Integrand size = 198, antiderivative size = 30 \[ \int \frac {e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (5-e^x+x+\left (5+e^x (-1-x)+2 x\right ) \log (x)+\left (x^2-e^x x^2\right ) \log ^2(x)\right )}{e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (125 x^2+5 e^{2 x} x^2+50 x^3+5 x^4+e^x \left (-50 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-250 x^2-10 e^{2 x} x^2-100 x^3-10 x^4+e^x \left (100 x^2+20 x^3\right )\right ) \log ^2(x)} \, dx=\log \left (4 \left (-2+e^{\frac {x+\frac {1}{\log (x)}}{5 \left (-5+e^x-x\right ) x}}\right )\right ) \]
\[ \int \frac {e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (5-e^x+x+\left (5+e^x (-1-x)+2 x\right ) \log (x)+\left (x^2-e^x x^2\right ) \log ^2(x)\right )}{e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (125 x^2+5 e^{2 x} x^2+50 x^3+5 x^4+e^x \left (-50 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-250 x^2-10 e^{2 x} x^2-100 x^3-10 x^4+e^x \left (100 x^2+20 x^3\right )\right ) \log ^2(x)} \, dx=\int \frac {e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (5-e^x+x+\left (5+e^x (-1-x)+2 x\right ) \log (x)+\left (x^2-e^x x^2\right ) \log ^2(x)\right )}{e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (125 x^2+5 e^{2 x} x^2+50 x^3+5 x^4+e^x \left (-50 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-250 x^2-10 e^{2 x} x^2-100 x^3-10 x^4+e^x \left (100 x^2+20 x^3\right )\right ) \log ^2(x)} \, dx \]
Integrate[(E^((1 + x*Log[x])/((-25*x + 5*E^x*x - 5*x^2)*Log[x]))*(5 - E^x + x + (5 + E^x*(-1 - x) + 2*x)*Log[x] + (x^2 - E^x*x^2)*Log[x]^2))/(E^((1 + x*Log[x])/((-25*x + 5*E^x*x - 5*x^2)*Log[x]))*(125*x^2 + 5*E^(2*x)*x^2 + 50*x^3 + 5*x^4 + E^x*(-50*x^2 - 10*x^3))*Log[x]^2 + (-250*x^2 - 10*E^(2*x )*x^2 - 100*x^3 - 10*x^4 + E^x*(100*x^2 + 20*x^3))*Log[x]^2),x]
Integrate[(E^((1 + x*Log[x])/((-25*x + 5*E^x*x - 5*x^2)*Log[x]))*(5 - E^x + x + (5 + E^x*(-1 - x) + 2*x)*Log[x] + (x^2 - E^x*x^2)*Log[x]^2))/(E^((1 + x*Log[x])/((-25*x + 5*E^x*x - 5*x^2)*Log[x]))*(125*x^2 + 5*E^(2*x)*x^2 + 50*x^3 + 5*x^4 + E^x*(-50*x^2 - 10*x^3))*Log[x]^2 + (-250*x^2 - 10*E^(2*x )*x^2 - 100*x^3 - 10*x^4 + E^x*(100*x^2 + 20*x^3))*Log[x]^2), 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.
