Integrand size = 218, antiderivative size = 38 \[ \int \frac {15 e^{4 e^x}+540 e^{4 x}+15 e^{\frac {2 e^{e^x}}{5}+4 x}+e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 e^x+2 x}-12 e^{e^x+5 x}\right )}{5 e^{4 e^x} x^2+5 e^{\frac {2 e^{e^x}}{5}+4 x} x^2+e^{2 e^x+2 x} \left (-40 x+60 x^2\right )+e^{4 x} \left (80-240 x+180 x^2\right )+e^{\frac {e^{e^x}}{5}} \left (-10 e^{2 e^x+2 x} x^2+e^{4 x} \left (40 x-60 x^2\right )\right )} \, dx=\frac {3}{\frac {4}{6-e^{\frac {e^{e^x}}{5}}+e^{2 e^x-2 x}}-x} \] Output:
3/(4/(6-exp(1/5*exp(exp(x)))+exp(exp(x))^2/exp(x)^2)-x)
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.45 \[ \int \frac {15 e^{4 e^x}+540 e^{4 x}+15 e^{\frac {2 e^{e^x}}{5}+4 x}+e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 e^x+2 x}-12 e^{e^x+5 x}\right )}{5 e^{4 e^x} x^2+5 e^{\frac {2 e^{e^x}}{5}+4 x} x^2+e^{2 e^x+2 x} \left (-40 x+60 x^2\right )+e^{4 x} \left (80-240 x+180 x^2\right )+e^{\frac {e^{e^x}}{5}} \left (-10 e^{2 e^x+2 x} x^2+e^{4 x} \left (40 x-60 x^2\right )\right )} \, dx=-\frac {3 \left (1-\frac {4 e^{2 x}}{e^{2 x} (4-6 x)-e^{2 e^x} x+e^{\frac {e^{e^x}}{5}+2 x} x}\right )}{x} \] Input:
Integrate[(15*E^(4*E^x) + 540*E^(4*x) + 15*E^((2*E^E^x)/5 + 4*x) + E^(2*E^ x)*(60*E^(2*x) + 120*E^(3*x)) + E^(E^E^x/5)*(-180*E^(4*x) - 30*E^(2*E^x + 2*x) - 12*E^(E^x + 5*x)))/(5*E^(4*E^x)*x^2 + 5*E^((2*E^E^x)/5 + 4*x)*x^2 + E^(2*E^x + 2*x)*(-40*x + 60*x^2) + E^(4*x)*(80 - 240*x + 180*x^2) + E^(E^ E^x/5)*(-10*E^(2*E^x + 2*x)*x^2 + E^(4*x)*(40*x - 60*x^2))),x]
Output:
(-3*(1 - (4*E^(2*x))/(E^(2*x)*(4 - 6*x) - E^(2*E^x)*x + E^(E^E^x/5 + 2*x)* x)))/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.
