Integrand size = 322, antiderivative size = 24 \[ \int \frac {e^{\frac {1}{9 x+6 x^2+x^3+\frac {e^{5+x} \left (-6 x-2 x^2\right ) (1+x \log (x))}{\log (x)}+\frac {e^{10+2 x} x (1+x \log (x))^2}{\log ^2(x)}}} \left ((3+3 x) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)+\frac {e^{5+x} (1+x \log (x)) \left (2+(-1-2 x) \log (x)+\left (-3 x-2 x^2\right ) \log ^2(x)\right )}{\log (x)}\right )}{\left (-27 x^2-27 x^3-9 x^4-x^5\right ) \log (x)+\left (-27 x^3-27 x^4-9 x^5-x^6\right ) \log ^2(x)+\frac {e^{15+3 x} (1+x \log (x))^3 \left (x^2 \log (x)+x^3 \log ^2(x)\right )}{\log ^3(x)}+\frac {e^{10+2 x} (1+x \log (x))^2 \left (\left (-9 x^2-3 x^3\right ) \log (x)+\left (-9 x^3-3 x^4\right ) \log ^2(x)\right )}{\log ^2(x)}+\frac {e^{5+x} (1+x \log (x)) \left (\left (27 x^2+18 x^3+3 x^4\right ) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)\right )}{\log (x)}} \, dx=e^{\frac {1}{x \left (3+x-e^{5+x} \left (x+\frac {1}{\log (x)}\right )\right )^2}} \] Output:
exp(1/x/(3-exp(x+ln(x+1/ln(x))+5)+x)^2)
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {e^{\frac {1}{9 x+6 x^2+x^3+\frac {e^{5+x} \left (-6 x-2 x^2\right ) (1+x \log (x))}{\log (x)}+\frac {e^{10+2 x} x (1+x \log (x))^2}{\log ^2(x)}}} \left ((3+3 x) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)+\frac {e^{5+x} (1+x \log (x)) \left (2+(-1-2 x) \log (x)+\left (-3 x-2 x^2\right ) \log ^2(x)\right )}{\log (x)}\right )}{\left (-27 x^2-27 x^3-9 x^4-x^5\right ) \log (x)+\left (-27 x^3-27 x^4-9 x^5-x^6\right ) \log ^2(x)+\frac {e^{15+3 x} (1+x \log (x))^3 \left (x^2 \log (x)+x^3 \log ^2(x)\right )}{\log ^3(x)}+\frac {e^{10+2 x} (1+x \log (x))^2 \left (\left (-9 x^2-3 x^3\right ) \log (x)+\left (-9 x^3-3 x^4\right ) \log ^2(x)\right )}{\log ^2(x)}+\frac {e^{5+x} (1+x \log (x)) \left (\left (27 x^2+18 x^3+3 x^4\right ) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)\right )}{\log (x)}} \, dx=e^{\frac {\log ^2(x)}{x \left (e^{5+x}+\left (-3+\left (-1+e^{5+x}\right ) x\right ) \log (x)\right )^2}} \] Input:
Integrate[(E^(9*x + 6*x^2 + x^3 + (E^(5 + x)*(-6*x - 2*x^2)*(1 + x*Log[x]) )/Log[x] + (E^(10 + 2*x)*x*(1 + x*Log[x])^2)/Log[x]^2)^(-1)*((3 + 3*x)*Log [x] + (3*x + 3*x^2)*Log[x]^2 + (E^(5 + x)*(1 + x*Log[x])*(2 + (-1 - 2*x)*L og[x] + (-3*x - 2*x^2)*Log[x]^2))/Log[x]))/((-27*x^2 - 27*x^3 - 9*x^4 - x^ 5)*Log[x] + (-27*x^3 - 27*x^4 - 9*x^5 - x^6)*Log[x]^2 + (E^(15 + 3*x)*(1 + x*Log[x])^3*(x^2*Log[x] + x^3*Log[x]^2))/Log[x]^3 + (E^(10 + 2*x)*(1 + x* Log[x])^2*((-9*x^2 - 3*x^3)*Log[x] + (-9*x^3 - 3*x^4)*Log[x]^2))/Log[x]^2 + (E^(5 + x)*(1 + x*Log[x])*((27*x^2 + 18*x^3 + 3*x^4)*Log[x] + (27*x^3 + 18*x^4 + 3*x^5)*Log[x]^2))/Log[x]),x]
Output:
E^(Log[x]^2/(x*(E^(5 + x) + (-3 + (-1 + E^(5 + x))*x)*Log[x])^2))
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.