| \(\displaystyle \int \frac {\left (\left (x^2-e^x x^2\right ) \log ^2(x)-e^x+x+\left (e^x (-x-1)+2 x+5\right ) \log (x)+5\right ) \exp \left (\frac {x \log (x)+1}{\left (-5 x^2+5 e^x x-25 x\right ) \log (x)}\right )}{\left (5 x^4+50 x^3+5 e^{2 x} x^2+125 x^2+e^x \left (-10 x^3-50 x^2\right )\right ) \log ^2(x) \exp \left (\frac {x \log (x)+1}{\left (-5 x^2+5 e^x x-25 x\right ) \log (x)}\right )+\left (-10 x^4-100 x^3-10 e^{2 x} x^2-250 x^2+e^x \left (20 x^3+100 x^2\right )\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-\left (\left (e^x-1\right ) x^2 \log ^2(x)\right )-e^x+x-\left (-2 x+e^x (x+1)-5\right ) \log (x)+5}{5 x^2 \left (x-e^x+5\right )^2 \left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) \log ^2(x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (1-e^x\right ) x^2 \log ^2(x)+\left (2 x-e^x (x+1)+5\right ) \log (x)-e^x+x+5}{x^2 \left (x-e^x+5\right )^2 \left (1-2 e^{\frac {1}{5 x \left (x-e^x+5\right ) \log (x)}} x^{\frac {1}{5 \left (x-e^x+5\right ) \log (x)}}\right ) \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {e^x}{\left (-x+e^x-5\right )^2 x \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {2}{x \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log (x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{\left (-x+e^x-5\right )^2 \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}+\frac {e^x}{\left (-x+e^x-5\right )^2 x^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {1}{x \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}-\frac {5}{x^2 \left (x-e^x+5\right )^2 \log ^2(x) \left (-1+2 e^{\frac {1}{5 x \log (x) \left (x-e^x+5\right )}+\frac {1}{5 \left (x-e^x+5\right )}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-\left (\left (-1+e^x\right ) x^2 \log ^2(x)\right )-\left (-2 x+e^x (x+1)-5\right ) \log (x)-e^x+x+5}{\left (1-2 e^{\frac {x \log (x)+1}{5 x \left (x-e^x+5\right ) \log (x)}}\right ) x^2 \left (x-e^x+5\right )^2 \log ^2(x)}dx\) |
Int[(E^((1 + x*Log[x])/((-25*x + 5*E^x*x - 5*x^2)*Log[x]))*(5 - E^x + x + (5 + E^x*(-1 - x) + 2*x)*Log[x] + (x^2 - E^x*x^2)*Log[x]^2))/(E^((1 + x*Lo g[x])/((-25*x + 5*E^x*x - 5*x^2)*Log[x]))*(125*x^2 + 5*E^(2*x)*x^2 + 50*x^ 3 + 5*x^4 + E^x*(-50*x^2 - 10*x^3))*Log[x]^2 + (-250*x^2 - 10*E^(2*x)*x^2 - 100*x^3 - 10*x^4 + E^x*(100*x^2 + 20*x^3))*Log[x]^2),x]
3.12.58.3.1 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 = 189.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\ln \left ({\mathrm e}^{\frac {x \ln \left (x \right )+1}{5 x \left ({\mathrm e}^{x}-5-x \right ) \ln \left (x \right )}}-2\right )\) | \(29\) |
| parallelrisch | \(\ln \left ({\mathrm e}^{\frac {x \ln \left (x \right )+1}{5 x \left ({\mathrm e}^{x}-5-x \right ) \ln \left (x \right )}}-2\right )\) | \(29\) |
