Integrand size = 163, antiderivative size = 21 \[ \int \frac {e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} (x-\log (x))^{\frac {1}{2+3 e^{x^2} x}} \left (2-2 x+e^{x^2} \left (3 x-3 x^2\right )+\left (e^{x^2} \left (3 x^2+6 x^4\right )+e^{x^2} \left (-3 x-6 x^3\right ) \log (x)\right ) \log (x-\log (x))\right )}{-4 x^2-12 e^{x^2} x^3-9 e^{2 x^2} x^4+\left (4 x+12 e^{x^2} x^2+9 e^{2 x^2} x^3\right ) \log (x)} \, dx=e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} \]
Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} (x-\log (x))^{\frac {1}{2+3 e^{x^2} x}} \left (2-2 x+e^{x^2} \left (3 x-3 x^2\right )+\left (e^{x^2} \left (3 x^2+6 x^4\right )+e^{x^2} \left (-3 x-6 x^3\right ) \log (x)\right ) \log (x-\log (x))\right )}{-4 x^2-12 e^{x^2} x^3-9 e^{2 x^2} x^4+\left (4 x+12 e^{x^2} x^2+9 e^{2 x^2} x^3\right ) \log (x)} \, dx=e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} \]
Integrate[(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(x - Log[x])^(2 + 3*E^x^2*x )^(-1)*(2 - 2*x + E^x^2*(3*x - 3*x^2) + (E^x^2*(3*x^2 + 6*x^4) + E^x^2*(-3 *x - 6*x^3)*Log[x])*Log[x - Log[x]]))/(-4*x^2 - 12*E^x^2*x^3 - 9*E^(2*x^2) *x^4 + (4*x + 12*E^x^2*x^2 + 9*E^(2*x^2)*x^3)*Log[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.
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \left (e^{x^2} \left (3 x-3 x^2\right )+\left (e^{x^2} \left (6 x^4+3 x^2\right )+e^{x^2} \left (-6 x^3-3 x\right ) \log (x)\right ) \log (x-\log (x))-2 x+2\right )}{-4 x^2-9 e^{2 x^2} x^4-12 e^{x^2} x^3+\left (12 e^{x^2} x^2+9 e^{2 x^2} x^3+4 x\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left ((x-1) \left (3 e^{x^2} x+2\right )-3 e^{x^2} x \left (2 x^2+1\right ) (x-\log (x)) \log (x-\log (x))\right )}{x \left (3 e^{x^2} x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (2 x^2+1\right ) e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}} \log (x-\log (x))}{x \left (3 e^{x^2} x+2\right )^2}-\frac {e^{(x-\log (x))^{\frac {1}{3 e^{x^2} x+2}}} (x-\log (x))^{\frac {1}{3 e^{x^2} x+2}-1} \left (2 x^3 \log (x-\log (x))-2 x^2 \log (x) \log (x-\log (x))-x+x \log (x-\log (x))-\log (x) \log (x-\log (x))+1\right )}{x \left (3 e^{x^2} x+2\right )}\right )dx\) |
Int[(E^(x - Log[x])^(2 + 3*E^x^2*x)^(-1)*(x - Log[x])^(2 + 3*E^x^2*x)^(-1) *(2 - 2*x + E^x^2*(3*x - 3*x^2) + (E^x^2*(3*x^2 + 6*x^4) + E^x^2*(-3*x - 6 *x^3)*Log[x])*Log[x - Log[x]]))/(-4*x^2 - 12*E^x^2*x^3 - 9*E^(2*x^2)*x^4 + (4*x + 12*E^x^2*x^2 + 9*E^(2*x^2)*x^3)*Log[x]),x]
3.24.13.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.58 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95
