Integrand size = 194, antiderivative size = 28 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\left (1-\left (4+\frac {e^x}{x^4}\right )^2-(1+x)^2-\log (5)\right )^x \]
Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\left (-\frac {e^{2 x}+8 e^x x^4+x^8 \left (16+2 x+x^2+\log (5)\right )}{x^8}\right )^x \]
Integrate[(((-E^(2*x) - 8*E^x*x^4 - 16*x^8 - 2*x^9 - x^10 - x^8*Log[5])/x^ 8)^x*(2*x^9 + 2*x^10 + E^(2*x)*(-8 + 2*x) + E^x*(-32*x^4 + 8*x^5) + (E^(2* x) + 8*E^x*x^4 + 16*x^8 + 2*x^9 + x^10 + x^8*Log[5])*Log[(-E^(2*x) - 8*E^x *x^4 - 16*x^8 - 2*x^9 - x^10 - x^8*Log[5])/x^8]))/(E^(2*x) + 8*E^x*x^4 + 1 6*x^8 + 2*x^9 + x^10 + x^8*Log[5]),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.
| \(\displaystyle \int \frac {\left (\frac {-x^{10}-2 x^9-16 x^8-x^8 \log (5)-8 e^x x^4-e^{2 x}}{x^8}\right )^x \left (2 x^{10}+2 x^9+e^x \left (8 x^5-32 x^4\right )+\left (x^{10}+2 x^9+16 x^8+x^8 \log (5)+8 e^x x^4+e^{2 x}\right ) \log \left (\frac {-x^{10}-2 x^9-16 x^8-x^8 \log (5)-8 e^x x^4-e^{2 x}}{x^8}\right )+e^{2 x} (2 x-8)\right )}{x^{10}+2 x^9+16 x^8+x^8 \log (5)+8 e^x x^4+e^{2 x}} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (\frac {-x^{10}-2 x^9-16 x^8-x^8 \log (5)-8 e^x x^4-e^{2 x}}{x^8}\right )^x \left (2 x^{10}+2 x^9+e^x \left (8 x^5-32 x^4\right )+\left (x^{10}+2 x^9+16 x^8+x^8 \log (5)+8 e^x x^4+e^{2 x}\right ) \log \left (\frac {-x^{10}-2 x^9-16 x^8-x^8 \log (5)-8 e^x x^4-e^{2 x}}{x^8}\right )+e^{2 x} (2 x-8)\right )}{x^{10}+2 x^9+x^8 (16+\log (5))+8 e^x x^4+e^{2 x}}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^x \left (2 x^{10}+2 x^9+e^x \left (8 x^5-32 x^4\right )+\left (x^{10}+2 x^9+16 x^8+x^8 \log (5)+8 e^x x^4+e^{2 x}\right ) \log \left (\frac {-x^{10}-2 x^9-16 x^8-x^8 \log (5)-8 e^x x^4-e^{2 x}}{x^8}\right )+e^{2 x} (2 x-8)\right )}{x^{10}+2 x^9+x^8 (16+\log (5))+8 e^x x^4+e^{2 x}}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^x+\log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^x-8 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^x+\frac {2 x^4 \left (-x^7+3 x^6-7 x^5 \left (1+\frac {\log (5)}{7}\right )+64 x^4 \left (1+\frac {\log (5)}{16}\right )-4 e^x x+16 e^x\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^x}{x^{10}+2 x^9+16 x^8 \left (1+\frac {\log (5)}{16}\right )+8 e^x x^4+e^{2 x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x^2 \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-x^2 \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-2 x \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-16 \left (1+\frac {\log (5)}{16}\right ) \log \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}-\frac {8 e^x \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+x-4\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^4}-\frac {e^{2 x} \left (\log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )+2 x-8\right ) \left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-\frac {e^{2 x}}{x^8}-\frac {8 e^x}{x^4}-x^2-2 x-16 \left (1+\frac {\log (5)}{16}\right )\right )^{x-1} \left (-2 \left ((x+1) x^9+4 e^x (x-4) x^4+e^{2 x} (x-4)\right )-\left (8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}\right ) \log \left (-\frac {8 e^x x^4+x^8 \left (x^2+2 x+16+\log (5)\right )+e^{2 x}}{x^8}\right )\right )}{x^8}dx\) |
