Integrand size = 224, antiderivative size = 30 \[ \int \frac {e^{4+4 x} \left (4 x^5-16 x^6+e^x \left (4 x^4-16 x^5\right )\right )+e^{4+4 x} \left (8 x^2+8 e^x x^2\right ) \log \left (e^x+x\right )+e^{4+4 x} \left (-8 x^2-16 x^3+e^x \left (-8 x-16 x^2\right )\right ) \log ^2\left (e^x+x\right )}{25 e^x x^6+25 x^7+e^{8+8 x} \left (e^x x^4+x^5\right )+e^{4+4 x} \left (-10 e^x x^5-10 x^6\right )+\left (50 e^x x^3+50 x^4+e^{4+4 x} \left (-10 e^x x^2-10 x^3\right )\right ) \log ^2\left (e^x+x\right )+\left (25 e^x+25 x\right ) \log ^4\left (e^x+x\right )} \, dx=\frac {4}{-5+\frac {e^{4 (1+x)}}{x+\frac {\log ^2\left (e^x+x\right )}{x^2}}} \]
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {e^{4+4 x} \left (4 x^5-16 x^6+e^x \left (4 x^4-16 x^5\right )\right )+e^{4+4 x} \left (8 x^2+8 e^x x^2\right ) \log \left (e^x+x\right )+e^{4+4 x} \left (-8 x^2-16 x^3+e^x \left (-8 x-16 x^2\right )\right ) \log ^2\left (e^x+x\right )}{25 e^x x^6+25 x^7+e^{8+8 x} \left (e^x x^4+x^5\right )+e^{4+4 x} \left (-10 e^x x^5-10 x^6\right )+\left (50 e^x x^3+50 x^4+e^{4+4 x} \left (-10 e^x x^2-10 x^3\right )\right ) \log ^2\left (e^x+x\right )+\left (25 e^x+25 x\right ) \log ^4\left (e^x+x\right )} \, dx=-\frac {4 e^{4+4 x} x^2}{5 \left (-e^{4+4 x} x^2+5 x^3+5 \log ^2\left (e^x+x\right )\right )} \]
Integrate[(E^(4 + 4*x)*(4*x^5 - 16*x^6 + E^x*(4*x^4 - 16*x^5)) + E^(4 + 4* x)*(8*x^2 + 8*E^x*x^2)*Log[E^x + x] + E^(4 + 4*x)*(-8*x^2 - 16*x^3 + E^x*( -8*x - 16*x^2))*Log[E^x + x]^2)/(25*E^x*x^6 + 25*x^7 + E^(8 + 8*x)*(E^x*x^ 4 + x^5) + E^(4 + 4*x)*(-10*E^x*x^5 - 10*x^6) + (50*E^x*x^3 + 50*x^4 + E^( 4 + 4*x)*(-10*E^x*x^2 - 10*x^3))*Log[E^x + x]^2 + (25*E^x + 25*x)*Log[E^x + x]^4),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.