int(((-exp(x)*x^2+x^2)*ln(x)^2+((-1-x)*exp(x)+5+2*x)*ln(x)+x-exp(x)+5)*exp ((x*ln(x)+1)/(5*exp(x)*x-5*x^2-25*x)/ln(x))/((5*exp(x)^2*x^2+(-10*x^3-50*x ^2)*exp(x)+5*x^4+50*x^3+125*x^2)*ln(x)^2*exp((x*ln(x)+1)/(5*exp(x)*x-5*x^2 -25*x)/ln(x))+(-10*exp(x)^2*x^2+(20*x^3+100*x^2)*exp(x)-10*x^4-100*x^3-250 *x^2)*ln(x)^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (5-e^x+x+\left (5+e^x (-1-x)+2 x\right ) \log (x)+\left (x^2-e^x x^2\right ) \log ^2(x)\right )}{e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (125 x^2+5 e^{2 x} x^2+50 x^3+5 x^4+e^x \left (-50 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-250 x^2-10 e^{2 x} x^2-100 x^3-10 x^4+e^x \left (100 x^2+20 x^3\right )\right ) \log ^2(x)} \, dx=\log \left (e^{\left (-\frac {x \log \left (x\right ) + 1}{5 \, {\left (x^{2} - x e^{x} + 5 \, x\right )} \log \left (x\right )}\right )} - 2\right ) \]
integrate(((-exp(x)*x^2+x^2)*log(x)^2+((-1-x)*exp(x)+5+2*x)*log(x)+x-exp(x )+5)*exp((x*log(x)+1)/(5*exp(x)*x-5*x^2-25*x)/log(x))/((5*exp(x)^2*x^2+(-1 0*x^3-50*x^2)*exp(x)+5*x^4+50*x^3+125*x^2)*log(x)^2*exp((x*log(x)+1)/(5*ex p(x)*x-5*x^2-25*x)/log(x))+(-10*exp(x)^2*x^2+(20*x^3+100*x^2)*exp(x)-10*x^ 4-100*x^3-250*x^2)*log(x)^2),x, algorithm=\
Time = 1.54 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (5-e^x+x+\left (5+e^x (-1-x)+2 x\right ) \log (x)+\left (x^2-e^x x^2\right ) \log ^2(x)\right )}{e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (125 x^2+5 e^{2 x} x^2+50 x^3+5 x^4+e^x \left (-50 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-250 x^2-10 e^{2 x} x^2-100 x^3-10 x^4+e^x \left (100 x^2+20 x^3\right )\right ) \log ^2(x)} \, dx=\log {\left (e^{\frac {x \log {\left (x \right )} + 1}{\left (- 5 x^{2} + 5 x e^{x} - 25 x\right ) \log {\left (x \right )}}} - 2 \right )} \]
integrate(((-exp(x)*x**2+x**2)*ln(x)**2+((-1-x)*exp(x)+5+2*x)*ln(x)+x-exp( x)+5)*exp((x*ln(x)+1)/(5*exp(x)*x-5*x**2-25*x)/ln(x))/((5*exp(x)**2*x**2+( -10*x**3-50*x**2)*exp(x)+5*x**4+50*x**3+125*x**2)*ln(x)**2*exp((x*ln(x)+1) /(5*exp(x)*x-5*x**2-25*x)/ln(x))+(-10*exp(x)**2*x**2+(20*x**3+100*x**2)*ex p(x)-10*x**4-100*x**3-250*x**2)*ln(x)**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (26) = 52\).
Time = 0.36 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.80 \[ \int \frac {e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (5-e^x+x+\left (5+e^x (-1-x)+2 x\right ) \log (x)+\left (x^2-e^x x^2\right ) \log ^2(x)\right )}{e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (125 x^2+5 e^{2 x} x^2+50 x^3+5 x^4+e^x \left (-50 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-250 x^2-10 e^{2 x} x^2-100 x^3-10 x^4+e^x \left (100 x^2+20 x^3\right )\right ) \log ^2(x)} \, dx=\frac {{\left (x \log \left (x\right ) + 1\right )} e^{x} - 5 \, x \log \left (x\right ) - x - 5}{5 \, {\left (x e^{\left (2 \, x\right )} \log \left (x\right ) - {\left (x^{2} + 10 \, x\right )} e^{x} \log \left (x\right ) + 5 \, {\left (x^{2} + 5 \, x\right )} \log \left (x\right )\right )}} + \log \left ({\left (e^{\left (-\frac {1}{5 \, {\left (x - e^{x} + 5\right )}} - \frac {1}{5 \, {\left ({\left (x + 10\right )} e^{x} - 5 \, x - e^{\left (2 \, x\right )} - 25\right )} \log \left (x\right )} + \frac {1}{5 \, {\left (x e^{x} - 5 \, x\right )} \log \left (x\right )}\right )} - 2\right )} e^{\left (\frac {1}{5 \, {\left (x - e^{x} + 5\right )}} - \frac {1}{5 \, {\left (x e^{x} - 5 \, x\right )} \log \left (x\right )}\right )}\right ) \]
