Integrand size = 314, antiderivative size = 31 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\frac {x}{-4+\frac {\left (-e^x-\frac {4}{x^2}\right ) x^2}{\left (-2+\log \left (e^2+x\right )\right )^2}} \] Output:
x/((-4/x^2-exp(x))*x^2/(ln(x+exp(2))-2)^2-4)
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=-\frac {x \left (-2+\log \left (e^2+x\right )\right )^2}{20+e^x x^2-16 \log \left (e^2+x\right )+4 \log ^2\left (e^2+x\right )} \] Input:
Integrate[(-80*E^2 - 64*x + E^x*(8*x^3 + 4*x^4 + E^2*(4*x^2 + 4*x^3)) + (1 44*E^2 + 136*x + E^x*(-6*x^3 - 4*x^4 + E^2*(-4*x^2 - 4*x^3)))*Log[E^2 + x] + (-100*E^2 - 100*x + E^x*(x^3 + x^4 + E^2*(x^2 + x^3)))*Log[E^2 + x]^2 + (32*E^2 + 32*x)*Log[E^2 + x]^3 + (-4*E^2 - 4*x)*Log[E^2 + x]^4)/(400*E^2 + 400*x + E^x*(40*E^2*x^2 + 40*x^3) + E^(2*x)*(E^2*x^4 + x^5) + (-640*E^2 - 640*x + E^x*(-32*E^2*x^2 - 32*x^3))*Log[E^2 + x] + (416*E^2 + 416*x + E^ x*(8*E^2*x^2 + 8*x^3))*Log[E^2 + x]^2 + (-128*E^2 - 128*x)*Log[E^2 + x]^3 + (16*E^2 + 16*x)*Log[E^2 + x]^4),x]
Output:
-((x*(-2 + Log[E^2 + x])^2)/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + 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.
| \(\displaystyle \int \frac {e^x \left (4 x^4+8 x^3+e^2 \left (4 x^3+4 x^2\right )\right )+\left (e^x \left (x^4+x^3+e^2 \left (x^3+x^2\right )\right )-100 x-100 e^2\right ) \log ^2\left (x+e^2\right )+\left (e^x \left (-4 x^4-6 x^3+e^2 \left (-4 x^3-4 x^2\right )\right )+136 x+144 e^2\right ) \log \left (x+e^2\right )-64 x+\left (-4 x-4 e^2\right ) \log ^4\left (x+e^2\right )+\left (32 x+32 e^2\right ) \log ^3\left (x+e^2\right )-80 e^2}{e^{2 x} \left (x^5+e^2 x^4\right )+e^x \left (40 x^3+40 e^2 x^2\right )+\left (e^x \left (8 x^3+8 e^2 x^2\right )+416 x+416 e^2\right ) \log ^2\left (x+e^2\right )+\left (e^x \left (-32 x^3-32 e^2 x^2\right )-640 x-640 e^2\right ) \log \left (x+e^2\right )+400 x+\left (16 x+16 e^2\right ) \log ^4\left (x+e^2\right )+\left (-128 x-128 e^2\right ) \log ^3\left (x+e^2\right )+400 e^2} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-\log \left (x+e^2\right )\right ) \left (2 e^x (x+2) x^3+2 e^{x+2} (x+1) x^2-\left (x+e^2\right ) \left (e^x x^2 (x+1)-52\right ) \log \left (x+e^2\right )-32 x+4 \left (x+e^2\right ) \log ^3\left (x+e^2\right )-24 \left (x+e^2\right ) \log ^2\left (x+e^2\right )-40 e^2\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (-5 x^2+x^2 \left (-\log ^2\left (x+e^2\right )\right )+4 x^2 \log \left (x+e^2\right )-14 \left (1+\frac {5 e^2}{14}\right ) x-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (x+e^2\right )-2 e^2 \log ^2\left (x+e^2\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (x+e^2\right )+8 e^2 \log \left (x+e^2\right )-10 e^2\right ) \left (2-\log \left (x+e^2\right )\right )^2}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )^2}+\frac {\left (2 x^2+x^2 \left (-\log \left (x+e^2\right )\right )+4 \left (1+\frac {e^2}{2}\right ) x-\left (1+e^2\right ) x \log \left (x+e^2\right )-e^2 \log \left (x+e^2\right )+2 e^2\right ) \left (2-\log \left (x+e^2\right )\right )}{\left (x+e^2\right ) \left (e^x x^2+4 \log ^2\left (x+e^2\right )-16 \log \left (x+e^2\right )+20\right )}\right )dx\) |
Input:
Int[(-80*E^2 - 64*x + E^x*(8*x^3 + 4*x^4 + E^2*(4*x^2 + 4*x^3)) + (144*E^2 + 136*x + E^x*(-6*x^3 - 4*x^4 + E^2*(-4*x^2 - 4*x^3)))*Log[E^2 + x] + (-1 00*E^2 - 100*x + E^x*(x^3 + x^4 + E^2*(x^2 + x^3)))*Log[E^2 + x]^2 + (32*E ^2 + 32*x)*Log[E^2 + x]^3 + (-4*E^2 - 4*x)*Log[E^2 + x]^4)/(400*E^2 + 400* x + E^x*(40*E^2*x^2 + 40*x^3) + E^(2*x)*(E^2*x^4 + x^5) + (-640*E^2 - 640* x + E^x*(-32*E^2*x^2 - 32*x^3))*Log[E^2 + x] + (416*E^2 + 416*x + E^x*(8*E ^2*x^2 + 8*x^3))*Log[E^2 + x]^2 + (-128*E^2 - 128*x)*Log[E^2 + x]^3 + (16* E^2 + 16*x)*Log[E^2 + x]^4),x]
Output:
$Aborted
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35
\[-\frac {x}{4}+\frac {x \left ({\mathrm e}^{x} x^{2}+4\right )}{4 \,{\mathrm e}^{x} x^{2}+16 \ln \left (x +{\mathrm e}^{2}\right )^{2}-64 \ln \left (x +{\mathrm e}^{2}\right )+80}\]
Input:
int(((-4*exp(2)-4*x)*ln(x+exp(2))^4+(32*exp(2)+32*x)*ln(x+exp(2))^3+(((x^3 +x^2)*exp(2)+x^4+x^3)*exp(x)-100*exp(2)-100*x)*ln(x+exp(2))^2+(((-4*x^3-4* x^2)*exp(2)-4*x^4-6*x^3)*exp(x)+144*exp(2)+136*x)*ln(x+exp(2))+((4*x^3+4*x ^2)*exp(2)+4*x^4+8*x^3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x)*ln(x+exp( 2))^4+(-128*exp(2)-128*x)*ln(x+exp(2))^3+((8*x^2*exp(2)+8*x^3)*exp(x)+416* exp(2)+416*x)*ln(x+exp(2))^2+((-32*x^2*exp(2)-32*x^3)*exp(x)-640*exp(2)-64 0*x)*ln(x+exp(2))+(x^4*exp(2)+x^5)*exp(x)^2+(40*x^2*exp(2)+40*x^3)*exp(x)+ 400*exp(2)+400*x),x)
Output:
-1/4*x+1/4*x*(exp(x)*x^2+4)/(exp(x)*x^2+4*ln(x+exp(2))^2-16*ln(x+exp(2))+2 0)
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=-\frac {x \log \left (x + e^{2}\right )^{2} - 4 \, x \log \left (x + e^{2}\right ) + 4 \, x}{x^{2} e^{x} + 4 \, \log \left (x + e^{2}\right )^{2} - 16 \, \log \left (x + e^{2}\right ) + 20} \] Input:
integrate(((-4*exp(2)-4*x)*log(x+exp(2))^4+(32*exp(2)+32*x)*log(x+exp(2))^ 3+(((x^3+x^2)*exp(2)+x^4+x^3)*exp(x)-100*exp(2)-100*x)*log(x+exp(2))^2+((( -4*x^3-4*x^2)*exp(2)-4*x^4-6*x^3)*exp(x)+144*exp(2)+136*x)*log(x+exp(2))+( (4*x^3+4*x^2)*exp(2)+4*x^4+8*x^3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x) *log(x+exp(2))^4+(-128*exp(2)-128*x)*log(x+exp(2))^3+((8*x^2*exp(2)+8*x^3) *exp(x)+416*exp(2)+416*x)*log(x+exp(2))^2+((-32*x^2*exp(2)-32*x^3)*exp(x)- 640*exp(2)-640*x)*log(x+exp(2))+(x^4*exp(2)+x^5)*exp(x)^2+(40*x^2*exp(2)+4 0*x^3)*exp(x)+400*exp(2)+400*x),x, algorithm="fricas")
Output:
-(x*log(x + e^2)^2 - 4*x*log(x + e^2) + 4*x)/(x^2*e^x + 4*log(x + e^2)^2 - 16*log(x + e^2) + 20)
Time = 0.54 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\frac {- x \log {\left (x + e^{2} \right )}^{2} + 4 x \log {\left (x + e^{2} \right )} - 4 x}{x^{2} e^{x} + 4 \log {\left (x + e^{2} \right )}^{2} - 16 \log {\left (x + e^{2} \right )} + 20} \] Input:
integrate(((-4*exp(2)-4*x)*ln(x+exp(2))**4+(32*exp(2)+32*x)*ln(x+exp(2))** 3+(((x**3+x**2)*exp(2)+x**4+x**3)*exp(x)-100*exp(2)-100*x)*ln(x+exp(2))**2 +(((-4*x**3-4*x**2)*exp(2)-4*x**4-6*x**3)*exp(x)+144*exp(2)+136*x)*ln(x+ex p(2))+((4*x**3+4*x**2)*exp(2)+4*x**4+8*x**3)*exp(x)-80*exp(2)-64*x)/((16*e xp(2)+16*x)*ln(x+exp(2))**4+(-128*exp(2)-128*x)*ln(x+exp(2))**3+((8*x**2*e xp(2)+8*x**3)*exp(x)+416*exp(2)+416*x)*ln(x+exp(2))**2+((-32*x**2*exp(2)-3 2*x**3)*exp(x)-640*exp(2)-640*x)*ln(x+exp(2))+(x**4*exp(2)+x**5)*exp(x)**2 +(40*x**2*exp(2)+40*x**3)*exp(x)+400*exp(2)+400*x),x)
Output:
(-x*log(x + exp(2))**2 + 4*x*log(x + exp(2)) - 4*x)/(x**2*exp(x) + 4*log(x + exp(2))**2 - 16*log(x + exp(2)) + 20)
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=-\frac {x \log \left (x + e^{2}\right )^{2} - 4 \, x \log \left (x + e^{2}\right ) + 4 \, x}{x^{2} e^{x} + 4 \, \log \left (x + e^{2}\right )^{2} - 16 \, \log \left (x + e^{2}\right ) + 20} \] Input:
integrate(((-4*exp(2)-4*x)*log(x+exp(2))^4+(32*exp(2)+32*x)*log(x+exp(2))^ 3+(((x^3+x^2)*exp(2)+x^4+x^3)*exp(x)-100*exp(2)-100*x)*log(x+exp(2))^2+((( -4*x^3-4*x^2)*exp(2)-4*x^4-6*x^3)*exp(x)+144*exp(2)+136*x)*log(x+exp(2))+( (4*x^3+4*x^2)*exp(2)+4*x^4+8*x^3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x) *log(x+exp(2))^4+(-128*exp(2)-128*x)*log(x+exp(2))^3+((8*x^2*exp(2)+8*x^3) *exp(x)+416*exp(2)+416*x)*log(x+exp(2))^2+((-32*x^2*exp(2)-32*x^3)*exp(x)- 640*exp(2)-640*x)*log(x+exp(2))+(x^4*exp(2)+x^5)*exp(x)^2+(40*x^2*exp(2)+4 0*x^3)*exp(x)+400*exp(2)+400*x),x, algorithm="maxima")