| \(\displaystyle \int \frac {e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+15 e^{4 e^x}+540 e^{4 x}+15 e^{4 x+\frac {2 e^{e^x}}{5}}+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 x+2 e^x}-12 e^{5 x+e^x}\right )}{5 e^{4 e^x} x^2+5 e^{4 x+\frac {2 e^{e^x}}{5}} x^2+e^{2 x+2 e^x} \left (60 x^2-40 x\right )+e^{4 x} \left (180 x^2-240 x+80\right )+e^{\frac {e^{e^x}}{5}} \left (e^{4 x} \left (40 x-60 x^2\right )-10 e^{2 x+2 e^x} x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+15 e^{4 e^x}+540 e^{4 x}+15 e^{4 x+\frac {2 e^{e^x}}{5}}+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 x+2 e^x}-12 e^{5 x+e^x}\right )}{5 \left (-e^{2 e^x} x-6 e^{2 x} x+e^{2 x+\frac {e^{e^x}}{5}} x+4 e^{2 x}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {3 \left (20 e^{2 e^x} \left (e^{2 x}+2 e^{3 x}\right )+5 e^{4 e^x}+180 e^{4 x}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-2 e^{\frac {e^{e^x}}{5}} \left (30 e^{4 x}+5 e^{2 x+2 e^x}+2 e^{5 x+e^x}\right )\right )}{\left (-e^{2 e^x} x-6 e^{2 x} x+e^{2 x+\frac {e^{e^x}}{5}} x+4 e^{2 x}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \int \frac {20 e^{2 e^x} \left (e^{2 x}+2 e^{3 x}\right )+5 e^{4 e^x}+180 e^{4 x}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-2 e^{\frac {e^{e^x}}{5}} \left (30 e^{4 x}+5 e^{2 x+2 e^x}+2 e^{5 x+e^x}\right )}{\left (-e^{2 e^x} x-6 e^{2 x} x+e^{2 x+\frac {e^{e^x}}{5}} x+4 e^{2 x}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {3}{5} \int \frac {5 e^{4 e^x}+180 e^{4 x}+20 e^{2 \left (x+e^x\right )}-10 e^{2 x+\frac {e^{e^x}}{5}+2 e^x}+40 e^{3 x+2 e^x}-60 e^{4 x+\frac {e^{e^x}}{5}}+5 e^{4 x+\frac {2 e^{e^x}}{5}}-4 e^{5 x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{2 x} (4-6 x)-e^{2 e^x} x+e^{2 x+\frac {e^{e^x}}{5}} x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {3}{5} \int \left (\frac {5 \left (-6+e^{\frac {e^{e^x}}{5}}\right )^2}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}-\frac {4 e^{4 e^x} \left (10 e^{\frac {e^{e^x}}{5}} x^2+60 e^x x^2-10 e^{x+\frac {e^{e^x}}{5}} x^2+e^{x+\frac {e^{e^x}}{5}+e^x} x^2-60 x^2-40 e^x x+40 x-20\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )^2}+\frac {8 e^{2 e^x} \left (5 e^{\frac {e^{e^x}}{5}} x+30 e^x x-5 e^{x+\frac {e^{e^x}}{5}} x+e^{x+\frac {e^{e^x}}{5}+e^x} x-30 x+5 e^{\frac {e^{e^x}}{5}}-20 e^x-10\right )}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2 \left (e^{2 e^x} x+6 e^{2 x} x-e^{2 x+\frac {e^{e^x}}{5}} x-4 e^{2 x}\right )}-\frac {4 e^{x+\frac {e^{e^x}}{5}+e^x}}{\left (e^{\frac {e^{e^x}}{5}} x-6 x+4\right )^2}\right )dx\) |
Input:
Int[(15*E^(4*E^x) + 540*E^(4*x) + 15*E^((2*E^E^x)/5 + 4*x) + E^(2*E^x)*(60 *E^(2*x) + 120*E^(3*x)) + E^(E^E^x/5)*(-180*E^(4*x) - 30*E^(2*E^x + 2*x) - 12*E^(E^x + 5*x)))/(5*E^(4*E^x)*x^2 + 5*E^((2*E^E^x)/5 + 4*x)*x^2 + E^(2* E^x + 2*x)*(-40*x + 60*x^2) + E^(4*x)*(80 - 240*x + 180*x^2) + E^(E^E^x/5) *(-10*E^(2*E^x + 2*x)*x^2 + E^(4*x)*(40*x - 60*x^2))),x]
Output:
$Aborted
Time = 4.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(-\frac {3}{x}+\frac {12 \,{\mathrm e}^{2 x}}{x \left (x \,{\mathrm e}^{2 x +\frac {{\mathrm e}^{{\mathrm e}^{x}}}{5}}-6 x \,{\mathrm e}^{2 x}-x \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+4 \,{\mathrm e}^{2 x}\right )}\) | \(52\) |
| parallelrisch | \(-\frac {15 \,{\mathrm e}^{2 x} {\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{x}}}{5}}-90 \,{\mathrm e}^{2 x}-15 \,{\mathrm e}^{2 \,{\mathrm e}^{x}}}{5 \left ({\mathrm e}^{2 x} {\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{x}}}{5}} x -6 x \,{\mathrm e}^{2 x}-x \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+4 \,{\mathrm e}^{2 x}\right )}\) | \(65\) |
Input:
int((15*exp(x)^4*exp(1/5*exp(exp(x)))^2+(-30*exp(x)^2*exp(exp(x))^2-12*exp (x)^5*exp(exp(x))-180*exp(x)^4)*exp(1/5*exp(exp(x)))+15*exp(exp(x))^4+(120 *exp(x)^3+60*exp(x)^2)*exp(exp(x))^2+540*exp(x)^4)/(5*x^2*exp(x)^4*exp(1/5 *exp(exp(x)))^2+(-10*x^2*exp(x)^2*exp(exp(x))^2+(-60*x^2+40*x)*exp(x)^4)*e xp(1/5*exp(exp(x)))+5*x^2*exp(exp(x))^4+(60*x^2-40*x)*exp(x)^2*exp(exp(x)) ^2+(180*x^2-240*x+80)*exp(x)^4),x,method=_RETURNVERBOSE)
Output:
-3/x+12/x*exp(2*x)/(x*exp(2*x+1/5*exp(exp(x)))-6*x*exp(2*x)-x*exp(2*exp(x) )+4*exp(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (29) = 58\).
