Integrand size = 281, antiderivative size = 33 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 e^{2 x^2}}{\left (-x+\frac {\left (-e^{2 e^x}+\log (x)\right )^2}{x}\right )^2} \]
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 e^{2 x^2} x^2}{\left (e^{4 e^x}-x^2-2 e^{2 e^x} \log (x)+\log ^2(x)\right )^2} \]
Integrate[(E^(4*E^x + 2*x^2)*(6*x - 24*E^x*x^2 + 12*x^3) + E^(2*x^2)*(6*x^ 3 - 12*x^5) - 12*E^(2*x^2)*x*Log[x] + E^(2*x^2)*(6*x + 12*x^3)*Log[x]^2 + E^(2*E^x)*(12*E^(2*x^2)*x + E^(2*x^2)*(-12*x + 24*E^x*x^2 - 24*x^3)*Log[x] ))/(E^(12*E^x) - x^6 - 6*E^(10*E^x)*Log[x] + 3*x^4*Log[x]^2 - 3*x^2*Log[x] ^4 + Log[x]^6 + E^(8*E^x)*(-3*x^2 + 15*Log[x]^2) + E^(6*E^x)*(12*x^2*Log[x ] - 20*Log[x]^3) + E^(4*E^x)*(3*x^4 - 18*x^2*Log[x]^2 + 15*Log[x]^4) + E^( 2*E^x)*(-6*x^4*Log[x] + 12*x^2*Log[x]^3 - 6*Log[x]^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.
| \(\displaystyle \int \frac {-12 e^{2 x^2} x \log (x)+e^{2 x^2+4 e^x} \left (12 x^3-24 e^x x^2+6 x\right )+e^{2 x^2} \left (12 x^3+6 x\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-24 x^3+24 e^x x^2-12 x\right ) \log (x)\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )}{-x^6+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{8 e^x} \left (15 \log ^2(x)-3 x^2\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{12 e^x}+\log ^6(x)-6 e^{10 e^x} \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )+e^{4 e^x} \left (2 x^2+1\right )+\left (2 x^2+1\right ) \log ^2(x)-\left (e^{2 e^x} \left (4 x^2+2\right )-4 e^{x+2 e^x} x+2\right ) \log (x)-4 e^{x+4 e^x} x+2 e^{2 e^x}\right )}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (\left (1-2 x^2\right ) x^2-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-2 \left (-2 e^{x+2 e^x} x+e^{2 e^x} \left (2 x^2+1\right )+1\right ) \log (x)\right )}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 6 \int \left (\frac {e^{2 x^2} \left (2 x^2-1\right ) x^3}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}-\frac {4 e^{2 x^2+x+2 e^x} \left (e^{2 e^x}-\log (x)\right ) x^2}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {e^{2 x^2} \left (2 x^2+1\right ) \log ^2(x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2} \log (x) x}{\left (x+e^{2 e^x}-\log (x)\right )^3 \left (x-e^{2 e^x}+\log (x)\right )^3}+\frac {2 e^{2 x^2+2 e^x} x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}+\frac {e^{2 x^2+4 e^x} \left (2 x^2+1\right ) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}-\frac {2 e^{2 x^2+2 e^x} \left (2 x^2+1\right ) \log (x) x}{\left (-x^2+e^{4 e^x}+\log ^2(x)-2 e^{2 e^x} \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 6 \int \frac {e^{2 x^2} x \left (-\left (\left (2 x^2-1\right ) x^2\right )-4 e^{x+4 e^x} x+2 e^{2 e^x}+\left (2 x^2+1\right ) \log ^2(x)+e^{4 e^x} \left (2 x^2+1\right )-\left (-4 e^{x+2 e^x} x+e^{2 e^x} \left (4 x^2+2\right )+2\right ) \log (x)\right )}{\left (-x+e^{2 e^x}-\log (x)\right )^3 \left (x+e^{2 e^x}-\log (x)\right )^3}dx\) |
