^4}{5 x^2+30 x+5 e+45}\right )}{5 \left (9+e+6 e^5+e^{10}\right )^3 \left (e^5-x\right )}+\frac {\left (4 \left (3+e^5\right ) x+3 e^{10}+30 e^5-e+63\right ) \left (e^4 \left (-4 x^2-12 x\right )+e^{\frac {e^4}{5 x^2+30 x+5 e+45}} \left (5 x^4+60 x^3+270 x^2+e \left (10 x^2+60 x+90\right )+540 x+5 e^2+405\right )+e^9 (4 x+12)\right ) \exp \left (2 e^{-\frac {e^4}{5 x^2+30 x+5 e+45}}-\frac {e^4}{5 x^2+30 x+5 e+45}\right )}{5 \left (9+e+6 e^5+e^{10}\right )^3 \left (x^2+6 x+e+9\right )}+\frac {\left (e^4 \left (-4 x^2-12 x\right )+e^{\frac {e^4}{5 x^2+30 x+5 e+45}} \left (5 x^4+60 x^3+270 x^2+e \left (10 x^2+60 x+90\right )+540 x+5 e^2+405\right )+e^9 (4 x+12)\right ) \exp \left (2 e^{-\frac {e^4}{5 x^2+30 x+5 e+45}}-\frac {e^4}{5 x^2+30 x+5 e+45}\right )}{5 \left (9+e+6 e^5+e^{10}\right )^2 \left (e^5-x\right )^2}+\frac {\left (2 \left (3+e^5\right ) x+e^{10}+12 e^5-e+27\right ) \left (e^4 \left (-4 x^2-12 x\right )+e^{\frac {e^4}{5 x^2+30 x+5 e+45}} \left (5 x^4+60 x^3+270 x^2+e \left (10 x^2+60 x+90\right )+540 x+5 e^2+405\right )+e^9 (4 x+12)\right ) \exp \left (2 e^{-\frac {e^4}{5 x^2+30 x+5 e+45}}-\frac {e^4}{5 x^2+30 x+5 e+45}\right )}{5 \left (9+e+6 e^5+e^{10}\right )^2 \left (x^2+6 x+e+9\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (5 e^{\frac {e^4}{5 \left ((x+3)^2+e\right )}} (x+3)^4+10 e^{\frac {e^4}{5 \left ((x+3)^2+e\right )}+1} (x+3)^2-4 e^4 x (x+3)+4 e^9 (x+3)+5 e^{\frac {e^4}{5 \left ((x+3)^2+e\right )}+2}\right ) \exp \left (2 e^{-\frac {e^4}{5 \left ((x+3)^2+e\right )}}-\frac {e^4}{5 \left ((x+3)^2+e\right )}\right )}{5 \left (e^5-x\right )^2 \left (x^2+6 x+e+9\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (2 e^{-\frac {e^4}{5 \left ((x+3)^2+e\right )}}-\frac {e^4}{5 \left ((x+3)^2+e\right )}\right ) \left (5 e^{\frac {e^4}{5 \left ((x+3)^2+e\right )}} (x+3)^4+10 e^{1+\frac {e^4}{5 \left ((x+3)^2+e\right )}} (x+3)^2-4 e^4 x (x+3)+4 e^9 (x+3)+5 e^{2+\frac {e^4}{5 \left ((x+3)^2+e\right )}}\right )}{\left (e^5-x\right )^2 \left (x^2+6 x+e+9\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (-\frac {4 \exp \left (2 e^{-\frac {e^4}{5 \left ((x+3)^2+e\right )}}+4-\frac {e^4}{5 \left ((x+3)^2+e\right )}\right ) x (x+3)}{\left (e^5-x\right )^2 \left (x^2+6 x+e+9\right )^2}+\frac {4 \exp \left (2 e^{-\frac {e^4}{5 \left ((x+3)^2+e\right )}}+9-\frac {e^4}{5 \left ((x+3)^2+e\right )}\right ) (x+3)}{\left (e^5-x\right )^2 \left (x^2+6 x+e+9\right )^2}+\frac {5 \exp \left (2 e^{-\frac {e^4}{5 \left ((x+3)^2+e\right )}}-\frac {e^4}{5 \left ((x+3)^2+e\right )}+\frac {e^4}{5 \left (x^2+6 x+e+9\right )}\right )}{\left (e^5-x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {1}{5} \int \left (-\frac {4 \exp \left (2 e^{-\frac {e^4}{5 \left ((x+3)^2+e\right )}}+4-\frac {e^4}{5 \left ((x+3)^2+e\right )}\right ) x (x+3)}{\left (e^5-x\right )^2 \left (x^2+6 x+e+9\right )^2}+\frac {4 \exp \left (2 e^{-\frac {e^4}{5 \left ((x+3)^2+e\right )}}+9-\frac {e^4}{5 \left ((x+3)^2+e\right )}\right ) (x+3)}{\left (e^5-x\right )^2 \left (x^2+6 x+e+9\right )^2}+\frac {5 e^{2 e^{-\frac {e^4}{5 \left (x^2+6 x+e+9\right )}}}}{\left (e^5-x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {1}{5} \int \left (-\frac {4 \exp \left (2 e^{-\frac {e^4}{5 \left ((x+3)^2+e\right )}}+4-\frac {e^4}{5 \left ((x+3)^2+e\right )}\right ) x (x+3)}{\left (e^5-x\right )^2 \left (x^2+6 x+e+9\right )^2}+\frac {4 \exp \left (2 e^{-\frac {e^4}{5 \left ((x+3)^2+e\right )}}+9-\frac {e^4}{5 \left ((x+3)^2+e\right )}\right ) (x+3)}{\left (e^5-x\right )^2 \left (x^2+6 x+e+9\right )^2}+\frac {5 e^{2 e^{-\frac {e^4}{5 \left (x^2+6 x+e+9\right )}}}}{\left (e^5-x\right )^2}\right )dx\) |
Input:
Int[(E^(2/E^(E^4/(45 + 5*E + 30*x + 5*x^2)) - E^4/(45 + 5*E + 30*x + 5*x^2 ))*(E^9*(12 + 4*x) + E^4*(-12*x - 4*x^2) + E^(E^4/(45 + 5*E + 30*x + 5*x^2 ))*(405 + 5*E^2 + 540*x + 270*x^2 + 60*x^3 + 5*x^4 + E*(90 + 60*x + 10*x^2 ))))/(405*x^2 + 5*E^2*x^2 + 540*x^3 + 270*x^4 + 60*x^5 + 5*x^6 + E*(90*x^2 + 60*x^3 + 10*x^4) + E^10*(405 + 5*E^2 + 540*x + 270*x^2 + 60*x^3 + 5*x^4 + E*(90 + 60*x + 10*x^2)) + E^5*(-810*x - 10*E^2*x - 1080*x^2 - 540*x^3 - 120*x^4 - 10*x^5 + E*(-180*x - 120*x^2 - 20*x^3))),x]
Output:
$Aborted
Time = 22.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(\frac {{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \left (x^{2}+{\mathrm e}+6 x +9\right )}}}}{{\mathrm e}^{5}-x}\) | \(30\) |
| parallelrisch | \(\frac {5 \,{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \left (x^{2}+{\mathrm e}+6 x +9\right )}}} x^{2}+5 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \left (x^{2}+{\mathrm e}+6 x +9\right )}}}+30 x \,{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \left (x^{2}+{\mathrm e}+6 x +9\right )}}}+45 \,{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \left (x^{2}+{\mathrm e}+6 x +9\right )}}}}{5 \left ({\mathrm e}^{5}-x \right ) \left (x^{2}+{\mathrm e}+6 x +9\right )}\) | \(126\) |
| norman | \(\frac {\left (\left (9+{\mathrm e}\right ) {\mathrm e}^{\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}} {\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}}}+{\mathrm e}^{\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}} x^{2} {\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}}}+6 \,{\mathrm e}^{\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}} x \,{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}}}\right ) {\mathrm e}^{-\frac {{\mathrm e}^{4}}{5 \,{\mathrm e}+5 x^{2}+30 x +45}}}{\left ({\mathrm e}^{5}-x \right ) \left (x^{2}+{\mathrm e}+6 x +9\right )}\) | \(192\) |
Input:
int(((5*exp(1)^2+(10*x^2+60*x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+405)*e xp(exp(4)/(5*exp(1)+5*x^2+30*x+45))+(4*x+12)*exp(4)*exp(5)+(-4*x^2-12*x)*e xp(4))*exp(2/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)))/((5*exp(1)^2+(10*x^2+60 *x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+405)*exp(5)^2+(-10*x*exp(1)^2+(-2 0*x^3-120*x^2-180*x)*exp(1)-10*x^5-120*x^4-540*x^3-1080*x^2-810*x)*exp(5)+ 5*x^2*exp(1)^2+(10*x^4+60*x^3+90*x^2)*exp(1)+5*x^6+60*x^5+270*x^4+540*x^3+ 405*x^2)/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)),x,method=_RETURNVERBOSE)
Output:
1/(exp(5)-x)*exp(2*exp(-1/5*exp(4)/(x^2+exp(1)+6*x+9)))
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (28) = 56\).