| \(\displaystyle \int \frac {\left (\left (3 x^2+3 x\right ) \log ^2(x)+\frac {e^{x+5} (x \log (x)+1) \left (\left (-2 x^2-3 x\right ) \log ^2(x)+(-2 x-1) \log (x)+2\right )}{\log (x)}+(3 x+3) \log (x)\right ) \exp \left (\frac {1}{x^3+6 x^2+\frac {e^{x+5} \left (-2 x^2-6 x\right ) (x \log (x)+1)}{\log (x)}+9 x+\frac {e^{2 x+10} x (x \log (x)+1)^2}{\log ^2(x)}}\right )}{\frac {e^{3 x+15} \left (x^3 \log ^2(x)+x^2 \log (x)\right ) (x \log (x)+1)^3}{\log ^3(x)}+\frac {e^{2 x+10} \left (\left (-3 x^4-9 x^3\right ) \log ^2(x)+\left (-3 x^3-9 x^2\right ) \log (x)\right ) (x \log (x)+1)^2}{\log ^2(x)}+\left (-x^6-9 x^5-27 x^4-27 x^3\right ) \log ^2(x)+\frac {e^{x+5} \left (\left (3 x^5+18 x^4+27 x^3\right ) \log ^2(x)+\left (3 x^4+18 x^3+27 x^2\right ) \log (x)\right ) (x \log (x)+1)}{\log (x)}+\left (-x^5-9 x^4-27 x^3-27 x^2\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (x) \left (-\left (\left (2 e^{x+5} x^2+3 \left (e^{x+5}-1\right ) x-3\right ) \log ^2(x)\right )+2 e^{x+5}-e^{x+5} (2 x+1) \log (x)\right ) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\left (2 x^2 \log ^2(x)+3 x \log ^2(x)+2 x \log (x)+\log (x)-2\right ) \log (x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^2}-\frac {2 \left (x^3 \log ^2(x)+3 x^2 \log ^2(x)+x^2 \log (x)-x+3 x \log ^2(x)+2 x \log (x)-3\right ) \log ^2(x) \exp \left (\frac {\log ^2(x)}{x \left (e^{x+5}+\left (\left (e^{x+5}-1\right ) x-3\right ) \log (x)\right )^2}\right )}{x^2 (x \log (x)+1) \left (e^{x+5}+e^{x+5} x \log (x)-x \log (x)-3 \log (x)\right )^3}\right )dx\) |
Input:
Int[(E^(9*x + 6*x^2 + x^3 + (E^(5 + x)*(-6*x - 2*x^2)*(1 + x*Log[x]))/Log[ x] + (E^(10 + 2*x)*x*(1 + x*Log[x])^2)/Log[x]^2)^(-1)*((3 + 3*x)*Log[x] + (3*x + 3*x^2)*Log[x]^2 + (E^(5 + x)*(1 + x*Log[x])*(2 + (-1 - 2*x)*Log[x] + (-3*x - 2*x^2)*Log[x]^2))/Log[x]))/((-27*x^2 - 27*x^3 - 9*x^4 - x^5)*Log [x] + (-27*x^3 - 27*x^4 - 9*x^5 - x^6)*Log[x]^2 + (E^(15 + 3*x)*(1 + x*Log [x])^3*(x^2*Log[x] + x^3*Log[x]^2))/Log[x]^3 + (E^(10 + 2*x)*(1 + x*Log[x] )^2*((-9*x^2 - 3*x^3)*Log[x] + (-9*x^3 - 3*x^4)*Log[x]^2))/Log[x]^2 + (E^( 5 + x)*(1 + x*Log[x])*((27*x^2 + 18*x^3 + 3*x^4)*Log[x] + (27*x^3 + 18*x^4 + 3*x^5)*Log[x]^2))/Log[x]),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 578, normalized size of antiderivative = 24.08