Int[(E^(4*E^x + 2*x^2)*(6*x - 24*E^x*x^2 + 12*x^3) + E^(2*x^2)*(6*x^3 - 12 *x^5) - 12*E^(2*x^2)*x*Log[x] + E^(2*x^2)*(6*x + 12*x^3)*Log[x]^2 + E^(2*E ^x)*(12*E^(2*x^2)*x + E^(2*x^2)*(-12*x + 24*E^x*x^2 - 24*x^3)*Log[x]))/(E^ (12*E^x) - x^6 - 6*E^(10*E^x)*Log[x] + 3*x^4*Log[x]^2 - 3*x^2*Log[x]^4 + L og[x]^6 + E^(8*E^x)*(-3*x^2 + 15*Log[x]^2) + E^(6*E^x)*(12*x^2*Log[x] - 20 *Log[x]^3) + E^(4*E^x)*(3*x^4 - 18*x^2*Log[x]^2 + 15*Log[x]^4) + E^(2*E^x) *(-6*x^4*Log[x] + 12*x^2*Log[x]^3 - 6*Log[x]^5)),x]
3.24.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21
\[\frac {3 x^{2} {\mathrm e}^{2 x^{2}}}{\left (-{\mathrm e}^{4 \,{\mathrm e}^{x}}+2 \ln \left (x \right ) {\mathrm e}^{2 \,{\mathrm e}^{x}}+x^{2}-\ln \left (x \right )^{2}\right )^{2}}\]
int(((-24*exp(x)*x^2+12*x^3+6*x)*exp(x^2)^2*exp(exp(x))^4+((24*exp(x)*x^2- 24*x^3-12*x)*exp(x^2)^2*ln(x)+12*x*exp(x^2)^2)*exp(exp(x))^2+(12*x^3+6*x)* exp(x^2)^2*ln(x)^2-12*x*exp(x^2)^2*ln(x)+(-12*x^5+6*x^3)*exp(x^2)^2)/(exp( exp(x))^12-6*ln(x)*exp(exp(x))^10+(15*ln(x)^2-3*x^2)*exp(exp(x))^8+(-20*ln (x)^3+12*x^2*ln(x))*exp(exp(x))^6+(15*ln(x)^4-18*x^2*ln(x)^2+3*x^4)*exp(ex p(x))^4+(-6*ln(x)^5+12*x^2*ln(x)^3-6*x^4*ln(x))*exp(exp(x))^2+ln(x)^6-3*x^ 2*ln(x)^4+3*x^4*ln(x)^2-x^6),x)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.45 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 \, x^{2} e^{\left (2 \, x^{2}\right )}}{x^{4} - 2 \, x^{2} \log \left (x\right )^{2} + \log \left (x\right )^{4} - 2 \, {\left (x^{2} - 3 \, \log \left (x\right )^{2}\right )} e^{\left (4 \, e^{x}\right )} + 4 \, {\left (x^{2} \log \left (x\right ) - \log \left (x\right )^{3}\right )} e^{\left (2 \, e^{x}\right )} - 4 \, e^{\left (6 \, e^{x}\right )} \log \left (x\right ) + e^{\left (8 \, e^{x}\right )}} \]
integrate(((-24*exp(x)*x^2+12*x^3+6*x)*exp(x^2)^2*exp(exp(x))^4+((24*exp(x )*x^2-24*x^3-12*x)*exp(x^2)^2*log(x)+12*x*exp(x^2)^2)*exp(exp(x))^2+(12*x^ 3+6*x)*exp(x^2)^2*log(x)^2-12*x*exp(x^2)^2*log(x)+(-12*x^5+6*x^3)*exp(x^2) ^2)/(exp(exp(x))^12-6*log(x)*exp(exp(x))^10+(15*log(x)^2-3*x^2)*exp(exp(x) )^8+(-20*log(x)^3+12*x^2*log(x))*exp(exp(x))^6+(15*log(x)^4-18*x^2*log(x)^ 2+3*x^4)*exp(exp(x))^4+(-6*log(x)^5+12*x^2*log(x)^3-6*x^4*log(x))*exp(exp( x))^2+log(x)^6-3*x^2*log(x)^4+3*x^4*log(x)^2-x^6),x, algorithm=\
3*x^2*e^(2*x^2)/(x^4 - 2*x^2*log(x)^2 + log(x)^4 - 2*(x^2 - 3*log(x)^2)*e^ (4*e^x) + 4*(x^2*log(x) - log(x)^3)*e^(2*e^x) - 4*e^(6*e^x)*log(x) + e^(8* e^x))
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (24) = 48\).
