Integrand size = 188, antiderivative size = 29 \[ \int \frac {8 x+32 x^3+e^{e^5} \left (-4-24 x^2\right )+4 (i \pi +\log (5))}{x^4+4 x^6+4 x^8+\left (2 x^3+4 x^5\right ) (i \pi +\log (5))+x^2 (i \pi +\log (5))^2+e^{2 e^5} \left (x^2+4 x^4+4 x^6+\left (2 x+4 x^3\right ) (i \pi +\log (5))+(i \pi +\log (5))^2\right )+e^{e^5} \left (-2 x^3-8 x^5-8 x^7+\left (-4 x^2-8 x^4\right ) (i \pi +\log (5))-2 x (i \pi +\log (5))^2\right )} \, dx=\frac {4}{\left (e^{e^5}-x\right ) \left (i \pi +x+2 x^3+\log (5)\right )} \]
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {8 x+32 x^3+e^{e^5} \left (-4-24 x^2\right )+4 (i \pi +\log (5))}{x^4+4 x^6+4 x^8+\left (2 x^3+4 x^5\right ) (i \pi +\log (5))+x^2 (i \pi +\log (5))^2+e^{2 e^5} \left (x^2+4 x^4+4 x^6+\left (2 x+4 x^3\right ) (i \pi +\log (5))+(i \pi +\log (5))^2\right )+e^{e^5} \left (-2 x^3-8 x^5-8 x^7+\left (-4 x^2-8 x^4\right ) (i \pi +\log (5))-2 x (i \pi +\log (5))^2\right )} \, dx=\frac {4}{\left (e^{e^5}-x\right ) \left (i \pi +x+2 x^3+\log (5)\right )} \]
Integrate[(8*x + 32*x^3 + E^E^5*(-4 - 24*x^2) + 4*(I*Pi + Log[5]))/(x^4 + 4*x^6 + 4*x^8 + (2*x^3 + 4*x^5)*(I*Pi + Log[5]) + x^2*(I*Pi + Log[5])^2 + E^(2*E^5)*(x^2 + 4*x^4 + 4*x^6 + (2*x + 4*x^3)*(I*Pi + Log[5]) + (I*Pi + L og[5])^2) + E^E^5*(-2*x^3 - 8*x^5 - 8*x^7 + (-4*x^2 - 8*x^4)*(I*Pi + Log[5 ]) - 2*x*(I*Pi + Log[5])^2)),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {32 x^3+e^{e^5} \left (-24 x^2-4\right )+8 x+4 (\log (5)+i \pi )}{4 x^8+4 x^6+x^4+x^2 (\log (5)+i \pi )^2+\left (4 x^5+2 x^3\right ) (\log (5)+i \pi )+e^{2 e^5} \left (4 x^6+4 x^4+\left (4 x^3+2 x\right ) (\log (5)+i \pi )+x^2+(\log (5)+i \pi )^2\right )+e^{e^5} \left (-8 x^7-8 x^5-2 x^3+\left (-8 x^4-4 x^2\right ) (\log (5)+i \pi )-2 x (\log (5)+i \pi )^2\right )} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-e^{e^5} (6 \log (5)+6 i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {8 \left (x+2 e^{e^5}\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (2 x^3+x+i \pi +\log (5)\right )}+\frac {4 \left (2 \left (1+6 e^{2 e^5}\right ) x^2-2 x \left (2 e^{e^5}+3 i \pi +\log (125)\right )+2 e^{2 e^5}+1-6 e^{e^5} (\log (5)+i \pi )\right )}{\left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right ) \left (-2 i x^3-i x+\pi -i \log (5)\right )^2}+\frac {4}{\left (e^{e^5}-x\right )^2 \left (e^{e^5}+2 e^{3 e^5}+i \pi +\log (5)\right )}\right )dx\) |
Int[(8*x + 32*x^3 + E^E^5*(-4 - 24*x^2) + 4*(I*Pi + Log[5]))/(x^4 + 4*x^6 + 4*x^8 + (2*x^3 + 4*x^5)*(I*Pi + Log[5]) + x^2*(I*Pi + Log[5])^2 + E^(2*E ^5)*(x^2 + 4*x^4 + 4*x^6 + (2*x + 4*x^3)*(I*Pi + Log[5]) + (I*Pi + Log[5]) ^2) + E^E^5*(-2*x^3 - 8*x^5 - 8*x^7 + (-4*x^2 - 8*x^4)*(I*Pi + Log[5]) - 2 *x*(I*Pi + Log[5])^2)),x]
3.20.5.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (26 ) = 52\).