\[{\mathrm e}^{\left (x -\ln \left (x \right )\right )^{\frac {1}{3 \,{\mathrm e}^{x^{2}} x +2}}}\]
int((((-6*x^3-3*x)*exp(x^2)*ln(x)+(6*x^4+3*x^2)*exp(x^2))*ln(x-ln(x))+(-3* x^2+3*x)*exp(x^2)-2*x+2)*exp(ln(x-ln(x))/(3*exp(x^2)*x+2))*exp(exp(ln(x-ln (x))/(3*exp(x^2)*x+2)))/((9*x^3*exp(x^2)^2+12*x^2*exp(x^2)+4*x)*ln(x)-9*x^ 4*exp(x^2)^2-12*x^3*exp(x^2)-4*x^2),x)
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} (x-\log (x))^{\frac {1}{2+3 e^{x^2} x}} \left (2-2 x+e^{x^2} \left (3 x-3 x^2\right )+\left (e^{x^2} \left (3 x^2+6 x^4\right )+e^{x^2} \left (-3 x-6 x^3\right ) \log (x)\right ) \log (x-\log (x))\right )}{-4 x^2-12 e^{x^2} x^3-9 e^{2 x^2} x^4+\left (4 x+12 e^{x^2} x^2+9 e^{2 x^2} x^3\right ) \log (x)} \, dx=e^{\left ({\left (x - \log \left (x\right )\right )}^{\left (\frac {1}{3 \, x e^{\left (x^{2}\right )} + 2}\right )}\right )} \]
integrate((((-6*x^3-3*x)*exp(x^2)*log(x)+(6*x^4+3*x^2)*exp(x^2))*log(x-log (x))+(-3*x^2+3*x)*exp(x^2)-2*x+2)*exp(log(x-log(x))/(3*exp(x^2)*x+2))*exp( exp(log(x-log(x))/(3*exp(x^2)*x+2)))/((9*x^3*exp(x^2)^2+12*x^2*exp(x^2)+4* x)*log(x)-9*x^4*exp(x^2)^2-12*x^3*exp(x^2)-4*x^2),x, algorithm=\
Timed out. \[ \int \frac {e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} (x-\log (x))^{\frac {1}{2+3 e^{x^2} x}} \left (2-2 x+e^{x^2} \left (3 x-3 x^2\right )+\left (e^{x^2} \left (3 x^2+6 x^4\right )+e^{x^2} \left (-3 x-6 x^3\right ) \log (x)\right ) \log (x-\log (x))\right )}{-4 x^2-12 e^{x^2} x^3-9 e^{2 x^2} x^4+\left (4 x+12 e^{x^2} x^2+9 e^{2 x^2} x^3\right ) \log (x)} \, dx=\text {Timed out} \]
integrate((((-6*x**3-3*x)*exp(x**2)*ln(x)+(6*x**4+3*x**2)*exp(x**2))*ln(x- ln(x))+(-3*x**2+3*x)*exp(x**2)-2*x+2)*exp(ln(x-ln(x))/(3*exp(x**2)*x+2))*e xp(exp(ln(x-ln(x))/(3*exp(x**2)*x+2)))/((9*x**3*exp(x**2)**2+12*x**2*exp(x **2)+4*x)*ln(x)-9*x**4*exp(x**2)**2-12*x**3*exp(x**2)-4*x**2),x)
Time = 0.33 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} (x-\log (x))^{\frac {1}{2+3 e^{x^2} x}} \left (2-2 x+e^{x^2} \left (3 x-3 x^2\right )+\left (e^{x^2} \left (3 x^2+6 x^4\right )+e^{x^2} \left (-3 x-6 x^3\right ) \log (x)\right ) \log (x-\log (x))\right )}{-4 x^2-12 e^{x^2} x^3-9 e^{2 x^2} x^4+\left (4 x+12 e^{x^2} x^2+9 e^{2 x^2} x^3\right ) \log (x)} \, dx=e^{\left ({\left (x - \log \left (x\right )\right )}^{\left (\frac {1}{3 \, x e^{\left (x^{2}\right )} + 2}\right )}\right )} \]
integrate((((-6*x^3-3*x)*exp(x^2)*log(x)+(6*x^4+3*x^2)*exp(x^2))*log(x-log (x))+(-3*x^2+3*x)*exp(x^2)-2*x+2)*exp(log(x-log(x))/(3*exp(x^2)*x+2))*exp( exp(log(x-log(x))/(3*exp(x^2)*x+2)))/((9*x^3*exp(x^2)^2+12*x^2*exp(x^2)+4* x)*log(x)-9*x^4*exp(x^2)^2-12*x^3*exp(x^2)-4*x^2),x, algorithm=\
