Integrand size = 199, antiderivative size = 36 \[ \int \frac {e^{\frac {-5 x+10 e^{2 x} x+e^x \left (15+10 x^2\right )+\left (4 e^x x+4 x^2\right ) \log ^2(x)}{5 e^{2 x} x+5 e^x x^2}} \left (5 e^{3 x} x+5 x^2+5 x^3+e^{2 x} \left (-15-15 x+10 x^2\right )+e^x \left (-30 x+10 x^2+5 x^3\right )+\left (8 e^{2 x} x+16 e^x x^2+8 x^3\right ) \log (x)+\left (-4 e^{2 x} x^2-8 e^x x^3-4 x^4\right ) \log ^2(x)\right )}{5 e^{3 x} x+10 e^{2 x} x^2+5 e^x x^3} \, dx=e^{2+e^{-x} \left (\frac {3}{x}+\frac {4}{5} \left (-\frac {5}{e^x+x}+\log ^2(x)\right )\right )} x \] Output:
exp(2+(3/x+4/5*ln(x)^2-4/(exp(x)+x))/exp(x))*x
Time = 0.39 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {-5 x+10 e^{2 x} x+e^x \left (15+10 x^2\right )+\left (4 e^x x+4 x^2\right ) \log ^2(x)}{5 e^{2 x} x+5 e^x x^2}} \left (5 e^{3 x} x+5 x^2+5 x^3+e^{2 x} \left (-15-15 x+10 x^2\right )+e^x \left (-30 x+10 x^2+5 x^3\right )+\left (8 e^{2 x} x+16 e^x x^2+8 x^3\right ) \log (x)+\left (-4 e^{2 x} x^2-8 e^x x^3-4 x^4\right ) \log ^2(x)\right )}{5 e^{3 x} x+10 e^{2 x} x^2+5 e^x x^3} \, dx=e^{2+\frac {4}{x^2}-\frac {e^{-x}}{x}-\frac {4}{x^2 \left (1+e^{-x} x\right )}+\frac {4}{5} e^{-x} \log ^2(x)} x \] Input:
Integrate[(E^((-5*x + 10*E^(2*x)*x + E^x*(15 + 10*x^2) + (4*E^x*x + 4*x^2) *Log[x]^2)/(5*E^(2*x)*x + 5*E^x*x^2))*(5*E^(3*x)*x + 5*x^2 + 5*x^3 + E^(2* x)*(-15 - 15*x + 10*x^2) + E^x*(-30*x + 10*x^2 + 5*x^3) + (8*E^(2*x)*x + 1 6*E^x*x^2 + 8*x^3)*Log[x] + (-4*E^(2*x)*x^2 - 8*E^x*x^3 - 4*x^4)*Log[x]^2) )/(5*E^(3*x)*x + 10*E^(2*x)*x^2 + 5*E^x*x^3),x]
Output:
E^(2 + 4/x^2 - 1/(E^x*x) - 4/(x^2*(1 + x/E^x)) + (4*Log[x]^2)/(5*E^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.