Output:
-(x*log(x + e^2)^2 - 4*x*log(x + e^2) + 4*x)/(x^2*e^x + 4*log(x + e^2)^2 - 16*log(x + e^2) + 20)
Exception generated. \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((-4*exp(2)-4*x)*log(x+exp(2))^4+(32*exp(2)+32*x)*log(x+exp(2))^ 3+(((x^3+x^2)*exp(2)+x^4+x^3)*exp(x)-100*exp(2)-100*x)*log(x+exp(2))^2+((( -4*x^3-4*x^2)*exp(2)-4*x^4-6*x^3)*exp(x)+144*exp(2)+136*x)*log(x+exp(2))+( (4*x^3+4*x^2)*exp(2)+4*x^4+8*x^3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x) *log(x+exp(2))^4+(-128*exp(2)-128*x)*log(x+exp(2))^3+((8*x^2*exp(2)+8*x^3) *exp(x)+416*exp(2)+416*x)*log(x+exp(2))^2+((-32*x^2*exp(2)-32*x^3)*exp(x)- 640*exp(2)-640*x)*log(x+exp(2))+(x^4*exp(2)+x^5)*exp(x)^2+(40*x^2*exp(2)+4 0*x^3)*exp(x)+400*exp(2)+400*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 = 3.66 (sec) , antiderivative size = 454, normalized size of antiderivative = 14.65 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\frac {32\,x^4\,{\mathrm {e}}^x+64\,x\,{\mathrm {e}}^2+8\,x^6\,{\mathrm {e}}^{2\,x}+4\,x^7\,{\mathrm {e}}^{2\,x}+x^8\,{\mathrm {e}}^{2\,x}+x^8\,{\mathrm {e}}^{3\,x}+x^9\,{\mathrm {e}}^{3\,x}+\frac {x^{10}\,{\mathrm {e}}^{3\,x}}{4}+32\,x^3\,{\mathrm {e}}^{x+2}+16\,x^5\,{\mathrm {e}}^{2\,x+2}+12\,x^4\,{\mathrm {e}}^{2\,x+4}+12\,x^6\,{\mathrm {e}}^{2\,x+2}+4\,x^3\,{\mathrm {e}}^{2\,x+6}+12\,x^5\,{\mathrm {e}}^{2\,x+4}+3\,x^7\,{\mathrm {e}}^{2\,x+2}+4\,x^4\,{\mathrm {e}}^{2\,x+6}+3\,x^6\,{\mathrm {e}}^{2\,x+4}+3\,x^7\,{\mathrm {e}}^{3\,x+2}+x^5\,{\mathrm {e}}^{2\,x+6}+3\,x^6\,{\mathrm {e}}^{3\,x+4}+3\,x^8\,{\mathrm {e}}^{3\,x+2}+x^5\,{\mathrm {e}}^{3\,x+6}+3\,x^7\,{\mathrm {e}}^{3\,x+4}+\frac {3\,x^9\,{\mathrm {e}}^{3\,x+2}}{4}+x^6\,{\mathrm {e}}^{3\,x+6}+\frac {3\,x^8\,{\mathrm {e}}^{3\,x+4}}{4}+\frac {x^7\,{\mathrm {e}}^{3\,x+6}}{4}+64\,x^2}{\left (4\,{\ln \left (x+{\mathrm {e}}^2\right )}^2-16\,\ln \left (x+{\mathrm {e}}^2\right )+x^2\,{\mathrm {e}}^x+20\right )\,\left (64\,x+64\,{\mathrm {e}}^2+16\,x^3\,{\mathrm {e}}^x+4\,x^5\,{\mathrm {e}}^{2\,x}+4\,x^6\,{\mathrm {e}}^{2\,x}+x^7\,{\mathrm {e}}^{2\,x}+16\,x^2\,{\mathrm {e}}^{x+2}+12\,x^4\,{\mathrm {e}}^{2\,x+2}+12\,x^3\,{\mathrm {e}}^{2\,x+4}+12\,x^5\,{\mathrm {e}}^{2\,x+2}+4\,x^2\,{\mathrm {e}}^{2\,x+6}+12\,x^4\,{\mathrm {e}}^{2\,x+4}+3\,x^6\,{\mathrm {e}}^{2\,x+2}+4\,x^3\,{\mathrm {e}}^{2\,x+6}+3\,x^5\,{\mathrm {e}}^{2\,x+4}+x^4\,{\mathrm {e}}^{2\,x+6}\right )}-\frac {x}{4} \] Input:
int(-(64*x + 80*exp(2) + log(x + exp(2))^4*(4*x + 4*exp(2)) - log(x + exp( 2))^3*(32*x + 32*exp(2)) + log(x + exp(2))^2*(100*x + 100*exp(2) - exp(x)* (exp(2)*(x^2 + x^3) + x^3 + x^4)) - exp(x)*(exp(2)*(4*x^2 + 4*x^3) + 8*x^3 + 4*x^4) - log(x + exp(2))*(136*x + 144*exp(2) - exp(x)*(exp(2)*(4*x^2 + 4*x^3) + 6*x^3 + 4*x^4)))/(400*x + 400*exp(2) + log(x + exp(2))^4*(16*x + 16*exp(2)) - log(x + exp(2))^3*(128*x + 128*exp(2)) + exp(2*x)*(x^4*exp(2) + x^5) + exp(x)*(40*x^2*exp(2) + 40*x^3) - log(x + exp(2))*(640*x + 640*e xp(2) + exp(x)*(32*x^2*exp(2) + 32*x^3)) + log(x + exp(2))^2*(416*x + 416* exp(2) + exp(x)*(8*x^2*exp(2) + 8*x^3))),x)
Output:
(32*x^4*exp(x) + 64*x*exp(2) + 8*x^6*exp(2*x) + 4*x^7*exp(2*x) + x^8*exp(2 *x) + x^8*exp(3*x) + x^9*exp(3*x) + (x^10*exp(3*x))/4 + 32*x^3*exp(x + 2) + 16*x^5*exp(2*x + 2) + 12*x^4*exp(2*x + 4) + 12*x^6*exp(2*x + 2) + 4*x^3* exp(2*x + 6) + 12*x^5*exp(2*x + 4) + 3*x^7*exp(2*x + 2) + 4*x^4*exp(2*x + 6) + 3*x^6*exp(2*x + 4) + 3*x^7*exp(3*x + 2) + x^5*exp(2*x + 6) + 3*x^6*ex p(3*x + 4) + 3*x^8*exp(3*x + 2) + x^5*exp(3*x + 6) + 3*x^7*exp(3*x + 4) + (3*x^9*exp(3*x + 2))/4 + x^6*exp(3*x + 6) + (3*x^8*exp(3*x + 4))/4 + (x^7* exp(3*x + 6))/4 + 64*x^2)/((4*log(x + exp(2))^2 - 16*log(x + exp(2)) + x^2 *exp(x) + 20)*(64*x + 64*exp(2) + 16*x^3*exp(x) + 4*x^5*exp(2*x) + 4*x^6*e xp(2*x) + x^7*exp(2*x) + 16*x^2*exp(x + 2) + 12*x^4*exp(2*x + 2) + 12*x^3* exp(2*x + 4) + 12*x^5*exp(2*x + 2) + 4*x^2*exp(2*x + 6) + 12*x^4*exp(2*x + 4) + 3*x^6*exp(2*x + 2) + 4*x^3*exp(2*x + 6) + 3*x^5*exp(2*x + 4) + x^4*e xp(2*x + 6))) - x/4
Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\frac {x \left (-\mathrm {log}\left (e^{2}+x \right )^{2}+4 \,\mathrm {log}\left (e^{2}+x \right )-4\right )}{e^{x} x^{2}+4 \mathrm {log}\left (e^{2}+x \right )^{2}-16 \,\mathrm {log}\left (e^{2}+x \right )+20} \] Input:
int(((-4*exp(2)-4*x)*log(x+exp(2))^4+(32*exp(2)+32*x)*log(x+exp(2))^3+(((x ^3+x^2)*exp(2)+x^4+x^3)*exp(x)-100*exp(2)-100*x)*log(x+exp(2))^2+(((-4*x^3 -4*x^2)*exp(2)-4*x^4-6*x^3)*exp(x)+144*exp(2)+136*x)*log(x+exp(2))+((4*x^3 +4*x^2)*exp(2)+4*x^4+8*x^3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x)*log(x +exp(2))^4+(-128*exp(2)-128*x)*log(x+exp(2))^3+((8*x^2*exp(2)+8*x^3)*exp(x )+416*exp(2)+416*x)*log(x+exp(2))^2+((-32*x^2*exp(2)-32*x^3)*exp(x)-640*ex p(2)-640*x)*log(x+exp(2))+(x^4*exp(2)+x^5)*exp(x)^2+(40*x^2*exp(2)+40*x^3) *exp(x)+400*exp(2)+400*x),x)
Output:
(x*( - log(e**2 + x)**2 + 4*log(e**2 + x) - 4))/(e**x*x**2 + 4*log(e**2 + x)**2 - 16*log(e**2 + x) + 20)