Int[(((-E^(2*x) - 8*E^x*x^4 - 16*x^8 - 2*x^9 - x^10 - x^8*Log[5])/x^8)^x*( 2*x^9 + 2*x^10 + E^(2*x)*(-8 + 2*x) + E^x*(-32*x^4 + 8*x^5) + (E^(2*x) + 8 *E^x*x^4 + 16*x^8 + 2*x^9 + x^10 + x^8*Log[5])*Log[(-E^(2*x) - 8*E^x*x^4 - 16*x^8 - 2*x^9 - x^10 - x^8*Log[5])/x^8]))/(E^(2*x) + 8*E^x*x^4 + 16*x^8 + 2*x^9 + x^10 + x^8*Log[5]),x]
3.15.42.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 754, normalized size of antiderivative = 26.93
\[\text {Expression too large to display}\]
int(((exp(x)^2+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)*ln((-exp(x)^2-8*e xp(x)*x^4-x^8*ln(5)-x^10-2*x^9-16*x^8)/x^8)+(2*x-8)*exp(x)^2+(8*x^5-32*x^4 )*exp(x)+2*x^10+2*x^9)*exp(x*ln((-exp(x)^2-8*exp(x)*x^4-x^8*ln(5)-x^10-2*x ^9-16*x^8)/x^8))/(exp(x)^2+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8),x)
x^(-8*x)*(exp(2*x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)^x*exp(1/2*I*x *Pi*(2-csgn(I*(exp(2*x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)/x^8)*csg n(I/x^8)*csgn(I*(exp(2*x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8))+csgn( I*x^2)*csgn(I*x)*csgn(I*x^3)+csgn(I*x)*csgn(I*x^3)*csgn(I*x^4)+csgn(I*x)*c sgn(I*x^4)*csgn(I*x^5)+csgn(I*x)*csgn(I*x^5)*csgn(I*x^6)+csgn(I*x)*csgn(I* x^6)*csgn(I*x^7)+csgn(I*x)*csgn(I*x^7)*csgn(I*x^8)-2*csgn(I*(exp(2*x)+8*ex p(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)/x^8)^2+csgn(I*(exp(2*x)+8*exp(x)*x^4 +x^8*ln(5)+x^10+2*x^9+16*x^8)/x^8)^3+csgn(I*x^2)^3+csgn(I*x^3)^3+csgn(I*x^ 4)^3+csgn(I*x^5)^3+csgn(I*x^6)^3+csgn(I*x^7)^3+csgn(I*x^8)^3+csgn(I*(exp(2 *x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)/x^8)^2*csgn(I/x^8)+csgn(I*(e xp(2*x)+8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8)/x^8)^2*csgn(I*(exp(2*x)+ 8*exp(x)*x^4+x^8*ln(5)+x^10+2*x^9+16*x^8))-2*csgn(I*x^2)^2*csgn(I*x)+csgn( I*x^2)*csgn(I*x)^2-csgn(I*x^2)*csgn(I*x^3)^2-csgn(I*x)*csgn(I*x^3)^2-csgn( I*x)*csgn(I*x^4)^2-csgn(I*x)*csgn(I*x^5)^2-csgn(I*x)*csgn(I*x^6)^2-csgn(I* x)*csgn(I*x^7)^2-csgn(I*x)*csgn(I*x^8)^2-csgn(I*x^3)*csgn(I*x^4)^2-csgn(I* x^4)*csgn(I*x^5)^2-csgn(I*x^5)*csgn(I*x^6)^2-csgn(I*x^6)*csgn(I*x^7)^2-csg n(I*x^7)*csgn(I*x^8)^2))
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\left (-\frac {x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}}{x^{8}}\right )^{x} \]
integrate(((exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+16*x^8)*log((-exp( x)^2-8*exp(x)*x^4-x^8*log(5)-x^10-2*x^9-16*x^8)/x^8)+(2*x-8)*exp(x)^2+(8*x ^5-32*x^4)*exp(x)+2*x^10+2*x^9)*exp(x*log((-exp(x)^2-8*exp(x)*x^4-x^8*log( 5)-x^10-2*x^9-16*x^8)/x^8))/(exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+1 6*x^8),x, algorithm=\
Timed out. \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\text {Timed out} \]