\[ \int \frac {e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} (x-\log (x))^{\frac {1}{2+3 e^{x^2} x}} \left (2-2 x+e^{x^2} \left (3 x-3 x^2\right )+\left (e^{x^2} \left (3 x^2+6 x^4\right )+e^{x^2} \left (-3 x-6 x^3\right ) \log (x)\right ) \log (x-\log (x))\right )}{-4 x^2-12 e^{x^2} x^3-9 e^{2 x^2} x^4+\left (4 x+12 e^{x^2} x^2+9 e^{2 x^2} x^3\right ) \log (x)} \, dx=\int { \frac {{\left (3 \, {\left (x^{2} - x\right )} e^{\left (x^{2}\right )} + 3 \, {\left ({\left (2 \, x^{3} + x\right )} e^{\left (x^{2}\right )} \log \left (x\right ) - {\left (2 \, x^{4} + x^{2}\right )} e^{\left (x^{2}\right )}\right )} \log \left (x - \log \left (x\right )\right ) + 2 \, x - 2\right )} {\left (x - \log \left (x\right )\right )}^{\left (\frac {1}{3 \, x e^{\left (x^{2}\right )} + 2}\right )} e^{\left ({\left (x - \log \left (x\right )\right )}^{\left (\frac {1}{3 \, x e^{\left (x^{2}\right )} + 2}\right )}\right )}}{9 \, x^{4} e^{\left (2 \, x^{2}\right )} + 12 \, x^{3} e^{\left (x^{2}\right )} + 4 \, x^{2} - {\left (9 \, x^{3} e^{\left (2 \, x^{2}\right )} + 12 \, x^{2} e^{\left (x^{2}\right )} + 4 \, x\right )} \log \left (x\right )} \,d x } \]
integrate((((-6*x^3-3*x)*exp(x^2)*log(x)+(6*x^4+3*x^2)*exp(x^2))*log(x-log (x))+(-3*x^2+3*x)*exp(x^2)-2*x+2)*exp(log(x-log(x))/(3*exp(x^2)*x+2))*exp( exp(log(x-log(x))/(3*exp(x^2)*x+2)))/((9*x^3*exp(x^2)^2+12*x^2*exp(x^2)+4* x)*log(x)-9*x^4*exp(x^2)^2-12*x^3*exp(x^2)-4*x^2),x, algorithm=\
integrate((3*(x^2 - x)*e^(x^2) + 3*((2*x^3 + x)*e^(x^2)*log(x) - (2*x^4 + x^2)*e^(x^2))*log(x - log(x)) + 2*x - 2)*(x - log(x))^(1/(3*x*e^(x^2) + 2) )*e^((x - log(x))^(1/(3*x*e^(x^2) + 2)))/(9*x^4*e^(2*x^2) + 12*x^3*e^(x^2) + 4*x^2 - (9*x^3*e^(2*x^2) + 12*x^2*e^(x^2) + 4*x)*log(x)), x)
Time = 9.54 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{(x-\log (x))^{\frac {1}{2+3 e^{x^2} x}}} (x-\log (x))^{\frac {1}{2+3 e^{x^2} x}} \left (2-2 x+e^{x^2} \left (3 x-3 x^2\right )+\left (e^{x^2} \left (3 x^2+6 x^4\right )+e^{x^2} \left (-3 x-6 x^3\right ) \log (x)\right ) \log (x-\log (x))\right )}{-4 x^2-12 e^{x^2} x^3-9 e^{2 x^2} x^4+\left (4 x+12 e^{x^2} x^2+9 e^{2 x^2} x^3\right ) \log (x)} \, dx={\mathrm {e}}^{{\left (x-\ln \left (x\right )\right )}^{\frac {1}{3\,x\,{\mathrm {e}}^{x^2}+2}}} \]
int(-(exp(log(x - log(x))/(3*x*exp(x^2) + 2))*exp(exp(log(x - log(x))/(3*x *exp(x^2) + 2)))*(exp(x^2)*(3*x - 3*x^2) - 2*x + log(x - log(x))*(exp(x^2) *(3*x^2 + 6*x^4) - exp(x^2)*log(x)*(3*x + 6*x^3)) + 2))/(12*x^3*exp(x^2) - log(x)*(4*x + 12*x^2*exp(x^2) + 9*x^3*exp(2*x^2)) + 9*x^4*exp(2*x^2) + 4* x^2),x)