integrate(((-exp(x)*x^2+x^2)*log(x)^2+((-1-x)*exp(x)+5+2*x)*log(x)+x-exp(x )+5)*exp((x*log(x)+1)/(5*exp(x)*x-5*x^2-25*x)/log(x))/((5*exp(x)^2*x^2+(-1 0*x^3-50*x^2)*exp(x)+5*x^4+50*x^3+125*x^2)*log(x)^2*exp((x*log(x)+1)/(5*ex p(x)*x-5*x^2-25*x)/log(x))+(-10*exp(x)^2*x^2+(20*x^3+100*x^2)*exp(x)-10*x^ 4-100*x^3-250*x^2)*log(x)^2),x, algorithm=\
1/5*((x*log(x) + 1)*e^x - 5*x*log(x) - x - 5)/(x*e^(2*x)*log(x) - (x^2 + 1 0*x)*e^x*log(x) + 5*(x^2 + 5*x)*log(x)) + log((e^(-1/5/(x - e^x + 5) - 1/5 /(((x + 10)*e^x - 5*x - e^(2*x) - 25)*log(x)) + 1/5/((x*e^x - 5*x)*log(x)) ) - 2)*e^(1/5/(x - e^x + 5) - 1/5/((x*e^x - 5*x)*log(x))))
Exception generated. \[ \int \frac {e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (5-e^x+x+\left (5+e^x (-1-x)+2 x\right ) \log (x)+\left (x^2-e^x x^2\right ) \log ^2(x)\right )}{e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (125 x^2+5 e^{2 x} x^2+50 x^3+5 x^4+e^x \left (-50 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-250 x^2-10 e^{2 x} x^2-100 x^3-10 x^4+e^x \left (100 x^2+20 x^3\right )\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]
integrate(((-exp(x)*x^2+x^2)*log(x)^2+((-1-x)*exp(x)+5+2*x)*log(x)+x-exp(x )+5)*exp((x*log(x)+1)/(5*exp(x)*x-5*x^2-25*x)/log(x))/((5*exp(x)^2*x^2+(-1 0*x^3-50*x^2)*exp(x)+5*x^4+50*x^3+125*x^2)*log(x)^2*exp((x*log(x)+1)/(5*ex p(x)*x-5*x^2-25*x)/log(x))+(-10*exp(x)^2*x^2+(20*x^3+100*x^2)*exp(x)-10*x^ 4-100*x^3-250*x^2)*log(x)^2),x, algorithm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:exp(sageVARx)^2=exp(2*sageVARx)exp( sageVARx)^2=exp(2*sageVARx)exp(sageVARx)^2=exp(2*sageVARx)exp(sageVARx)^2= exp(2*sag
Time = 8.39 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (5-e^x+x+\left (5+e^x (-1-x)+2 x\right ) \log (x)+\left (x^2-e^x x^2\right ) \log ^2(x)\right )}{e^{\frac {1+x \log (x)}{\left (-25 x+5 e^x x-5 x^2\right ) \log (x)}} \left (125 x^2+5 e^{2 x} x^2+50 x^3+5 x^4+e^x \left (-50 x^2-10 x^3\right )\right ) \log ^2(x)+\left (-250 x^2-10 e^{2 x} x^2-100 x^3-10 x^4+e^x \left (100 x^2+20 x^3\right )\right ) \log ^2(x)} \, dx=\ln \left ({\mathrm {e}}^{-\frac {1}{5\,x-5\,{\mathrm {e}}^x+25}}\,{\mathrm {e}}^{-\frac {1}{5\,x^2\,\ln \left (x\right )+25\,x\,\ln \left (x\right )-5\,x\,{\mathrm {e}}^x\,\ln \left (x\right )}}-2\right ) \]
int(-(exp(-(x*log(x) + 1)/(log(x)*(25*x - 5*x*exp(x) + 5*x^2)))*(x - exp(x ) - log(x)^2*(x^2*exp(x) - x^2) + log(x)*(2*x - exp(x)*(x + 1) + 5) + 5))/ (log(x)^2*(10*x^2*exp(2*x) - exp(x)*(100*x^2 + 20*x^3) + 250*x^2 + 100*x^3 + 10*x^4) - exp(-(x*log(x) + 1)/(log(x)*(25*x - 5*x*exp(x) + 5*x^2)))*log (x)^2*(5*x^2*exp(2*x) - exp(x)*(50*x^2 + 10*x^3) + 125*x^2 + 50*x^3 + 5*x^ 4)),x)