integrate(((exp(x)**2+8*exp(x)*x**4+x**8*ln(5)+x**10+2*x**9+16*x**8)*ln((- exp(x)**2-8*exp(x)*x**4-x**8*ln(5)-x**10-2*x**9-16*x**8)/x**8)+(2*x-8)*exp (x)**2+(8*x**5-32*x**4)*exp(x)+2*x**10+2*x**9)*exp(x*ln((-exp(x)**2-8*exp( x)*x**4-x**8*ln(5)-x**10-2*x**9-16*x**8)/x**8))/(exp(x)**2+8*exp(x)*x**4+x **8*ln(5)+x**10+2*x**9+16*x**8),x)
Time = 0.37 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=e^{\left (x \log \left (-x^{10} - 2 \, x^{9} - x^{8} {\left (\log \left (5\right ) + 16\right )} - 8 \, x^{4} e^{x} - e^{\left (2 \, x\right )}\right ) - 8 \, x \log \left (x\right )\right )} \]
integrate(((exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+16*x^8)*log((-exp( x)^2-8*exp(x)*x^4-x^8*log(5)-x^10-2*x^9-16*x^8)/x^8)+(2*x-8)*exp(x)^2+(8*x ^5-32*x^4)*exp(x)+2*x^10+2*x^9)*exp(x*log((-exp(x)^2-8*exp(x)*x^4-x^8*log( 5)-x^10-2*x^9-16*x^8)/x^8))/(exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+1 6*x^8),x, algorithm=\
\[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx=\int { \frac {{\left (2 \, x^{10} + 2 \, x^{9} + 2 \, {\left (x - 4\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{5} - 4 \, x^{4}\right )} e^{x} + {\left (x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}\right )} \log \left (-\frac {x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}}{x^{8}}\right )\right )} \left (-\frac {x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}}{x^{8}}\right )^{x}}{x^{10} + 2 \, x^{9} + x^{8} \log \left (5\right ) + 16 \, x^{8} + 8 \, x^{4} e^{x} + e^{\left (2 \, x\right )}} \,d x } \]
integrate(((exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+16*x^8)*log((-exp( x)^2-8*exp(x)*x^4-x^8*log(5)-x^10-2*x^9-16*x^8)/x^8)+(2*x-8)*exp(x)^2+(8*x ^5-32*x^4)*exp(x)+2*x^10+2*x^9)*exp(x*log((-exp(x)^2-8*exp(x)*x^4-x^8*log( 5)-x^10-2*x^9-16*x^8)/x^8))/(exp(x)^2+8*exp(x)*x^4+x^8*log(5)+x^10+2*x^9+1 6*x^8),x, algorithm=\
integrate((2*x^10 + 2*x^9 + 2*(x - 4)*e^(2*x) + 8*(x^5 - 4*x^4)*e^x + (x^1 0 + 2*x^9 + x^8*log(5) + 16*x^8 + 8*x^4*e^x + e^(2*x))*log(-(x^10 + 2*x^9 + x^8*log(5) + 16*x^8 + 8*x^4*e^x + e^(2*x))/x^8))*(-(x^10 + 2*x^9 + x^8*l og(5) + 16*x^8 + 8*x^4*e^x + e^(2*x))/x^8)^x/(x^10 + 2*x^9 + x^8*log(5) + 16*x^8 + 8*x^4*e^x + e^(2*x)), x)
Time = 12.75 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {\left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )^x \left (2 x^9+2 x^{10}+e^{2 x} (-8+2 x)+e^x \left (-32 x^4+8 x^5\right )+\left (e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)\right ) \log \left (\frac {-e^{2 x}-8 e^x x^4-16 x^8-2 x^9-x^{10}-x^8 \log (5)}{x^8}\right )\right )}{e^{2 x}+8 e^x x^4+16 x^8+2 x^9+x^{10}+x^8 \log (5)} \, dx={\left (\frac {1}{x^8}\right )}^x\,{\left (-{\mathrm {e}}^{2\,x}-8\,x^4\,{\mathrm {e}}^x-x^8\,\ln \left (5\right )-16\,x^8-2\,x^9-x^{10}\right )}^x \]
int((exp(x*log(-(exp(2*x) + 8*x^4*exp(x) + x^8*log(5) + 16*x^8 + 2*x^9 + x ^10)/x^8))*(log(-(exp(2*x) + 8*x^4*exp(x) + x^8*log(5) + 16*x^8 + 2*x^9 + x^10)/x^8)*(exp(2*x) + 8*x^4*exp(x) + x^8*log(5) + 16*x^8 + 2*x^9 + x^10) - exp(x)*(32*x^4 - 8*x^5) + exp(2*x)*(2*x - 8) + 2*x^9 + 2*x^10))/(exp(2*x ) + 8*x^4*exp(x) + x^8*log(5) + 16*x^8 + 2*x^9 + x^10),x)