| \(\displaystyle \int \frac {\left (5 x^3+5 x^2+e^{2 x} \left (10 x^2-15 x-15\right )+e^x \left (5 x^3+10 x^2-30 x\right )+\left (8 x^3+16 e^x x^2+8 e^{2 x} x\right ) \log (x)+\left (-4 x^4-8 e^x x^3-4 e^{2 x} x^2\right ) \log ^2(x)+5 e^{3 x} x\right ) \exp \left (\frac {e^x \left (10 x^2+15\right )+\left (4 x^2+4 e^x x\right ) \log ^2(x)+10 e^{2 x} x-5 x}{5 e^x x^2+5 e^{2 x} x}\right )}{5 e^x x^3+10 e^{2 x} x^2+5 e^{3 x} x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (5 x^3+5 x^2+e^{2 x} \left (10 x^2-15 x-15\right )+e^x \left (5 x^3+10 x^2-30 x\right )+\left (8 x^3+16 e^x x^2+8 e^{2 x} x\right ) \log (x)+\left (-4 x^4-8 e^x x^3-4 e^{2 x} x^2\right ) \log ^2(x)+5 e^{3 x} x\right ) \exp \left (\frac {e^{-x} \left (e^x \left (10 x^2+15\right )+\left (4 x^2+4 e^x x\right ) \log ^2(x)+10 e^{2 x} x-5 x\right )}{5 x \left (x+e^x\right )}-x\right )}{5 x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (-x-\frac {e^{-x} \left (-4 \left (x^2+e^x x\right ) \log ^2(x)-10 e^{2 x} x+5 x-5 e^x \left (2 x^2+3\right )\right )}{5 \left (x+e^x\right ) x}\right ) \left (5 x^3+5 x^2+5 e^{3 x} x-4 \left (x^4+2 e^x x^3+e^{2 x} x^2\right ) \log ^2(x)-5 e^{2 x} \left (-2 x^2+3 x+3\right )-5 e^x \left (-x^3-2 x^2+6 x\right )+8 \left (x^3+2 e^x x^2+e^{2 x} x\right ) \log (x)\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (-x-\frac {e^{-x} \left (-4 \left (x^2+e^x x\right ) \log ^2(x)-10 e^{2 x} x+5 x-5 e^x \left (2 x^2+3\right )\right )}{5 \left (x+e^x\right ) x}\right ) x}{x+e^x}-\frac {20 \exp \left (-x-\frac {e^{-x} \left (-4 \left (x^2+e^x x\right ) \log ^2(x)-10 e^{2 x} x+5 x-5 e^x \left (2 x^2+3\right )\right )}{5 \left (x+e^x\right ) x}\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (-\frac {e^{-x} \left (-4 \left (x^2+e^x x\right ) \log ^2(x)-10 e^{2 x} x+5 x-5 e^x \left (2 x^2+3\right )\right )}{5 x \left (x+e^x\right )}\right )+\frac {\exp \left (-x-\frac {e^{-x} \left (-4 \left (x^2+e^x x\right ) \log ^2(x)-10 e^{2 x} x+5 x-5 e^x \left (2 x^2+3\right )\right )}{5 \left (x+e^x\right ) x}\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {40 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) x}{x+e^x}-\frac {20 \exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) (x-1) x}{\left (x+e^x\right )^2}+5 \exp \left (x+\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right )+\frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x)+8 x \log (x)-15 x-15\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {1}{5} e^{-x} \left (4 \log ^2(x)-\frac {5 \left (e^{2 x} (x-2) x+x+e^x \left (x^3-2 x^2-3\right )\right )}{x \left (x+e^x\right )}\right )\right ) \left (-4 x^2 \log ^2(x) \left (x+e^x\right )^2+8 x \log (x) \left (x+e^x\right )^2+5 \left ((x+1) x^2+e^{3 x} x+e^x \left (x^2+2 x-6\right ) x+e^{2 x} \left (2 x^2-3 x-3\right )\right )\right )}{x \left (x+e^x\right )^2}dx\) |