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.94 \[ \int \frac {e^{2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}} \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{405 x^2+5 e^2 x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx=-\frac {e^{\left (\frac {{\left (10 \, x^{2} + 60 \, x + 10 \, e - e^{\left (\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}} + 4\right )} + 90\right )} e^{\left (-\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}}\right )}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}} + \frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}}\right )}}{x - e^{5}} \] Input:
integrate(((5*exp(1)^2+(10*x^2+60*x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+ 405)*exp(exp(4)/(5*exp(1)+5*x^2+30*x+45))+(4*x+12)*exp(4)*exp(5)+(-4*x^2-1 2*x)*exp(4))*exp(2/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)))/((5*exp(1)^2+(10* x^2+60*x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+405)*exp(5)^2+(-10*x*exp(1) ^2+(-20*x^3-120*x^2-180*x)*exp(1)-10*x^5-120*x^4-540*x^3-1080*x^2-810*x)*e xp(5)+5*x^2*exp(1)^2+(10*x^4+60*x^3+90*x^2)*exp(1)+5*x^6+60*x^5+270*x^4+54 0*x^3+405*x^2)/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)),x, algorithm="fricas")
Output:
-e^(1/5*(10*x^2 + 60*x + 10*e - e^(1/5*e^4/(x^2 + 6*x + e + 9) + 4) + 90)* e^(-1/5*e^4/(x^2 + 6*x + e + 9))/(x^2 + 6*x + e + 9) + 1/5*e^4/(x^2 + 6*x + e + 9))/(x - e^5)
Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}} \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{405 x^2+5 e^2 x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx=- \frac {e^{2 e^{- \frac {e^{4}}{5 x^{2} + 30 x + 5 e + 45}}}}{x - e^{5}} \] Input:
integrate(((5*exp(1)**2+(10*x**2+60*x+90)*exp(1)+5*x**4+60*x**3+270*x**2+5 40*x+405)*exp(exp(4)/(5*exp(1)+5*x**2+30*x+45))+(4*x+12)*exp(4)*exp(5)+(-4 *x**2-12*x)*exp(4))*exp(2/exp(exp(4)/(5*exp(1)+5*x**2+30*x+45)))/((5*exp(1 )**2+(10*x**2+60*x+90)*exp(1)+5*x**4+60*x**3+270*x**2+540*x+405)*exp(5)**2 +(-10*x*exp(1)**2+(-20*x**3-120*x**2-180*x)*exp(1)-10*x**5-120*x**4-540*x* *3-1080*x**2-810*x)*exp(5)+5*x**2*exp(1)**2+(10*x**4+60*x**3+90*x**2)*exp( 1)+5*x**6+60*x**5+270*x**4+540*x**3+405*x**2)/exp(exp(4)/(5*exp(1)+5*x**2+ 30*x+45)),x)
Output:
-exp(2*exp(-exp(4)/(5*x**2 + 30*x + 5*E + 45)))/(x - exp(5))
\[ \int \frac {e^{2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}} \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{405 x^2+5 e^2 x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx=\int { \frac {{\left (4 \, {\left (x + 3\right )} e^{9} - 4 \, {\left (x^{2} + 3 \, x\right )} e^{4} + 5 \, {\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 2 \, {\left (x^{2} + 6 \, x + 9\right )} e + 108 \, x + e^{2} + 81\right )} e^{\left (\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}}\right )}\right )} e^{\left (-\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}} + 2 \, e^{\left (-\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}}\right )}\right )}}{5 \, {\left (x^{6} + 12 \, x^{5} + 54 \, x^{4} + 108 \, x^{3} + x^{2} e^{2} + 81 \, x^{2} + {\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 2 \, {\left (x^{2} + 6 \, x + 9\right )} e + 108 \, x + e^{2} + 81\right )} e^{10} - 2 \, {\left (x^{5} + 12 \, x^{4} + 54 \, x^{3} + 108 \, x^{2} + x e^{2} + 2 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} e + 81 \, x\right )} e^{5} + 2 \, {\left (x^{4} + 6 \, x^{3} + 9 \, x^{2}\right )} e\right )}} \,d x } \] Input:
integrate(((5*exp(1)^2+(10*x^2+60*x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+ 405)*exp(exp(4)/(5*exp(1)+5*x^2+30*x+45))+(4*x+12)*exp(4)*exp(5)+(-4*x^2-1 2*x)*exp(4))*exp(2/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)))/((5*exp(1)^2+(10* x^2+60*x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+405)*exp(5)^2+(-10*x*exp(1) ^2+(-20*x^3-120*x^2-180*x)*exp(1)-10*x^5-120*x^4-540*x^3-1080*x^2-810*x)*e xp(5)+5*x^2*exp(1)^2+(10*x^4+60*x^3+90*x^2)*exp(1)+5*x^6+60*x^5+270*x^4+54 0*x^3+405*x^2)/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)),x, algorithm="maxima")
Output:
1/5*integrate((4*(x + 3)*e^9 - 4*(x^2 + 3*x)*e^4 + 5*(x^4 + 12*x^3 + 54*x^ 2 + 2*(x^2 + 6*x + 9)*e + 108*x + e^2 + 81)*e^(1/5*e^4/(x^2 + 6*x + e + 9) ))*e^(-1/5*e^4/(x^2 + 6*x + e + 9) + 2*e^(-1/5*e^4/(x^2 + 6*x + e + 9)))/( x^6 + 12*x^5 + 54*x^4 + 108*x^3 + x^2*e^2 + 81*x^2 + (x^4 + 12*x^3 + 54*x^ 2 + 2*(x^2 + 6*x + 9)*e + 108*x + e^2 + 81)*e^10 - 2*(x^5 + 12*x^4 + 54*x^ 3 + 108*x^2 + x*e^2 + 2*(x^3 + 6*x^2 + 9*x)*e + 81*x)*e^5 + 2*(x^4 + 6*x^3 + 9*x^2)*e), x)
\[ \int \frac {e^{2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}} \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{405 x^2+5 e^2 x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx=\int { \frac {{\left (4 \, {\left (x + 3\right )} e^{9} - 4 \, {\left (x^{2} + 3 \, x\right )} e^{4} + 5 \, {\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 2 \, {\left (x^{2} + 6 \, x + 9\right )} e + 108 \, x + e^{2} + 81\right )} e^{\left (\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}}\right )}\right )} e^{\left (-\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}} + 2 \, e^{\left (-\frac {e^{4}}{5 \, {\left (x^{2} + 6 \, x + e + 9\right )}}\right )}\right )}}{5 \, {\left (x^{6} + 12 \, x^{5} + 54 \, x^{4} + 108 \, x^{3} + x^{2} e^{2} + 81 \, x^{2} + {\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 2 \, {\left (x^{2} + 6 \, x + 9\right )} e + 108 \, x + e^{2} + 81\right )} e^{10} - 2 \, {\left (x^{5} + 12 \, x^{4} + 54 \, x^{3} + 108 \, x^{2} + x e^{2} + 2 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} e + 81 \, x\right )} e^{5} + 2 \, {\left (x^{4} + 6 \, x^{3} + 9 \, x^{2}\right )} e\right )}} \,d x } \] Input:
integrate(((5*exp(1)^2+(10*x^2+60*x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+ 405)*exp(exp(4)/(5*exp(1)+5*x^2+30*x+45))+(4*x+12)*exp(4)*exp(5)+(-4*x^2-1 2*x)*exp(4))*exp(2/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)))/((5*exp(1)^2+(10* x^2+60*x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+405)*exp(5)^2+(-10*x*exp(1) ^2+(-20*x^3-120*x^2-180*x)*exp(1)-10*x^5-120*x^4-540*x^3-1080*x^2-810*x)*e xp(5)+5*x^2*exp(1)^2+(10*x^4+60*x^3+90*x^2)*exp(1)+5*x^6+60*x^5+270*x^4+54 0*x^3+405*x^2)/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)),x, algorithm="giac")