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.79 \[ \int \frac {15 e^{4 e^x}+540 e^{4 x}+15 e^{\frac {2 e^{e^x}}{5}+4 x}+e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 e^x+2 x}-12 e^{e^x+5 x}\right )}{5 e^{4 e^x} x^2+5 e^{\frac {2 e^{e^x}}{5}+4 x} x^2+e^{2 e^x+2 x} \left (-40 x+60 x^2\right )+e^{4 x} \left (80-240 x+180 x^2\right )+e^{\frac {e^{e^x}}{5}} \left (-10 e^{2 e^x+2 x} x^2+e^{4 x} \left (40 x-60 x^2\right )\right )} \, dx=-\frac {3 \, {\left (6 \, e^{\left (12 \, x\right )} - e^{\left (12 \, x + \frac {1}{5} \, e^{\left (e^{x}\right )}\right )} + e^{\left (10 \, x + 2 \, e^{x}\right )}\right )}}{2 \, {\left (3 \, x - 2\right )} e^{\left (12 \, x\right )} - x e^{\left (12 \, x + \frac {1}{5} \, e^{\left (e^{x}\right )}\right )} + x e^{\left (10 \, x + 2 \, e^{x}\right )}} \] Input:
integrate((15*exp(x)^4*exp(1/5*exp(exp(x)))^2+(-30*exp(x)^2*exp(exp(x))^2- 12*exp(x)^5*exp(exp(x))-180*exp(x)^4)*exp(1/5*exp(exp(x)))+15*exp(exp(x))^ 4+(120*exp(x)^3+60*exp(x)^2)*exp(exp(x))^2+540*exp(x)^4)/(5*x^2*exp(x)^4*e xp(1/5*exp(exp(x)))^2+(-10*x^2*exp(x)^2*exp(exp(x))^2+(-60*x^2+40*x)*exp(x )^4)*exp(1/5*exp(exp(x)))+5*x^2*exp(exp(x))^4+(60*x^2-40*x)*exp(x)^2*exp(e xp(x))^2+(180*x^2-240*x+80)*exp(x)^4),x, algorithm="fricas")
Output:
-3*(6*e^(12*x) - e^(12*x + 1/5*e^(e^x)) + e^(10*x + 2*e^x))/(2*(3*x - 2)*e ^(12*x) - x*e^(12*x + 1/5*e^(e^x)) + x*e^(10*x + 2*e^x))
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42 \[ \int \frac {15 e^{4 e^x}+540 e^{4 x}+15 e^{\frac {2 e^{e^x}}{5}+4 x}+e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 e^x+2 x}-12 e^{e^x+5 x}\right )}{5 e^{4 e^x} x^2+5 e^{\frac {2 e^{e^x}}{5}+4 x} x^2+e^{2 e^x+2 x} \left (-40 x+60 x^2\right )+e^{4 x} \left (80-240 x+180 x^2\right )+e^{\frac {e^{e^x}}{5}} \left (-10 e^{2 e^x+2 x} x^2+e^{4 x} \left (40 x-60 x^2\right )\right )} \, dx=\frac {12 e^{2 x}}{x^{2} e^{2 x} e^{\frac {e^{e^{x}}}{5}} - 6 x^{2} e^{2 x} - x^{2} e^{2 e^{x}} + 4 x e^{2 x}} - \frac {3}{x} \] Input:
integrate((15*exp(x)**4*exp(1/5*exp(exp(x)))**2+(-30*exp(x)**2*exp(exp(x)) **2-12*exp(x)**5*exp(exp(x))-180*exp(x)**4)*exp(1/5*exp(exp(x)))+15*exp(ex p(x))**4+(120*exp(x)**3+60*exp(x)**2)*exp(exp(x))**2+540*exp(x)**4)/(5*x** 2*exp(x)**4*exp(1/5*exp(exp(x)))**2+(-10*x**2*exp(x)**2*exp(exp(x))**2+(-6 0*x**2+40*x)*exp(x)**4)*exp(1/5*exp(exp(x)))+5*x**2*exp(exp(x))**4+(60*x** 2-40*x)*exp(x)**2*exp(exp(x))**2+(180*x**2-240*x+80)*exp(x)**4),x)
Output:
12*exp(2*x)/(x**2*exp(2*x)*exp(exp(exp(x))/5) - 6*x**2*exp(2*x) - x**2*exp (2*exp(x)) + 4*x*exp(2*x)) - 3/x
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \[ \int \frac {15 e^{4 e^x}+540 e^{4 x}+15 e^{\frac {2 e^{e^x}}{5}+4 x}+e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 e^x+2 x}-12 e^{e^x+5 x}\right )}{5 e^{4 e^x} x^2+5 e^{\frac {2 e^{e^x}}{5}+4 x} x^2+e^{2 e^x+2 x} \left (-40 x+60 x^2\right )+e^{4 x} \left (80-240 x+180 x^2\right )+e^{\frac {e^{e^x}}{5}} \left (-10 e^{2 e^x+2 x} x^2+e^{4 x} \left (40 x-60 x^2\right )\right )} \, dx=-\frac {3 \, {\left (6 \, e^{\left (2 \, x\right )} - e^{\left (2 \, x + \frac {1}{5} \, e^{\left (e^{x}\right )}\right )} + e^{\left (2 \, e^{x}\right )}\right )}}{2 \, {\left (3 \, x - 2\right )} e^{\left (2 \, x\right )} - x e^{\left (2 \, x + \frac {1}{5} \, e^{\left (e^{x}\right )}\right )} + x e^{\left (2 \, e^{x}\right )}} \] Input:
integrate((15*exp(x)^4*exp(1/5*exp(exp(x)))^2+(-30*exp(x)^2*exp(exp(x))^2- 12*exp(x)^5*exp(exp(x))-180*exp(x)^4)*exp(1/5*exp(exp(x)))+15*exp(exp(x))^ 4+(120*exp(x)^3+60*exp(x)^2)*exp(exp(x))^2+540*exp(x)^4)/(5*x^2*exp(x)^4*e xp(1/5*exp(exp(x)))^2+(-10*x^2*exp(x)^2*exp(exp(x))^2+(-60*x^2+40*x)*exp(x )^4)*exp(1/5*exp(exp(x)))+5*x^2*exp(exp(x))^4+(60*x^2-40*x)*exp(x)^2*exp(e xp(x))^2+(180*x^2-240*x+80)*exp(x)^4),x, algorithm="maxima")
Output:
-3*(6*e^(2*x) - e^(2*x + 1/5*e^(e^x)) + e^(2*e^x))/(2*(3*x - 2)*e^(2*x) - x*e^(2*x + 1/5*e^(e^x)) + x*e^(2*e^x))
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.63 \[ \int \frac {15 e^{4 e^x}+540 e^{4 x}+15 e^{\frac {2 e^{e^x}}{5}+4 x}+e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 e^x+2 x}-12 e^{e^x+5 x}\right )}{5 e^{4 e^x} x^2+5 e^{\frac {2 e^{e^x}}{5}+4 x} x^2+e^{2 e^x+2 x} \left (-40 x+60 x^2\right )+e^{4 x} \left (80-240 x+180 x^2\right )+e^{\frac {e^{e^x}}{5}} \left (-10 e^{2 