\[\text {Expression too large to display}\]
Input:
int((((-2*x^2-3*x)*ln(x)^2+(-1-2*x)*ln(x)+2)*exp(ln((x*ln(x)+1)/ln(x))+5+x )+(3*x^2+3*x)*ln(x)^2+(3*x+3)*ln(x))*exp(1/(x*exp(ln((x*ln(x)+1)/ln(x))+5+ x)^2+(-2*x^2-6*x)*exp(ln((x*ln(x)+1)/ln(x))+5+x)+x^3+6*x^2+9*x))/((x^3*ln( x)^2+x^2*ln(x))*exp(ln((x*ln(x)+1)/ln(x))+5+x)^3+((-3*x^4-9*x^3)*ln(x)^2+( -3*x^3-9*x^2)*ln(x))*exp(ln((x*ln(x)+1)/ln(x))+5+x)^2+((3*x^5+18*x^4+27*x^ 3)*ln(x)^2+(3*x^4+18*x^3+27*x^2)*ln(x))*exp(ln((x*ln(x)+1)/ln(x))+5+x)+(-x ^6-9*x^5-27*x^4-27*x^3)*ln(x)^2+(-x^5-9*x^4-27*x^3-27*x^2)*ln(x)),x)
Output:
exp(ln(x)^2/x/(-2*ln(x)^2*exp(5+x)*exp(-1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1)) ^3)*exp(1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1))^2*csgn(I/ln(x)))*exp(1/2*I*Pi*c sgn(I/ln(x)*(x*ln(x)+1))^2*csgn(I*(x*ln(x)+1)))*exp(-1/2*I*Pi*csgn(I/ln(x) *(x*ln(x)+1))*csgn(I/ln(x))*csgn(I*(x*ln(x)+1)))*x^2-6*ln(x)^2*exp(5+x)*ex p(-1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1))^3)*exp(1/2*I*Pi*csgn(I/ln(x)*(x*ln(x )+1))^2*csgn(I/ln(x)))*exp(1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1))^2*csgn(I*(x* ln(x)+1)))*exp(-1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1))*csgn(I/ln(x))*csgn(I*(x *ln(x)+1)))*x-2*ln(x)*exp(5+x)*exp(-1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1))^3)* exp(1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1))^2*csgn(I/ln(x)))*exp(1/2*I*Pi*csgn( I/ln(x)*(x*ln(x)+1))^2*csgn(I*(x*ln(x)+1)))*exp(-1/2*I*Pi*csgn(I/ln(x)*(x* ln(x)+1))*csgn(I/ln(x))*csgn(I*(x*ln(x)+1)))*x-6*exp(5+x)*exp(-1/2*I*Pi*cs gn(I/ln(x)*(x*ln(x)+1))^3)*exp(1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1))^2*csgn(I /ln(x)))*exp(1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1))^2*csgn(I*(x*ln(x)+1)))*exp (-1/2*I*Pi*csgn(I/ln(x)*(x*ln(x)+1))*csgn(I/ln(x))*csgn(I*(x*ln(x)+1)))*ln (x)+ln(x)^2*exp(2*x+10)*x^2+x^2*ln(x)^2+6*x*ln(x)^2+2*ln(x)*exp(2*x+10)*x+ 9*ln(x)^2+exp(2*x+10)))
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (23) = 46\).