Output:
integrate(1/5*(4*(x + 3)*e^9 - 4*(x^2 + 3*x)*e^4 + 5*(x^4 + 12*x^3 + 54*x^ 2 + 2*(x^2 + 6*x + 9)*e + 108*x + e^2 + 81)*e^(1/5*e^4/(x^2 + 6*x + e + 9) ))*e^(-1/5*e^4/(x^2 + 6*x + e + 9) + 2*e^(-1/5*e^4/(x^2 + 6*x + e + 9)))/( x^6 + 12*x^5 + 54*x^4 + 108*x^3 + x^2*e^2 + 81*x^2 + (x^4 + 12*x^3 + 54*x^ 2 + 2*(x^2 + 6*x + 9)*e + 108*x + e^2 + 81)*e^10 - 2*(x^5 + 12*x^4 + 54*x^ 3 + 108*x^2 + x*e^2 + 2*(x^3 + 6*x^2 + 9*x)*e + 81*x)*e^5 + 2*(x^4 + 6*x^3 + 9*x^2)*e), x)
Time = 9.60 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^{2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}} \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{405 x^2+5 e^2 x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx=-\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^4}{5\,\left (x^2+6\,x+\mathrm {e}+9\right )}}}}{x-{\mathrm {e}}^5} \] Input:
int((exp(2*exp(-exp(4)/(30*x + 5*exp(1) + 5*x^2 + 45)))*exp(-exp(4)/(30*x + 5*exp(1) + 5*x^2 + 45))*(exp(exp(4)/(30*x + 5*exp(1) + 5*x^2 + 45))*(540 *x + 5*exp(2) + exp(1)*(60*x + 10*x^2 + 90) + 270*x^2 + 60*x^3 + 5*x^4 + 4 05) - exp(4)*(12*x + 4*x^2) + exp(9)*(4*x + 12)))/(exp(10)*(540*x + 5*exp( 2) + exp(1)*(60*x + 10*x^2 + 90) + 270*x^2 + 60*x^3 + 5*x^4 + 405) - exp(5 )*(810*x + 10*x*exp(2) + exp(1)*(180*x + 120*x^2 + 20*x^3) + 1080*x^2 + 54 0*x^3 + 120*x^4 + 10*x^5) + 5*x^2*exp(2) + exp(1)*(90*x^2 + 60*x^3 + 10*x^ 4) + 405*x^2 + 540*x^3 + 270*x^4 + 60*x^5 + 5*x^6),x)
Output:
-exp(2*exp(-exp(4)/(5*(6*x + exp(1) + x^2 + 9))))/(x - exp(5))
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^{2 e^{-\frac {e^4}{45+5 e+30 x+5 x^2}}-\frac {e^4}{45+5 e+30 x+5 x^2}} \left (e^9 (12+4 x)+e^4 \left (-12 x-4 x^2\right )+e^{\frac {e^4}{45+5 e+30 x+5 x^2}} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )\right )}{405 x^2+5 e^2 x^2+540 x^3+270 x^4+60 x^5+5 x^6+e \left (90 x^2+60 x^3+10 x^4\right )+e^{10} \left (405+5 e^2+540 x+270 x^2+60 x^3+5 x^4+e \left (90+60 x+10 x^2\right )\right )+e^5 \left (-810 x-10 e^2 x-1080 x^2-540 x^3-120 x^4-10 x^5+e \left (-180 x-120 x^2-20 x^3\right )\right )} \, dx=\frac {e^{\frac {2}{e^{\frac {e^{4}}{5 x^{2}+5 e +30 x +45}}}}}{e^{5}-x} \] Input:
int(((5*exp(1)^2+(10*x^2+60*x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+405)*e xp(exp(4)/(5*exp(1)+5*x^2+30*x+45))+(4*x+12)*exp(4)*exp(5)+(-4*x^2-12*x)*e xp(4))*exp(2/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)))/((5*exp(1)^2+(10*x^2+60 *x+90)*exp(1)+5*x^4+60*x^3+270*x^2+540*x+405)*exp(5)^2+(-10*x*exp(1)^2+(-2 0*x^3-120*x^2-180*x)*exp(1)-10*x^5-120*x^4-540*x^3-1080*x^2-810*x)*exp(5)+ 5*x^2*exp(1)^2+(10*x^4+60*x^3+90*x^2)*exp(1)+5*x^6+60*x^5+270*x^4+540*x^3+ 405*x^2)/exp(exp(4)/(5*exp(1)+5*x^2+30*x+45)),x)
Output:
e**(2/e**(e**4/(5*e + 5*x**2 + 30*x + 45)))/(e**5 - x)