e^x+2 x} x^2+e^{4 x} \left (40 x-60 x^2\right )\right )} \, dx=-\frac {3 \, {\left (6 \, e^{\left (2 \, x\right )} - e^{\left (2 \, x + \frac {1}{5} \, e^{\left (e^{x}\right )}\right )} + e^{\left (2 \, e^{x}\right )}\right )}}{6 \, x e^{\left (2 \, x\right )} - x e^{\left (2 \, x + \frac {1}{5} \, e^{\left (e^{x}\right )}\right )} + x e^{\left (2 \, e^{x}\right )} - 4 \, e^{\left (2 \, x\right )}} \] Input:
integrate((15*exp(x)^4*exp(1/5*exp(exp(x)))^2+(-30*exp(x)^2*exp(exp(x))^2- 12*exp(x)^5*exp(exp(x))-180*exp(x)^4)*exp(1/5*exp(exp(x)))+15*exp(exp(x))^ 4+(120*exp(x)^3+60*exp(x)^2)*exp(exp(x))^2+540*exp(x)^4)/(5*x^2*exp(x)^4*e xp(1/5*exp(exp(x)))^2+(-10*x^2*exp(x)^2*exp(exp(x))^2+(-60*x^2+40*x)*exp(x )^4)*exp(1/5*exp(exp(x)))+5*x^2*exp(exp(x))^4+(60*x^2-40*x)*exp(x)^2*exp(e xp(x))^2+(180*x^2-240*x+80)*exp(x)^4),x, algorithm="giac")
Output:
-3*(6*e^(2*x) - e^(2*x + 1/5*e^(e^x)) + e^(2*e^x))/(6*x*e^(2*x) - x*e^(2*x + 1/5*e^(e^x)) + x*e^(2*e^x) - 4*e^(2*x))
Time = 1.85 (sec) , antiderivative size = 191, normalized size of antiderivative = 5.03 \[ \int \frac {15 e^{4 e^x}+540 e^{4 x}+15 e^{\frac {2 e^{e^x}}{5}+4 x}+e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 e^x+2 x}-12 e^{e^x+5 x}\right )}{5 e^{4 e^x} x^2+5 e^{\frac {2 e^{e^x}}{5}+4 x} x^2+e^{2 e^x+2 x} \left (-40 x+60 x^2\right )+e^{4 x} \left (80-240 x+180 x^2\right )+e^{\frac {e^{e^x}}{5}} \left (-10 e^{2 e^x+2 x} x^2+e^{4 x} \left (40 x-60 x^2\right )\right )} \, dx=\frac {12\,\left (20\,{\mathrm {e}}^{6\,x}-10\,x^2\,{\mathrm {e}}^{4\,x+2\,{\mathrm {e}}^x}+10\,x^2\,{\mathrm {e}}^{5\,x+2\,{\mathrm {e}}^x}-x^2\,{\mathrm {e}}^{5\,x+3\,{\mathrm {e}}^x}+4\,x\,{\mathrm {e}}^{7\,x+{\mathrm {e}}^x}-6\,x^2\,{\mathrm {e}}^{7\,x+{\mathrm {e}}^x}\right )}{x\,\left (4\,{\mathrm {e}}^{2\,x}-6\,x\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x+\frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{5}}-x\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}\right )\,\left (20\,{\mathrm {e}}^{4\,x}-10\,x^2\,{\mathrm {e}}^{2\,x+2\,{\mathrm {e}}^x}+10\,x^2\,{\mathrm {e}}^{3\,x+2\,{\mathrm {e}}^x}-x^2\,{\mathrm {e}}^{3\,x+3\,{\mathrm {e}}^x}+4\,x\,{\mathrm {e}}^{5\,x+{\mathrm {e}}^x}-6\,x^2\,{\mathrm {e}}^{5\,x+{\mathrm {e}}^x}\right )}-\frac {3}{x} \] Input:
int((540*exp(4*x) + 15*exp(4*exp(x)) - exp(exp(exp(x))/5)*(180*exp(4*x) + 30*exp(2*x)*exp(2*exp(x)) + 12*exp(5*x)*exp(exp(x))) + exp(2*exp(x))*(60*e xp(2*x) + 120*exp(3*x)) + 15*exp((2*exp(exp(x)))/5)*exp(4*x))/(exp(4*x)*(1 80*x^2 - 240*x + 80) + 5*x^2*exp(4*exp(x)) + exp(exp(exp(x))/5)*(exp(4*x)* (40*x - 60*x^2) - 10*x^2*exp(2*x)*exp(2*exp(x))) + 5*x^2*exp((2*exp(exp(x) ))/5)*exp(4*x) - exp(2*x)*exp(2*exp(x))*(40*x - 60*x^2)),x)
Output:
(12*(20*exp(6*x) - 10*x^2*exp(4*x + 2*exp(x)) + 10*x^2*exp(5*x + 2*exp(x)) - x^2*exp(5*x + 3*exp(x)) + 4*x*exp(7*x + exp(x)) - 6*x^2*exp(7*x + exp(x ))))/(x*(4*exp(2*x) - 6*x*exp(2*x) + x*exp(2*x + exp(exp(x))/5) - x*exp(2* exp(x)))*(20*exp(4*x) - 10*x^2*exp(2*x + 2*exp(x)) + 10*x^2*exp(3*x + 2*ex p(x)) - x^2*exp(3*x + 3*exp(x)) + 4*x*exp(5*x + exp(x)) - 6*x^2*exp(5*x + exp(x)))) - 3/x
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.97 \[ \int \frac {15 e^{4 e^x}+540 e^{4 x}+15 e^{\frac {2 e^{e^x}}{5}+4 x}+e^{2 e^x} \left (60 e^{2 x}+120 e^{3 x}\right )+e^{\frac {e^{e^x}}{5}} \left (-180 e^{4 x}-30 e^{2 e^x+2 x}-12 e^{e^x+5 x}\right )}{5 e^{4 e^x} x^2+5 e^{\frac {2 e^{e^x}}{5}+4 x} x^2+e^{2 e^x+2 x} \left (-40 x+60 x^2\right )+e^{4 x} \left (80-240 x+180 x^2\right )+e^{\frac {e^{e^x}}{5}} \left (-10 e^{2 e^x+2 x} x^2+e^{4 x} \left (40 x-60 x^2\right )\right )} \, dx=\frac {-3 e^{2 e^{x}}+3 e^{\frac {e^{e^{x}}}{5}+2 x}-18 e^{2 x}}{e^{2 e^{x}} x -e^{\frac {e^{e^{x}}}{5}+2 x} x +6 e^{2 x} x -4 e^{2 x}} \] Input:
int((15*exp(x)^4*exp(1/5*exp(exp(x)))^2+(-30*exp(x)^2*exp(exp(x))^2-12*exp (x)^5*exp(exp(x))-180*exp(x)^4)*exp(1/5*exp(exp(x)))+15*exp(exp(x))^4+(120 *exp(x)^3+60*exp(x)^2)*exp(exp(x))^2+540*exp(x)^4)/(5*x^2*exp(x)^4*exp(1/5 *exp(exp(x)))^2+(-10*x^2*exp(x)^2*exp(exp(x))^2+(-60*x^2+40*x)*exp(x)^4)*e xp(1/5*exp(exp(x)))+5*x^2*exp(exp(x))^4+(60*x^2-40*x)*exp(x)^2*exp(exp(x)) ^2+(180*x^2-240*x+80)*exp(x)^4),x)
Output:
(3*( - e**(2*e**x) + e**((e**(e**x) + 10*x)/5) - 6*e**(2*x)))/(e**(2*e**x) *x - e**((e**(e**x) + 10*x)/5)*x + 6*e**(2*x)*x - 4*e**(2*x))