Time = 0.30 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.67 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 x^{2} e^{2 x^{2}}}{x^{4} - 2 x^{2} \log {\left (x \right )}^{2} + \left (- 2 x^{2} + 6 \log {\left (x \right )}^{2}\right ) e^{4 e^{x}} + \left (4 x^{2} \log {\left (x \right )} - 4 \log {\left (x \right )}^{3}\right ) e^{2 e^{x}} + e^{8 e^{x}} - 4 e^{6 e^{x}} \log {\left (x \right )} + \log {\left (x \right )}^{4}} \]
integrate(((-24*exp(x)*x**2+12*x**3+6*x)*exp(x**2)**2*exp(exp(x))**4+((24* exp(x)*x**2-24*x**3-12*x)*exp(x**2)**2*ln(x)+12*x*exp(x**2)**2)*exp(exp(x) )**2+(12*x**3+6*x)*exp(x**2)**2*ln(x)**2-12*x*exp(x**2)**2*ln(x)+(-12*x**5 +6*x**3)*exp(x**2)**2)/(exp(exp(x))**12-6*ln(x)*exp(exp(x))**10+(15*ln(x)* *2-3*x**2)*exp(exp(x))**8+(-20*ln(x)**3+12*x**2*ln(x))*exp(exp(x))**6+(15* ln(x)**4-18*x**2*ln(x)**2+3*x**4)*exp(exp(x))**4+(-6*ln(x)**5+12*x**2*ln(x )**3-6*x**4*ln(x))*exp(exp(x))**2+ln(x)**6-3*x**2*ln(x)**4+3*x**4*ln(x)**2 -x**6),x)
3*x**2*exp(2*x**2)/(x**4 - 2*x**2*log(x)**2 + (-2*x**2 + 6*log(x)**2)*exp( 4*exp(x)) + (4*x**2*log(x) - 4*log(x)**3)*exp(2*exp(x)) + exp(8*exp(x)) - 4*exp(6*exp(x))*log(x) + log(x)**4)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (29) = 58\).
Time = 0.74 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.45 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 \, x^{2} e^{\left (2 \, x^{2}\right )}}{x^{4} - 2 \, x^{2} \log \left (x\right )^{2} + \log \left (x\right )^{4} - 2 \, {\left (x^{2} - 3 \, \log \left (x\right )^{2}\right )} e^{\left (4 \, e^{x}\right )} + 4 \, {\left (x^{2} \log \left (x\right ) - \log \left (x\right )^{3}\right )} e^{\left (2 \, e^{x}\right )} - 4 \, e^{\left (6 \, e^{x}\right )} \log \left (x\right ) + e^{\left (8 \, e^{x}\right )}} \]
integrate(((-24*exp(x)*x^2+12*x^3+6*x)*exp(x^2)^2*exp(exp(x))^4+((24*exp(x )*x^2-24*x^3-12*x)*exp(x^2)^2*log(x)+12*x*exp(x^2)^2)*exp(exp(x))^2+(12*x^ 3+6*x)*exp(x^2)^2*log(x)^2-12*x*exp(x^2)^2*log(x)+(-12*x^5+6*x^3)*exp(x^2) ^2)/(exp(exp(x))^12-6*log(x)*exp(exp(x))^10+(15*log(x)^2-3*x^2)*exp(exp(x) )^8+(-20*log(x)^3+12*x^2*log(x))*exp(exp(x))^6+(15*log(x)^4-18*x^2*log(x)^ 2+3*x^4)*exp(exp(x))^4+(-6*log(x)^5+12*x^2*log(x)^3-6*x^4*log(x))*exp(exp( x))^2+log(x)^6-3*x^2*log(x)^4+3*x^4*log(x)^2-x^6),x, algorithm=\
3*x^2*e^(2*x^2)/(x^4 - 2*x^2*log(x)^2 + log(x)^4 - 2*(x^2 - 3*log(x)^2)*e^ (4*e^x) + 4*(x^2*log(x) - log(x)^3)*e^(2*e^x) - 4*e^(6*e^x)*log(x) + e^(8* e^x))
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (29) = 58\).