| \(\displaystyle \int \frac {e^{4 x+4} \left (8 e^x x^2+8 x^2\right ) \log \left (x+e^x\right )+e^{4 x+4} \left (-16 x^3-8 x^2+e^x \left (-16 x^2-8 x\right )\right ) \log ^2\left (x+e^x\right )+e^{4 x+4} \left (-16 x^6+4 x^5+e^x \left (4 x^4-16 x^5\right )\right )}{25 x^7+25 e^x x^6+e^{4 x+4} \left (-10 x^6-10 e^x x^5\right )+e^{8 x+8} \left (x^5+e^x x^4\right )+\left (50 x^4+50 e^x x^3+e^{4 x+4} \left (-10 x^3-10 e^x x^2\right )\right ) \log ^2\left (x+e^x\right )+\left (25 x+25 e^x\right ) \log ^4\left (x+e^x\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )+2 \left (e^x+1\right ) x \log \left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left ((1-4 x) \left (x+e^x\right ) x^3+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (-\frac {2 e^{4 x+8} (x-1) \left (x^3-e^x x^2+e^{2 x} x-e^{3 x}\right ) \log \left (x+e^x\right ) x^4}{\left (-5 x^3+e^{4 x+4} x^2-5 \log ^2\left (x+e^x\right )\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {2 e^{4 x+4} (x-1) \log \left (x+e^x\right ) x^2}{\left (x+e^x\right ) \left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right )^2}-\frac {e^{4 x+4} \left (4 e^4 x^{10}-e^4 x^9+4 e^4 \log ^2\left (x+e^x\right ) x^7-20 x^7+2 e^4 \log ^2\left (x+e^x\right ) x^6-2 e^{x+4} \log \left (x+e^x\right ) x^6-2 e^4 \log \left (x+e^x\right ) x^6+5 x^6+2 e^{x+4} \log \left (x+e^x\right ) x^5+2 e^{2 x+4} \log \left (x+e^x\right ) x^5-40 \log ^2\left (x+e^x\right ) x^4-2 e^{2 x+4} \log \left (x+e^x\right ) x^4-2 e^{3 x+4} \log \left (x+e^x\right ) x^4+10 \log \left (x+e^x\right ) x^4-5 \log ^2\left (x+e^x\right ) x^3+2 e^{3 x+4} \log \left (x+e^x\right ) x^3-20 \log ^4\left (x+e^x\right ) x+10 \log ^3\left (x+e^x\right ) x-10 \log ^4\left (x+e^x\right )\right ) x}{\left (e^4 x^6-5 x^3-5 \log ^2\left (x+e^x\right )\right ) \left (5 x^3-e^{4 x+4} x^2+5 \log ^2\left (x+e^x\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 4 \int \frac {e^{4 x+4} x \left (-\left (\left (x+e^x\right ) (4 x-1) x^3\right )+2 \left (1+e^x\right ) \log \left (x+e^x\right ) x-2 \left (x+e^x\right ) (2 x+1) \log ^2\left (x+e^x\right )\right )}{\left (x+e^x\right ) \left (\left (e^{4 x+4}-5 x\right ) x^2-5 \log ^2\left (x+e^x\right )\right )^2}dx\) |
Int[(E^(4 + 4*x)*(4*x^5 - 16*x^6 + E^x*(4*x^4 - 16*x^5)) + E^(4 + 4*x)*(8* x^2 + 8*E^x*x^2)*Log[E^x + x] + E^(4 + 4*x)*(-8*x^2 - 16*x^3 + E^x*(-8*x - 16*x^2))*Log[E^x + x]^2)/(25*E^x*x^6 + 25*x^7 + E^(8 + 8*x)*(E^x*x^4 + x^ 5) + E^(4 + 4*x)*(-10*E^x*x^5 - 10*x^6) + (50*E^x*x^3 + 50*x^4 + E^(4 + 4* x)*(-10*E^x*x^2 - 10*x^3))*Log[E^x + x]^2 + (25*E^x + 25*x)*Log[E^x + x]^4 ),x]