Input:
Int[(E^((-5*x + 10*E^(2*x)*x + E^x*(15 + 10*x^2) + (4*E^x*x + 4*x^2)*Log[x ]^2)/(5*E^(2*x)*x + 5*E^x*x^2))*(5*E^(3*x)*x + 5*x^2 + 5*x^3 + E^(2*x)*(-1 5 - 15*x + 10*x^2) + E^x*(-30*x + 10*x^2 + 5*x^3) + (8*E^(2*x)*x + 16*E^x* x^2 + 8*x^3)*Log[x] + (-4*E^(2*x)*x^2 - 8*E^x*x^3 - 4*x^4)*Log[x]^2))/(5*E ^(3*x)*x + 10*E^(2*x)*x^2 + 5*E^x*x^3),x]
Output:
$Aborted
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.67
\[x \,{\mathrm e}^{\frac {4 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )^{2}+10 \,{\mathrm e}^{x} x^{2}+10 x \,{\mathrm e}^{2 x}+15 \,{\mathrm e}^{x}-5 x}{5 x \left ({\mathrm e}^{x} x +{\mathrm e}^{2 x}\right )}}\]
Input:
int(((-4*exp(x)^2*x^2-8*exp(x)*x^3-4*x^4)*ln(x)^2+(8*x*exp(x)^2+16*exp(x)* x^2+8*x^3)*ln(x)+5*x*exp(x)^3+(10*x^2-15*x-15)*exp(x)^2+(5*x^3+10*x^2-30*x )*exp(x)+5*x^3+5*x^2)*exp(((4*exp(x)*x+4*x^2)*ln(x)^2+10*x*exp(x)^2+(10*x^ 2+15)*exp(x)-5*x)/(5*x*exp(x)^2+5*exp(x)*x^2))/(5*x*exp(x)^3+10*exp(x)^2*x ^2+5*exp(x)*x^3),x)
Output:
x*exp(1/5*(4*x*exp(x)*ln(x)^2+4*x^2*ln(x)^2+10*exp(x)*x^2+10*x*exp(2*x)+15 *exp(x)-5*x)/x/(exp(x)*x+exp(2*x)))
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {-5 x+10 e^{2 x} x+e^x \left (15+10 x^2\right )+\left (4 e^x x+4 x^2\right ) \log ^2(x)}{5 e^{2 x} x+5 e^x x^2}} \left (5 e^{3 x} x+5 x^2+5 x^3+e^{2 x} \left (-15-15 x+10 x^2\right )+e^x \left (-30 x+10 x^2+5 x^3\right )+\left (8 e^{2 x} x+16 e^x x^2+8 x^3\right ) \log (x)+\left (-4 e^{2 x} x^2-8 e^x x^3-4 x^4\right ) \log ^2(x)\right )}{5 e^{3 x} x+10 e^{2 x} x^2+5 e^x x^3} \, dx=x e^{\left (\frac {4 \, {\left (x^{2} + x e^{x}\right )} \log \left (x\right )^{2} + 10 \, x e^{\left (2 \, x\right )} + 5 \, {\left (2 \, x^{2} + 3\right )} e^{x} - 5 \, x}{5 \, {\left (x^{2} e^{x} + x e^{\left (2 \, x\right )}\right )}}\right )} \] Input:
integrate(((-4*exp(x)^2*x^2-8*exp(x)*x^3-4*x^4)*log(x)^2+(8*x*exp(x)^2+16* exp(x)*x^2+8*x^3)*log(x)+5*x*exp(x)^3+(10*x^2-15*x-15)*exp(x)^2+(5*x^3+10* x^2-30*x)*exp(x)+5*x^3+5*x^2)*exp(((4*exp(x)*x+4*x^2)*log(x)^2+10*x*exp(x) ^2+(10*x^2+15)*exp(x)-5*x)/(5*x*exp(x)^2+5*exp(x)*x^2))/(5*x*exp(x)^3+10*e xp(x)^2*x^2+5*exp(x)*x^3),x, algorithm="fricas")
Output:
x*e^(1/5*(4*(x^2 + x*e^x)*log(x)^2 + 10*x*e^(2*x) + 5*(2*x^2 + 3)*e^x - 5* x)/(x^2*e^x + x*e^(2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 38.64 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {e^{\frac {-5 x+10 e^{2 x} x+e^x \left (15+10 x^2\right )+\left (4 e^x x+4 x^2\right ) \log ^2(x)}{5 e^{2 x} x+5 e^x x^2}} \left (5 e^{3 x} x+5 x^2+5 x^3+e^{2 x} \left (-15-15 x+10 x^2\right )+e^x \left (-30 x+10 x^2+5 x^3\right )+\left (8 e^{2 x} x+16 e^x x^2+8 x^3\right ) \log (x)+\left (-4 e^{2 x} x^2-8 e^x x^3-4 x^4\right ) \log ^2(x)\right )}{5 e^{3 x} x+10 e^{2 x} x^2+5 e^x x^3} \, dx=x e^{\frac {10 x e^{2 x} - 5 x + \left (4 x^{2} + 4 x e^{x}\right ) \log {\left (x \right )}^{2} + \left (10 x^{2} + 15\right ) e^{x}}{5 x^{2} e^{x} + 5 x e^{2 x}}} \] Input:
integrate(((-4*exp(x)**2*x**2-8*exp(x)*x**3-4*x**4)*ln(x)**2+(8*x*exp(x)** 2+16*exp(x)*x**2+8*x**3)*ln(x)+5*x*exp(x)**3+(10*x**2-15*x-15)*exp(x)**2+( 5*x**3+10*x**2-30*x)*exp(x)+5*x**3+5*x**2)*exp(((4*exp(x)*x+4*x**2)*ln(x)* *2+10*x*exp(x)**2+(10*x**2+15)*exp(x)-5*x)/(5*x*exp(x)**2+5*exp(x)*x**2))/ (5*x*exp(x)**3+10*exp(x)**2*x**2+5*exp(x)*x**3),x)
Output:
x*exp((10*x*exp(2*x) - 5*x + (4*x**2 + 4*x*exp(x))*log(x)**2 + (10*x**2 + 15)*exp(x))/(5*x**2*exp(x) + 5*x*exp(2*x)))
\[ \int \frac {e^{\frac {-5 x+10 e^{2 x} x+e^x \left (15+10 x^2\right )+\left (4 e^x x+4 x^2\right ) \log ^2(x)}{5 e^{2 x} x+5 e^x x^2}} \left (5 e^{3 x} x+5 x^2+5 x^3+e^{2 x} \left (-15-15 x+10 x^2\right )+e^x \left (-30 x+10 x^2+5 x^3\right )+\left (8 e^{2 x} x+16 e^x x^2+8 x^3\right ) \log (x)+\left (-4 e^{2 x} x^2-8 e^x x^3-4 x^4\right ) \log ^2(x)\right )}{5 e^{3 x} x+10 e^{2 x} x^2+5 e^x x^3} \, dx=\int { \frac {{\left (5 \, x^{3} - 4 \, {\left (x^{4} + 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )} \log \left (x\right )^{2} + 5 \, x^{2} + 5 \, x e^{\left (3 \, x\right )} + 5 \, {\left (2 \, x^{2} - 3 \, x - 3\right )} e^{\left (2 \, x\right )} + 5 \, {\left (x^{3} + 2 \, x^{2} - 6 \, x\right )} e^{x} + 8 \, {\left (x^{3} + 2 \, x^{2} e^{x} + x e^{\left (2 \, x\right )}\right )} \log \left (x\right )\right )} e^{\left (\frac {4 \, {\left (x^{2} + x e^{x}\right )} \log \left (x\right )^{2} + 10 \, x e^{\left (2 \, x\right )} + 5 \, {\left (2 \, x^{2} + 3\right )} e^{x} - 5 \, x}{5 \, {\left (x^{2} e^{x} + x e^{\left (2 \, x\right )}\right )}}\right )}}{5 \, {\left (x^{3} e^{x} + 2 \, x^{2} e^{\left (2 \, x\right )} + x e^{\left (3 \, x\right )}\right )}} \,d x } \] Input:
integrate(((-4*exp(x)^2*x^2-8*exp(x)*x^3-4*x^4)*log(x)^2+(8*x*exp(x)^2+16* exp(x)*x^2+8*x^3)*log(x)+5*x*exp(x)^3+(10*x^2-15*x-15)*exp(x)^2+(5*x^3+10* x^2-30*x)*exp(x)+5*x^3+5*x^2)*exp(((4*exp(x)*x+4*x^2)*log(x)^2+10*x*exp(x) ^2+(10*x^2+15)*exp(x)-5*x)/(5*x*exp(x)^2+5*exp(x)*x^2))/(5*x*exp(x)^3+10*e xp(x)^2*x^2+5*exp(x)*x^3),x, algorithm="maxima")
Output:
1/5*integrate((5*x^3 - 4*(x^4 + 2*x^3*e^x + x^2*e^(2*x))*log(x)^2 + 5*x^2 + 5*x*e^(3*x) + 5*(2*x^2 - 3*x - 3)*e^(2*x) + 5*(x^3 + 2*x^2 - 6*x)*e^x + 8*(x^3 + 2*x^2*e^x + x*e^(2*x))*log(x))*e^(1/5*(4*(x^2 + x*e^x)*log(x)^2 + 10*x*e^(2*x) + 5*(2*x^2 + 3)*e^x - 5*x)/(x^2*e^x + x*e^(2*x)))/(x^3*e^x + 2*x^2*e^(2*x) + x*e^(3*x)), x)
\[ \int \frac {e^{\frac {-5 x+10 e^{2 x} x+e^x \left (15+10 x^2\right )+\left (4 e^x x+4 x^2\right ) \log ^2(x)}{5 e^{2 x} x+5 e^x x^2}} \left (5 e^{3 x} x+5 x^2+5 x^3+e^{2 x} \left (-15-15 x+10 x^2\right )+e^x \left (-30 x+10 x^2+5 x^3\right )+\left (8 e^{2 x} x+16 e^x x^2+8 x^3\right ) \log (x)+\left (-4 e^{2 x} x^2-8 e^x x^3-4 x^4\right ) \log ^2(x)\right )}{5 e^{3 x} x+10 e^{2 x} x^2+5 e^x x^3} \, dx=\int { \frac {{\left (5 \, x^{3} - 4 \, {\left (x^{4} + 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )} \log \left (x\right )^{2} + 5 \, x^{2} + 5 \, x e^{\left (3 \, x\right )} + 5 \, {\left (2 \, x^{2} - 3 \, x - 3\right )} e^{\left (2 \, x\right )} + 5 \, {\left (x^{3} + 2 \, x^{2} - 6 \, x\right )} e^{x} + 8 \, {\left (x^{3} + 2 \, x^{2} e^{x} + x e^{\left (2 \, x\right )}\right )} \log \left (x\right )\right )} e^{\left (\frac {4 \, {\left (x^{2} + x e^{x}\right )} \log \left (x\right )^{2} + 10 \, x e^{\left (2 \, x\right )} + 5 \, {\left (2 \, x^{2} + 3\right )} e^{x} - 5 \, x}{5 \, {\left (x^{2} e^{x} + x e^{\left (2 \, x\right )}\right )}}\right )}}{5 \, {\left (x^{3} e^{x} + 2 \, x^{2} e^{\left (2 \, x\right )} + x e^{\left (3 \, x\right )}\right )}} \,d x } \] Input:
integrate(((-4*exp(x)^2*x^2-8*exp(x)*x^3-4*x^4)*log(x)^2+(8*x*exp(x)^2+16* exp(x)*x^2+8*x^3)*log(x)+5*x*exp(x)^3+(10*x^2-15*x-15)*exp(x)^2+(5*x^3+10* x^2-30*x)*exp(x)+5*x^3+5*x^2)*exp(((4*exp(x)*x+4*x^2)*log(x)^2+10*x*exp(x) ^2+(10*x^2+15)*exp(x)-5*x)/(5*x*exp(x)^2+5*exp(x)*x^2))/(5*x*exp(x)^3+10*e xp(x)^2*x^2+5*exp(x)*x^3),x, algorithm="giac")
Output:
integrate(1/5*(5*x^3 - 4*(x^4 + 2*x^3*e^x + x^2*e^(2*x))*log(x)^2 + 5*x^2 + 5*x*e^(3*x) + 5*(2*x^2 - 3*x - 3)*e^(2*x) + 5*(x^3 + 2*x^2 - 6*x)*e^x + 8*(x^3 + 2*x^2*e^x + x*e^(2*x))*log(x))*e^(1/5*(4*(x^2 + x*e^x)*log(x)^2 + 10*x*e^(2*x) + 5*(2*x^2 + 3)*e^x - 5*x)/(x^2*e^x + x*e^(2*x)))/(x^3*e^x + 2*x^2*e^(2*x) + x*e^(3*x)), x)