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58 \[ \int \frac {e^{\frac {1}{9 x+6 x^2+x^3+\frac {e^{5+x} \left (-6 x-2 x^2\right ) (1+x \log (x))}{\log (x)}+\frac {e^{10+2 x} x (1+x \log (x))^2}{\log ^2(x)}}} \left ((3+3 x) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)+\frac {e^{5+x} (1+x \log (x)) \left (2+(-1-2 x) \log (x)+\left (-3 x-2 x^2\right ) \log ^2(x)\right )}{\log (x)}\right )}{\left (-27 x^2-27 x^3-9 x^4-x^5\right ) \log (x)+\left (-27 x^3-27 x^4-9 x^5-x^6\right ) \log ^2(x)+\frac {e^{15+3 x} (1+x \log (x))^3 \left (x^2 \log (x)+x^3 \log ^2(x)\right )}{\log ^3(x)}+\frac {e^{10+2 x} (1+x \log (x))^2 \left (\left (-9 x^2-3 x^3\right ) \log (x)+\left (-9 x^3-3 x^4\right ) \log ^2(x)\right )}{\log ^2(x)}+\frac {e^{5+x} (1+x \log (x)) \left (\left (27 x^2+18 x^3+3 x^4\right ) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)\right )}{\log (x)}} \, dx=e^{\left (\frac {1}{x^{3} + 6 \, x^{2} + x e^{\left (2 \, x + 2 \, \log \left (\frac {x \log \left (x\right ) + 1}{\log \left (x\right )}\right ) + 10\right )} - 2 \, {\left (x^{2} + 3 \, x\right )} e^{\left (x + \log \left (\frac {x \log \left (x\right ) + 1}{\log \left (x\right )}\right ) + 5\right )} + 9 \, x}\right )} \] Input:
integrate((((-2*x^2-3*x)*log(x)^2+(-1-2*x)*log(x)+2)*exp(log((x*log(x)+1)/ log(x))+5+x)+(3*x^2+3*x)*log(x)^2+(3*x+3)*log(x))*exp(1/(x*exp(log((x*log( x)+1)/log(x))+5+x)^2+(-2*x^2-6*x)*exp(log((x*log(x)+1)/log(x))+5+x)+x^3+6* x^2+9*x))/((x^3*log(x)^2+x^2*log(x))*exp(log((x*log(x)+1)/log(x))+5+x)^3+( (-3*x^4-9*x^3)*log(x)^2+(-3*x^3-9*x^2)*log(x))*exp(log((x*log(x)+1)/log(x) )+5+x)^2+((3*x^5+18*x^4+27*x^3)*log(x)^2+(3*x^4+18*x^3+27*x^2)*log(x))*exp (log((x*log(x)+1)/log(x))+5+x)+(-x^6-9*x^5-27*x^4-27*x^3)*log(x)^2+(-x^5-9 *x^4-27*x^3-27*x^2)*log(x)),x, algorithm="fricas")
Output:
e^(1/(x^3 + 6*x^2 + x*e^(2*x + 2*log((x*log(x) + 1)/log(x)) + 10) - 2*(x^2 + 3*x)*e^(x + log((x*log(x) + 1)/log(x)) + 5) + 9*x))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 3.72 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.54 \[ \int \frac {e^{\frac {1}{9 x+6 x^2+x^3+\frac {e^{5+x} \left (-6 x-2 x^2\right ) (1+x \log (x))}{\log (x)}+\frac {e^{10+2 x} x (1+x \log (x))^2}{\log ^2(x)}}} \left ((3+3 x) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)+\frac {e^{5+x} (1+x \log (x)) \left (2+(-1-2 x) \log (x)+\left (-3 x-2 x^2\right ) \log ^2(x)\right )}{\log (x)}\right )}{\left (-27 x^2-27 x^3-9 x^4-x^5\right ) \log (x)+\left (-27 x^3-27 x^4-9 x^5-x^6\right ) \log ^2(x)+\frac {e^{15+3 x} (1+x \log (x))^3 \left (x^2 \log (x)+x^3 \log ^2(x)\right )}{\log ^3(x)}+\frac {e^{10+2 x} (1+x \log (x))^2 \left (\left (-9 x^2-3 x^3\right ) \log (x)+\left (-9 x^3-3 x^4\right ) \log ^2(x)\right )}{\log ^2(x)}+\frac {e^{5+x} (1+x \log (x)) \left (\left (27 x^2+18 x^3+3 x^4\right ) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)\right )}{\log (x)}} \, dx=e^{\frac {1}{x^{3} + 6 x^{2} + \frac {x \left (x \log {\left (x \right )} + 1\right )^{2} e^{2 x + 10}}{\log {\left (x \right )}^{2}} + 9 x + \frac {\left (- 2 x^{2} - 6 x\right ) \left (x \log {\left (x \right )} + 1\right ) e^{x + 5}}{\log {\left (x \right )}}}} \] Input:
integrate((((-2*x**2-3*x)*ln(x)**2+(-2*x-1)*ln(x)+2)*exp(ln((x*ln(x)+1)/ln (x))+5+x)+(3*x**2+3*x)*ln(x)**2+(3*x+3)*ln(x))*exp(1/(x*exp(ln((x*ln(x)+1) /ln(x))+5+x)**2+(-2*x**2-6*x)*exp(ln((x*ln(x)+1)/ln(x))+5+x)+x**3+6*x**2+9 *x))/((x**3*ln(x)**2+x**2*ln(x))*exp(ln((x*ln(x)+1)/ln(x))+5+x)**3+((-3*x* *4-9*x**3)*ln(x)**2+(-3*x**3-9*x**2)*ln(x))*exp(ln((x*ln(x)+1)/ln(x))+5+x) **2+((3*x**5+18*x**4+27*x**3)*ln(x)**2+(3*x**4+18*x**3+27*x**2)*ln(x))*exp (ln((x*ln(x)+1)/ln(x))+5+x)+(-x**6-9*x**5-27*x**4-27*x**3)*ln(x)**2+(-x**5 -9*x**4-27*x**3-27*x**2)*ln(x)),x)
Output:
exp(1/(x**3 + 6*x**2 + x*(x*log(x) + 1)**2*exp(2*x + 10)/log(x)**2 + 9*x + (-2*x**2 - 6*x)*(x*log(x) + 1)*exp(x + 5)/log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (23) = 46\).
Time = 5.31 (sec) , antiderivative size = 500, normalized size of antiderivative = 20.83 \[ \int \frac {e^{\frac {1}{9 x+6 x^2+x^3+\frac {e^{5+x} \left (-6 x-2 x^2\right ) (1+x \log (x))}{\log (x)}+\frac {e^{10+2 x} x (1+x \log (x))^2}{\log ^2(x)}}} \left ((3+3 x) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)+\frac {e^{5+x} (1+x \log (x)) \left (2+(-1-2 x) \log (x)+\left (-3 x-2 x^2\right ) \log ^2(x)\right )}{\log (x)}\right )}{\left (-27 x^2-27 x^3-9 x^4-x^5\right ) \log (x)+\left (-27 x^3-27 x^4-9 x^5-x^6\right ) \log ^2(x)+\frac {e^{15+3 x} (1+x \log (x))^3 \left (x^2 \log (x)+x^3 \log ^2(x)\right )}{\log ^3(x)}+\frac {e^{10+2 x} (1+x \log (x))^2 \left (\left (-9 x^2-3 x^3\right ) \log (x)+\left (-9 x^3-3 x^4\right ) \log ^2(x)\right )}{\log ^2(x)}+\frac {e^{5+x} (1+x \log (x)) \left (\left (27 x^2+18 x^3+3 x^4\right ) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)\right )}{\log (x)}} \, dx =\text {Too large to display} \] Input:
integrate((((-2*x^2-3*x)*log(x)^2+(-1-2*x)*log(x)+2)*exp(log((x*log(x)+1)/ log(x))+5+x)+(3*x^2+3*x)*log(x)^2+(3*x+3)*log(x))*exp(1/(x*exp(log((x*log( x)+1)/log(x))+5+x)^2+(-2*x^2-6*x)*exp(log((x*log(x)+1)/log(x))+5+x)+x^3+6* x^2+9*x))/((x^3*log(x)^2+x^2*log(x))*exp(log((x*log(x)+1)/log(x))+5+x)^3+( (-3*x^4-9*x^3)*log(x)^2+(-3*x^3-9*x^2)*log(x))*exp(log((x*log(x)+1)/log(x) )+5+x)^2+((3*x^5+18*x^4+27*x^3)*log(x)^2+(3*x^4+18*x^3+27*x^2)*log(x))*exp (log((x*log(x)+1)/log(x))+5+x)+(-x^6-9*x^5-27*x^4-27*x^3)*log(x)^2+(-x^5-9 *x^4-27*x^3-27*x^2)*log(x)),x, algorithm="maxima")
Output:
e^(e^(x + 5)*log(x)^3/(3*(x^2*e^(2*x + 10) + x^2 - 2*(x^2*e^5 + 3*x*e^5)*e ^x + 6*x + 9)*log(x)^3 - (x^2*e^(3*x + 15) - 2*(x^2*e^10 + 6*x*e^10)*e^(2* x) + (x^2*e^5 + 12*x*e^5 + 27*e^5)*e^x)*log(x)^2 + ((2*x*e^10 + 9*e^10)*e^ (2*x) - 2*x*e^(3*x + 15))*log(x) - e^(3*x + 15)) - e^(x + 5)*log(x)^3/(9*( x*e^(x + 5) - x - 3)*log(x)^3 - 3*(2*x*e^(2*x + 10) - (2*x*e^5 + 9*e^5)*e^ x)*log(x)^2 - ((x*e^10 + 9*e^10)*e^(2*x) - x*e^(3*x + 15))*log(x) + e^(3*x + 15)) - log(x)^3/(3*(x^2*e^(2*x + 10) + x^2 - 2*(x^2*e^5 + 3*x*e^5)*e^x + 6*x + 9)*log(x)^3 - (x^2*e^(3*x + 15) - 2*(x^2*e^10 + 6*x*e^10)*e^(2*x) + (x^2*e^5 + 12*x*e^5 + 27*e^5)*e^x)*log(x)^2 + ((2*x*e^10 + 9*e^10)*e^(2* x) - 2*x*e^(3*x + 15))*log(x) - e^(3*x + 15)) + log(x)^3/(9*(x*e^(x + 5) - x - 3)*log(x)^3 - 3*(2*x*e^(2*x + 10) - (2*x*e^5 + 9*e^5)*e^x)*log(x)^2 - ((x*e^10 + 9*e^10)*e^(2*x) - x*e^(3*x + 15))*log(x) + e^(3*x + 15)) - log (x)^2/(6*x*e^(x + 5)*log(x) - 9*x*log(x)^2 - x*e^(2*x + 10)))
Exception generated. \[ \int \frac {e^{\frac {1}{9 x+6 x^2+x^3+\frac {e^{5+x} \left (-6 x-2 x^2\right ) (1+x \log (x))}{\log (x)}+\frac {e^{10+2 x} x (1+x \log (x))^2}{\log ^2(x)}}} \left ((3+3 x) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)+\frac {e^{5+x} (1+x \log (x)) \left (2+(-1-2 x) \log (x)+\left (-3 x-2 x^2\right ) \log ^2(x)\right )}{\log (x)}\right )}{\left (-27 x^2-27 x^3-9 x^4-x^5\right ) \log (x)+\left (-27 x^3-27 x^4-9 x^5-x^6\right ) \log ^2(x)+\frac {e^{15+3 x} (1+x \log (x))^3 \left (x^2 \log (x)+x^3 \log ^2(x)\right )}{\log ^3(x)}+\frac {e^{10+2 x} (1+x \log (x))^2 \left (\left (-9 x^2-3 x^3\right ) \log (x)+\left (-9 x^3-3 x^4\right ) \log ^2(x)\right )}{\log ^2(x)}+\frac {e^{5+x} (1+x \log (x)) \left (\left (27 x^2+18 x^3+3 x^4\right ) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)\right )}{\log (x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((-2*x^2-3*x)*log(x)^2+(-1-2*x)*log(x)+2)*exp(log((x*log(x)+1)/ log(x))+5+x)+(3*x^2+3*x)*log(x)^2+(3*x+3)*log(x))*exp(1/(x*exp(log((x*log( x)+1)/log(x))+5+x)^2+(-2*x^2-6*x)*exp(log((x*log(x)+1)/log(x))+5+x)+x^3+6* x^2+9*x))/((x^3*log(x)^2+x^2*log(x))*exp(log((x*log(x)+1)/log(x))+5+x)^3+( (-3*x^4-9*x^3)*log(x)^2+(-3*x^3-9*x^2)*log(x))*exp(log((x*log(x)+1)/log(x) )+5+x)^2+((3*x^5+18*x^4+27*x^3)*log(x)^2+(3*x^4+18*x^3+27*x^2)*log(x))*exp (log((x*log(x)+1)/log(x))+5+x)+(-x^6-9*x^5-27*x^4-27*x^3)*log(x)^2+(-x^5-9 *x^4-27*x^3-27*x^2)*log(x)),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument Value
Time = 1.67 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.92 \[ \int \frac {e^{\frac {1}{9 x+6 x^2+x^3+\frac {e^{5+x} \left (-6 x-2 x^2\right ) (1+x \log (x))}{\log (x)}+\frac {e^{10+2 x} x (1+x \log (x))^2}{\log ^2(x)}}} \left ((3+3 x) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)+\frac {e^{5+x} (1+x \log (x)) \left (2+(-1-2 x) \log (x)+\left (-3 x-2 x^2\right ) \log ^2(x)\right )}{\log (x)}\right )}{\left (-27 x^2-27 x^3-9 x^4-x^5\right ) \log (x)+\left (-27 x^3-27 x^4-9 x^5-x^6\right ) \log ^2(x)+\frac {e^{15+3 x} (1+x \log (x))^3 \left (x^2 \log (x)+x^3 \log ^2(x)\right )}{\log ^3(x)}+\frac {e^{10+2 x} (1+x \log (x))^2 \left (\left (-9 x^2-3 x^3\right ) \log (x)+\left (-9 x^3-3 x^4\right ) \log ^2(x)\right )}{\log ^2(x)}+\frac {e^{5+x} (1+x \log (x)) \left (\left (27 x^2+18 x^3+3 x^4\right ) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)\right )}{\log (x)}} \, dx={\mathrm {e}}^{\frac {1}{9\,x-6\,x^2\,{\mathrm {e}}^{x+5}-2\,x^3\,{\mathrm {e}}^{x+5}+x^3\,{\mathrm {e}}^{2\,x+10}+6\,x^2+x^3+\frac {2\,x^2\,{\mathrm {e}}^{2\,x+10}}{\ln \left (x\right )}-\frac {6\,x\,{\mathrm {e}}^{x+5}}{\ln \left (x\right )}+\frac {x\,{\mathrm {e}}^{2\,x+10}}{{\ln \left (x\right )}^2}-\frac {2\,x^2\,{\mathrm {e}}^{x+5}}{\ln \left (x\right )}}} \] Input:
int(-(exp(1/(9*x - exp(x + log((x*log(x) + 1)/log(x)) + 5)*(6*x + 2*x^2) + 6*x^2 + x^3 + x*exp(2*x + 2*log((x*log(x) + 1)/log(x)) + 10)))*(log(x)^2* (3*x + 3*x^2) + log(x)*(3*x + 3) - exp(x + log((x*log(x) + 1)/log(x)) + 5) *(log(x)^2*(3*x + 2*x^2) + log(x)*(2*x + 1) - 2)))/(log(x)*(27*x^2 + 27*x^ 3 + 9*x^4 + x^5) - exp(x + log((x*log(x) + 1)/log(x)) + 5)*(log(x)*(27*x^2 + 18*x^3 + 3*x^4) + log(x)^2*(27*x^3 + 18*x^4 + 3*x^5)) + exp(2*x + 2*log ((x*log(x) + 1)/log(x)) + 10)*(log(x)*(9*x^2 + 3*x^3) + log(x)^2*(9*x^3 + 3*x^4)) - exp(3*x + 3*log((x*log(x) + 1)/log(x)) + 15)*(x^2*log(x) + x^3*l og(x)^2) + log(x)^2*(27*x^3 + 27*x^4 + 9*x^5 + x^6)),x)