3.2.2.3.1 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 = 6.72 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30
| method | result | size |
| risch | \(\frac {4 x^{2} {\mathrm e}^{4+4 x}}{5 \left ({\mathrm e}^{4+4 x} x^{2}-5 x^{3}-5 \ln \left ({\mathrm e}^{x}+x \right )^{2}\right )}\) | \(39\) |
| parallelrisch | \(-\frac {4 \,{\mathrm e}^{4+4 x} x^{2}}{5 \left (5 x^{3}-{\mathrm e}^{4+4 x} x^{2}+5 \ln \left ({\mathrm e}^{x}+x \right )^{2}\right )}\) | \(40\) |
int((((-16*x^2-8*x)*exp(x)-16*x^3-8*x^2)*exp(4+4*x)*ln(exp(x)+x)^2+(8*exp( x)*x^2+8*x^2)*exp(4+4*x)*ln(exp(x)+x)+((-16*x^5+4*x^4)*exp(x)-16*x^6+4*x^5 )*exp(4+4*x))/((25*exp(x)+25*x)*ln(exp(x)+x)^4+((-10*exp(x)*x^2-10*x^3)*ex p(4+4*x)+50*exp(x)*x^3+50*x^4)*ln(exp(x)+x)^2+(exp(x)*x^4+x^5)*exp(4+4*x)^ 2+(-10*x^5*exp(x)-10*x^6)*exp(4+4*x)+25*x^6*exp(x)+25*x^7),x,method=_RETUR NVERBOSE)
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {e^{4+4 x} \left (4 x^5-16 x^6+e^x \left (4 x^4-16 x^5\right )\right )+e^{4+4 x} \left (8 x^2+8 e^x x^2\right ) \log \left (e^x+x\right )+e^{4+4 x} \left (-8 x^2-16 x^3+e^x \left (-8 x-16 x^2\right )\right ) \log ^2\left (e^x+x\right )}{25 e^x x^6+25 x^7+e^{8+8 x} \left (e^x x^4+x^5\right )+e^{4+4 x} \left (-10 e^x x^5-10 x^6\right )+\left (50 e^x x^3+50 x^4+e^{4+4 x} \left (-10 e^x x^2-10 x^3\right )\right ) \log ^2\left (e^x+x\right )+\left (25 e^x+25 x\right ) \log ^4\left (e^x+x\right )} \, dx=-\frac {4 \, x^{2} e^{\left (4 \, x + 4\right )}}{5 \, {\left (5 \, x^{3} - x^{2} e^{\left (4 \, x + 4\right )} + 5 \, \log \left (x + e^{x}\right )^{2}\right )}} \]
integrate((((-16*x^2-8*x)*exp(x)-16*x^3-8*x^2)*exp(4+4*x)*log(exp(x)+x)^2+ (8*exp(x)*x^2+8*x^2)*exp(4+4*x)*log(exp(x)+x)+((-16*x^5+4*x^4)*exp(x)-16*x ^6+4*x^5)*exp(4+4*x))/((25*exp(x)+25*x)*log(exp(x)+x)^4+((-10*exp(x)*x^2-1 0*x^3)*exp(4+4*x)+50*exp(x)*x^3+50*x^4)*log(exp(x)+x)^2+(exp(x)*x^4+x^5)*e xp(4+4*x)^2+(-10*x^5*exp(x)-10*x^6)*exp(4+4*x)+25*x^6*exp(x)+25*x^7),x, al gorithm=\
Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {e^{4+4 x} \left (4 x^5-16 x^6+e^x \left (4 x^4-16 x^5\right )\right )+e^{4+4 x} \left (8 x^2+8 e^x x^2\right ) \log \left (e^x+x\right )+e^{4+4 x} \left (-8 x^2-16 x^3+e^x \left (-8 x-16 x^2\right )\right ) \log ^2\left (e^x+x\right )}{25 e^x x^6+25 x^7+e^{8+8 x} \left (e^x x^4+x^5\right )+e^{4+4 x} \left (-10 e^x x^5-10 x^6\right )+\left (50 e^x x^3+50 x^4+e^{4+4 x} \left (-10 e^x x^2-10 x^3\right )\right ) \log ^2\left (e^x+x\right )+\left (25 e^x+25 x\right ) \log ^4\left (e^x+x\right )} \, dx=- \frac {4 x^{2} e^{4} e^{4 x}}{25 x^{3} - 5 x^{2} e^{4} e^{4 x} + 25 \log {\left (x + e^{x} \right )}^{2}} \]