Time = 0.34 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.67 \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\frac {3 \, x^{2} e^{\left (2 \, x^{2}\right )}}{x^{4} + 4 \, x^{2} e^{\left (2 \, e^{x}\right )} \log \left (x\right ) - 2 \, x^{2} \log \left (x\right )^{2} - 4 \, e^{\left (2 \, e^{x}\right )} \log \left (x\right )^{3} + \log \left (x\right )^{4} - 2 \, x^{2} e^{\left (4 \, e^{x}\right )} + 6 \, e^{\left (4 \, e^{x}\right )} \log \left (x\right )^{2} - 4 \, e^{\left (6 \, e^{x}\right )} \log \left (x\right ) + e^{\left (8 \, e^{x}\right )}} \]
integrate(((-24*exp(x)*x^2+12*x^3+6*x)*exp(x^2)^2*exp(exp(x))^4+((24*exp(x )*x^2-24*x^3-12*x)*exp(x^2)^2*log(x)+12*x*exp(x^2)^2)*exp(exp(x))^2+(12*x^ 3+6*x)*exp(x^2)^2*log(x)^2-12*x*exp(x^2)^2*log(x)+(-12*x^5+6*x^3)*exp(x^2) ^2)/(exp(exp(x))^12-6*log(x)*exp(exp(x))^10+(15*log(x)^2-3*x^2)*exp(exp(x) )^8+(-20*log(x)^3+12*x^2*log(x))*exp(exp(x))^6+(15*log(x)^4-18*x^2*log(x)^ 2+3*x^4)*exp(exp(x))^4+(-6*log(x)^5+12*x^2*log(x)^3-6*x^4*log(x))*exp(exp( x))^2+log(x)^6-3*x^2*log(x)^4+3*x^4*log(x)^2-x^6),x, algorithm=\
3*x^2*e^(2*x^2)/(x^4 + 4*x^2*e^(2*e^x)*log(x) - 2*x^2*log(x)^2 - 4*e^(2*e^ x)*log(x)^3 + log(x)^4 - 2*x^2*e^(4*e^x) + 6*e^(4*e^x)*log(x)^2 - 4*e^(6*e ^x)*log(x) + e^(8*e^x))
Timed out. \[ \int \frac {e^{4 e^x+2 x^2} \left (6 x-24 e^x x^2+12 x^3\right )+e^{2 x^2} \left (6 x^3-12 x^5\right )-12 e^{2 x^2} x \log (x)+e^{2 x^2} \left (6 x+12 x^3\right ) \log ^2(x)+e^{2 e^x} \left (12 e^{2 x^2} x+e^{2 x^2} \left (-12 x+24 e^x x^2-24 x^3\right ) \log (x)\right )}{e^{12 e^x}-x^6-6 e^{10 e^x} \log (x)+3 x^4 \log ^2(x)-3 x^2 \log ^4(x)+\log ^6(x)+e^{8 e^x} \left (-3 x^2+15 \log ^2(x)\right )+e^{6 e^x} \left (12 x^2 \log (x)-20 \log ^3(x)\right )+e^{4 e^x} \left (3 x^4-18 x^2 \log ^2(x)+15 \log ^4(x)\right )+e^{2 e^x} \left (-6 x^4 \log (x)+12 x^2 \log ^3(x)-6 \log ^5(x)\right )} \, dx=\int \frac {{\mathrm {e}}^{4\,{\mathrm {e}}^x+2\,x^2}\,\left (6\,x-24\,x^2\,{\mathrm {e}}^x+12\,x^3\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (12\,x\,{\mathrm {e}}^{2\,x^2}-{\mathrm {e}}^{2\,x^2}\,\ln \left (x\right )\,\left (12\,x-24\,x^2\,{\mathrm {e}}^x+24\,x^3\right )\right )+{\mathrm {e}}^{2\,x^2}\,\left (6\,x^3-12\,x^5\right )-12\,x\,{\mathrm {e}}^{2\,x^2}\,\ln \left (x\right )+{\mathrm {e}}^{2\,x^2}\,{\ln \left (x\right )}^2\,\left (12\,x^3+6\,x\right )}{{\mathrm {e}}^{12\,{\mathrm {e}}^x}+{\mathrm {e}}^{6\,{\mathrm {e}}^x}\,\left (12\,x^2\,\ln \left (x\right )-20\,{\ln \left (x\right )}^3\right )+{\mathrm {e}}^{8\,{\mathrm {e}}^x}\,\left (15\,{\ln \left (x\right )}^2-3\,x^2\right )+{\ln \left (x\right )}^6-3\,x^2\,{\ln \left (x\right )}^4+3\,x^4\,{\ln \left (x\right )}^2-{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (6\,x^4\,\ln \left (x\right )-12\,x^2\,{\ln \left (x\right )}^3+6\,{\ln \left (x\right )}^5\right )+{\mathrm {e}}^{4\,{\mathrm {e}}^x}\,\left (3\,x^4-18\,x^2\,{\ln \left (x\right )}^2+15\,{\ln \left (x\right )}^4\right )-x^6-6\,{\mathrm {e}}^{10\,{\mathrm {e}}^x}\,\ln \left (x\right )} \,d x \]
int((exp(2*exp(x))*(12*x*exp(2*x^2) - exp(2*x^2)*log(x)*(12*x - 24*x^2*exp (x) + 24*x^3)) + exp(2*x^2)*(6*x^3 - 12*x^5) + exp(2*x^2)*exp(4*exp(x))*(6 *x - 24*x^2*exp(x) + 12*x^3) - 12*x*exp(2*x^2)*log(x) + exp(2*x^2)*log(x)^ 2*(6*x + 12*x^3))/(exp(12*exp(x)) + exp(6*exp(x))*(12*x^2*log(x) - 20*log( x)^3) + exp(8*exp(x))*(15*log(x)^2 - 3*x^2) + log(x)^6 - 3*x^2*log(x)^4 + 3*x^4*log(x)^2 - exp(2*exp(x))*(6*x^4*log(x) + 6*log(x)^5 - 12*x^2*log(x)^ 3) + exp(4*exp(x))*(15*log(x)^4 - 18*x^2*log(x)^2 + 3*x^4) - x^6 - 6*exp(1 0*exp(x))*log(x)),x)
int((exp(4*exp(x) + 2*x^2)*(6*x - 24*x^2*exp(x) + 12*x^3) + exp(2*exp(x))* (12*x*exp(2*x^2) - exp(2*x^2)*log(x)*(12*x - 24*x^2*exp(x) + 24*x^3)) + ex p(2*x^2)*(6*x^3 - 12*x^5) - 12*x*exp(2*x^2)*log(x) + exp(2*x^2)*log(x)^2*( 6*x + 12*x^3))/(exp(12*exp(x)) + exp(6*exp(x))*(12*x^2*log(x) - 20*log(x)^ 3) + exp(8*exp(x))*(15*log(x)^2 - 3*x^2) + log(x)^6 - 3*x^2*log(x)^4 + 3*x ^4*log(x)^2 - exp(2*exp(x))*(6*x^4*log(x) + 6*log(x)^5 - 12*x^2*log(x)^3) + exp(4*exp(x))*(15*log(x)^4 - 18*x^2*log(x)^2 + 3*x^4) - x^6 - 6*exp(10*e xp(x))*log(x)), x)