Output:
exp(1/(9*x - 6*x^2*exp(x + 5) - 2*x^3*exp(x + 5) + x^3*exp(2*x + 10) + 6*x ^2 + x^3 + (2*x^2*exp(2*x + 10))/log(x) - (6*x*exp(x + 5))/log(x) + (x*exp (2*x + 10))/log(x)^2 - (2*x^2*exp(x + 5))/log(x)))
Time = 0.52 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.38 \[ \int \frac {e^{\frac {1}{9 x+6 x^2+x^3+\frac {e^{5+x} \left (-6 x-2 x^2\right ) (1+x \log (x))}{\log (x)}+\frac {e^{10+2 x} x (1+x \log (x))^2}{\log ^2(x)}}} \left ((3+3 x) \log (x)+\left (3 x+3 x^2\right ) \log ^2(x)+\frac {e^{5+x} (1+x \log (x)) \left (2+(-1-2 x) \log (x)+\left (-3 x-2 x^2\right ) \log ^2(x)\right )}{\log (x)}\right )}{\left (-27 x^2-27 x^3-9 x^4-x^5\right ) \log (x)+\left (-27 x^3-27 x^4-9 x^5-x^6\right ) \log ^2(x)+\frac {e^{15+3 x} (1+x \log (x))^3 \left (x^2 \log (x)+x^3 \log ^2(x)\right )}{\log ^3(x)}+\frac {e^{10+2 x} (1+x \log (x))^2 \left (\left (-9 x^2-3 x^3\right ) \log (x)+\left (-9 x^3-3 x^4\right ) \log ^2(x)\right )}{\log ^2(x)}+\frac {e^{5+x} (1+x \log (x)) \left (\left (27 x^2+18 x^3+3 x^4\right ) \log (x)+\left (27 x^3+18 x^4+3 x^5\right ) \log ^2(x)\right )}{\log (x)}} \, dx=e^{\frac {\mathrm {log}\left (x \right )^{2}}{e^{2 x} \mathrm {log}\left (x \right )^{2} e^{10} x^{3}+2 e^{2 x} \mathrm {log}\left (x \right ) e^{10} x^{2}+e^{2 x} e^{10} x -2 e^{x} \mathrm {log}\left (x \right )^{2} e^{5} x^{3}-6 e^{x} \mathrm {log}\left (x \right )^{2} e^{5} x^{2}-2 e^{x} \mathrm {log}\left (x \right ) e^{5} x^{2}-6 e^{x} \mathrm {log}\left (x \right ) e^{5} x +\mathrm {log}\left (x \right )^{2} x^{3}+6 \mathrm {log}\left (x \right )^{2} x^{2}+9 \mathrm {log}\left (x \right )^{2} x}} \] Input:
int((((-2*x^2-3*x)*log(x)^2+(-2*x-1)*log(x)+2)*exp(log((x*log(x)+1)/log(x) )+5+x)+(3*x^2+3*x)*log(x)^2+(3*x+3)*log(x))*exp(1/(x*exp(log((x*log(x)+1)/ log(x))+5+x)^2+(-2*x^2-6*x)*exp(log((x*log(x)+1)/log(x))+5+x)+x^3+6*x^2+9* x))/((x^3*log(x)^2+x^2*log(x))*exp(log((x*log(x)+1)/log(x))+5+x)^3+((-3*x^ 4-9*x^3)*log(x)^2+(-3*x^3-9*x^2)*log(x))*exp(log((x*log(x)+1)/log(x))+5+x) ^2+((3*x^5+18*x^4+27*x^3)*log(x)^2+(3*x^4+18*x^3+27*x^2)*log(x))*exp(log(( x*log(x)+1)/log(x))+5+x)+(-x^6-9*x^5-27*x^4-27*x^3)*log(x)^2+(-x^5-9*x^4-2 7*x^3-27*x^2)*log(x)),x)
Output:
e**(log(x)**2/(e**(2*x)*log(x)**2*e**10*x**3 + 2*e**(2*x)*log(x)*e**10*x** 2 + e**(2*x)*e**10*x - 2*e**x*log(x)**2*e**5*x**3 - 6*e**x*log(x)**2*e**5* x**2 - 2*e**x*log(x)*e**5*x**2 - 6*e**x*log(x)*e**5*x + log(x)**2*x**3 + 6 *log(x)**2*x**2 + 9*log(x)**2*x))