integrate((((-16*x**2-8*x)*exp(x)-16*x**3-8*x**2)*exp(4+4*x)*ln(exp(x)+x)* *2+(8*exp(x)*x**2+8*x**2)*exp(4+4*x)*ln(exp(x)+x)+((-16*x**5+4*x**4)*exp(x )-16*x**6+4*x**5)*exp(4+4*x))/((25*exp(x)+25*x)*ln(exp(x)+x)**4+((-10*exp( x)*x**2-10*x**3)*exp(4+4*x)+50*exp(x)*x**3+50*x**4)*ln(exp(x)+x)**2+(exp(x )*x**4+x**5)*exp(4+4*x)**2+(-10*x**5*exp(x)-10*x**6)*exp(4+4*x)+25*x**6*ex p(x)+25*x**7),x)
Time = 0.45 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {e^{4+4 x} \left (4 x^5-16 x^6+e^x \left (4 x^4-16 x^5\right )\right )+e^{4+4 x} \left (8 x^2+8 e^x x^2\right ) \log \left (e^x+x\right )+e^{4+4 x} \left (-8 x^2-16 x^3+e^x \left (-8 x-16 x^2\right )\right ) \log ^2\left (e^x+x\right )}{25 e^x x^6+25 x^7+e^{8+8 x} \left (e^x x^4+x^5\right )+e^{4+4 x} \left (-10 e^x x^5-10 x^6\right )+\left (50 e^x x^3+50 x^4+e^{4+4 x} \left (-10 e^x x^2-10 x^3\right )\right ) \log ^2\left (e^x+x\right )+\left (25 e^x+25 x\right ) \log ^4\left (e^x+x\right )} \, dx=-\frac {4 \, x^{2} e^{\left (4 \, x + 4\right )}}{5 \, {\left (5 \, x^{3} - x^{2} e^{\left (4 \, x + 4\right )} + 5 \, \log \left (x + e^{x}\right )^{2}\right )}} \]
integrate((((-16*x^2-8*x)*exp(x)-16*x^3-8*x^2)*exp(4+4*x)*log(exp(x)+x)^2+ (8*exp(x)*x^2+8*x^2)*exp(4+4*x)*log(exp(x)+x)+((-16*x^5+4*x^4)*exp(x)-16*x ^6+4*x^5)*exp(4+4*x))/((25*exp(x)+25*x)*log(exp(x)+x)^4+((-10*exp(x)*x^2-1 0*x^3)*exp(4+4*x)+50*exp(x)*x^3+50*x^4)*log(exp(x)+x)^2+(exp(x)*x^4+x^5)*e xp(4+4*x)^2+(-10*x^5*exp(x)-10*x^6)*exp(4+4*x)+25*x^6*exp(x)+25*x^7),x, al gorithm=\
Time = 5.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {e^{4+4 x} \left (4 x^5-16 x^6+e^x \left (4 x^4-16 x^5\right )\right )+e^{4+4 x} \left (8 x^2+8 e^x x^2\right ) \log \left (e^x+x\right )+e^{4+4 x} \left (-8 x^2-16 x^3+e^x \left (-8 x-16 x^2\right )\right ) \log ^2\left (e^x+x\right )}{25 e^x x^6+25 x^7+e^{8+8 x} \left (e^x x^4+x^5\right )+e^{4+4 x} \left (-10 e^x x^5-10 x^6\right )+\left (50 e^x x^3+50 x^4+e^{4+4 x} \left (-10 e^x x^2-10 x^3\right )\right ) \log ^2\left (e^x+x\right )+\left (25 e^x+25 x\right ) \log ^4\left (e^x+x\right )} \, dx=-\frac {8 \, x^{2} e^{\left (4 \, x + 4\right )}}{5 \, {\left (5 \, x^{3} - x^{2} e^{\left (4 \, x + 4\right )} + 5 \, \log \left (x + e^{x}\right )^{2}\right )}} \]