Time = 3.80 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.47 \[ \int \frac {e^{\frac {-5 x+10 e^{2 x} x+e^x \left (15+10 x^2\right )+\left (4 e^x x+4 x^2\right ) \log ^2(x)}{5 e^{2 x} x+5 e^x x^2}} \left (5 e^{3 x} x+5 x^2+5 x^3+e^{2 x} \left (-15-15 x+10 x^2\right )+e^x \left (-30 x+10 x^2+5 x^3\right )+\left (8 e^{2 x} x+16 e^x x^2+8 x^3\right ) \log (x)+\left (-4 e^{2 x} x^2-8 e^x x^3-4 x^4\right ) \log ^2(x)\right )}{5 e^{3 x} x+10 e^{2 x} x^2+5 e^x x^3} \, dx=x\,{\mathrm {e}}^{\frac {3}{x\,{\mathrm {e}}^x+x^2}}\,{\mathrm {e}}^{\frac {2\,x}{x+{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {1}{{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,x\,{\ln \left (x\right )}^2}{5\,{\mathrm {e}}^{2\,x}+5\,x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^x}{x+{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (x\right )}^2}{5\,x+5\,{\mathrm {e}}^x}} \] Input:
int((exp((10*x*exp(2*x) - 5*x + exp(x)*(10*x^2 + 15) + log(x)^2*(4*x*exp(x ) + 4*x^2))/(5*x*exp(2*x) + 5*x^2*exp(x)))*(log(x)*(8*x*exp(2*x) + 16*x^2* exp(x) + 8*x^3) - exp(2*x)*(15*x - 10*x^2 + 15) + 5*x*exp(3*x) - log(x)^2* (8*x^3*exp(x) + 4*x^2*exp(2*x) + 4*x^4) + 5*x^2 + 5*x^3 + exp(x)*(10*x^2 - 30*x + 5*x^3)))/(5*x*exp(3*x) + 5*x^3*exp(x) + 10*x^2*exp(2*x)),x)
Output:
x*exp(3/(x*exp(x) + x^2))*exp((2*x)/(x + exp(x)))*exp(-1/(exp(2*x) + x*exp (x)))*exp((4*x*log(x)^2)/(5*exp(2*x) + 5*x*exp(x)))*exp((2*exp(x))/(x + ex p(x)))*exp((4*log(x)^2)/(5*x + 5*exp(x)))
Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.92 \[ \int \frac {e^{\frac {-5 x+10 e^{2 x} x+e^x \left (15+10 x^2\right )+\left (4 e^x x+4 x^2\right ) \log ^2(x)}{5 e^{2 x} x+5 e^x x^2}} \left (5 e^{3 x} x+5 x^2+5 x^3+e^{2 x} \left (-15-15 x+10 x^2\right )+e^x \left (-30 x+10 x^2+5 x^3\right )+\left (8 e^{2 x} x+16 e^x x^2+8 x^3\right ) \log (x)+\left (-4 e^{2 x} x^2-8 e^x x^3-4 x^4\right ) \log ^2(x)\right )}{5 e^{3 x} x+10 e^{2 x} x^2+5 e^x x^3} \, dx=\frac {e^{\frac {4 e^{x} \mathrm {log}\left (x \right )^{2} x +15 e^{x}+4 \mathrm {log}\left (x \right )^{2} x^{2}}{5 e^{2 x} x +5 e^{x} x^{2}}} e^{2} x}{e^{\frac {1}{e^{2 x}+e^{x} x}}} \] Input:
int(((-4*exp(x)^2*x^2-8*exp(x)*x^3-4*x^4)*log(x)^2+(8*x*exp(x)^2+16*exp(x) *x^2+8*x^3)*log(x)+5*x*exp(x)^3+(10*x^2-15*x-15)*exp(x)^2+(5*x^3+10*x^2-30 *x)*exp(x)+5*x^3+5*x^2)*exp(((4*exp(x)*x+4*x^2)*log(x)^2+10*x*exp(x)^2+(10 *x^2+15)*exp(x)-5*x)/(5*x*exp(x)^2+5*exp(x)*x^2))/(5*x*exp(x)^3+10*exp(x)^ 2*x^2+5*exp(x)*x^3),x)
Output:
(e**((4*e**x*log(x)**2*x + 15*e**x + 4*log(x)**2*x**2)/(5*e**(2*x)*x + 5*e **x*x**2))*e**2*x)/e**(1/(e**(2*x) + e**x*x))