integrate((((-16*x^2-8*x)*exp(x)-16*x^3-8*x^2)*exp(4+4*x)*log(exp(x)+x)^2+ (8*exp(x)*x^2+8*x^2)*exp(4+4*x)*log(exp(x)+x)+((-16*x^5+4*x^4)*exp(x)-16*x ^6+4*x^5)*exp(4+4*x))/((25*exp(x)+25*x)*log(exp(x)+x)^4+((-10*exp(x)*x^2-1 0*x^3)*exp(4+4*x)+50*exp(x)*x^3+50*x^4)*log(exp(x)+x)^2+(exp(x)*x^4+x^5)*e xp(4+4*x)^2+(-10*x^5*exp(x)-10*x^6)*exp(4+4*x)+25*x^6*exp(x)+25*x^7),x, al gorithm=\
Timed out. \[ \int \frac {e^{4+4 x} \left (4 x^5-16 x^6+e^x \left (4 x^4-16 x^5\right )\right )+e^{4+4 x} \left (8 x^2+8 e^x x^2\right ) \log \left (e^x+x\right )+e^{4+4 x} \left (-8 x^2-16 x^3+e^x \left (-8 x-16 x^2\right )\right ) \log ^2\left (e^x+x\right )}{25 e^x x^6+25 x^7+e^{8+8 x} \left (e^x x^4+x^5\right )+e^{4+4 x} \left (-10 e^x x^5-10 x^6\right )+\left (50 e^x x^3+50 x^4+e^{4+4 x} \left (-10 e^x x^2-10 x^3\right )\right ) \log ^2\left (e^x+x\right )+\left (25 e^x+25 x\right ) \log ^4\left (e^x+x\right )} \, dx=\int \frac {-{\mathrm {e}}^{4\,x+4}\,\left ({\mathrm {e}}^x\,\left (16\,x^2+8\,x\right )+8\,x^2+16\,x^3\right )\,{\ln \left (x+{\mathrm {e}}^x\right )}^2+{\mathrm {e}}^{4\,x+4}\,\left (8\,x^2\,{\mathrm {e}}^x+8\,x^2\right )\,\ln \left (x+{\mathrm {e}}^x\right )+{\mathrm {e}}^{4\,x+4}\,\left ({\mathrm {e}}^x\,\left (4\,x^4-16\,x^5\right )+4\,x^5-16\,x^6\right )}{25\,x^6\,{\mathrm {e}}^x+{\ln \left (x+{\mathrm {e}}^x\right )}^2\,\left (50\,x^3\,{\mathrm {e}}^x-{\mathrm {e}}^{4\,x+4}\,\left (10\,x^2\,{\mathrm {e}}^x+10\,x^3\right )+50\,x^4\right )+{\mathrm {e}}^{8\,x+8}\,\left (x^4\,{\mathrm {e}}^x+x^5\right )+{\ln \left (x+{\mathrm {e}}^x\right )}^4\,\left (25\,x+25\,{\mathrm {e}}^x\right )-{\mathrm {e}}^{4\,x+4}\,\left (10\,x^5\,{\mathrm {e}}^x+10\,x^6\right )+25\,x^7} \,d x \]
int((exp(4*x + 4)*(exp(x)*(4*x^4 - 16*x^5) + 4*x^5 - 16*x^6) + log(x + exp (x))*exp(4*x + 4)*(8*x^2*exp(x) + 8*x^2) - log(x + exp(x))^2*exp(4*x + 4)* (exp(x)*(8*x + 16*x^2) + 8*x^2 + 16*x^3))/(25*x^6*exp(x) + log(x + exp(x)) ^2*(50*x^3*exp(x) - exp(4*x + 4)*(10*x^2*exp(x) + 10*x^3) + 50*x^4) + exp( 8*x + 8)*(x^4*exp(x) + x^5) + log(x + exp(x))^4*(25*x + 25*exp(x)) - exp(4 *x + 4)*(10*x^5*exp(x) + 10*x^6) + 25*x^7),x)
int((exp(4*x + 4)*(exp(x)*(4*x^4 - 16*x^5) + 4*x^5 - 16*x^6) + log(x + exp (x))*exp(4*x + 4)*(8*x^2*exp(x) + 8*x^2) - log(x + exp(x))^2*exp(4*x + 4)* (exp(x)*(8*x + 16*x^2) + 8*x^2 + 16*x^3))/(25*x^6*exp(x) + log(x + exp(x)) ^2*(50*x^3*exp(x) - exp(4*x + 4)*(10*x^2*exp(x) + 10*x^3) + 50*x^4) + exp( 8*x + 8)*(x^4*exp(x) + x^5) + log(x + exp(x))^4*(25*x + 25*exp(x)) - exp(4 *x + 4)*(10*x^5*exp(x) + 10*x^6) + 25*x^7), x)