Time = 6.38 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00
| method | result | size |
| risch | \(-\frac {4 i}{-2 i {\mathrm e}^{{\mathrm e}^{5}} x^{3}+2 i x^{4}-i \ln \left (5\right ) {\mathrm e}^{{\mathrm e}^{5}}+i x \ln \left (5\right )-i {\mathrm e}^{{\mathrm e}^{5}} x +i x^{2}+\pi \,{\mathrm e}^{{\mathrm e}^{5}}-\pi x}\) | \(58\) |
| parallelrisch | \(-\frac {4 i}{-2 i {\mathrm e}^{{\mathrm e}^{5}} x^{3}+2 i x^{4}-i \ln \left (5\right ) {\mathrm e}^{{\mathrm e}^{5}}+i x \ln \left (5\right )-i {\mathrm e}^{{\mathrm e}^{5}} x +i x^{2}+\pi \,{\mathrm e}^{{\mathrm e}^{5}}-\pi x}\) | \(58\) |
| norman | \(\frac {8 x^{3}+4 x -4 i \pi +4 \ln \left (5\right )}{\left ({\mathrm e}^{{\mathrm e}^{5}}-x \right ) \left (4 x^{6}+4 x^{3} \ln \left (5\right )+4 x^{4}+\pi ^{2}+\ln \left (5\right )^{2}+2 x \ln \left (5\right )+x^{2}\right )}\) | \(63\) |
| gosper | \(-\frac {4 \left (-6 x^{2} {\mathrm e}^{{\mathrm e}^{5}}+8 x^{3}+i \pi +\ln \left (5\right )-{\mathrm e}^{{\mathrm e}^{5}}+2 x \right ) \left (-2 i x^{3}-i \ln \left (5\right )-i x +\pi \right )}{\left (-4 \,{\mathrm e}^{{\mathrm e}^{5}} x^{6}+4 x^{7}+4 i x^{4} \pi +2 i x^{2} \pi -4 \ln \left (5\right ) {\mathrm e}^{{\mathrm e}^{5}} x^{3}+4 x^{4} \ln \left (5\right )-4 \,{\mathrm e}^{{\mathrm e}^{5}} x^{4}+4 x^{5}-2 i \pi \,{\mathrm e}^{{\mathrm e}^{5}} x +2 i x \ln \left (5\right ) \pi -4 i \pi \,{\mathrm e}^{{\mathrm e}^{5}} x^{3}-2 i \pi \ln \left (5\right ) {\mathrm e}^{{\mathrm e}^{5}}+\pi ^{2} {\mathrm e}^{{\mathrm e}^{5}}-x \,\pi ^{2}-\ln \left (5\right )^{2} {\mathrm e}^{{\mathrm e}^{5}}+x \ln \left (5\right )^{2}-2 \ln \left (5\right ) {\mathrm e}^{{\mathrm e}^{5}} x +2 x^{2} \ln \left (5\right )-x^{2} {\mathrm e}^{{\mathrm e}^{5}}+x^{3}\right ) \left (6 i {\mathrm e}^{{\mathrm e}^{5}} x^{2}-8 i x^{3}+\pi +i {\mathrm e}^{{\mathrm e}^{5}}-i \ln \left (5\right )-2 i x \right )}\) | \(230\) |
int(((-24*x^2-4)*exp(exp(5))+4*ln(5)+4*I*Pi+32*x^3+8*x)/(((ln(5)+I*Pi)^2+( 4*x^3+2*x)*(ln(5)+I*Pi)+4*x^6+4*x^4+x^2)*exp(exp(5))^2+(-2*x*(ln(5)+I*Pi)^ 2+(-8*x^4-4*x^2)*(ln(5)+I*Pi)-8*x^7-8*x^5-2*x^3)*exp(exp(5))+x^2*(ln(5)+I* Pi)^2+(4*x^5+2*x^3)*(ln(5)+I*Pi)+4*x^8+4*x^6+x^4),x,method=_RETURNVERBOSE)
-4*I/(-2*I*exp(exp(5))*x^3+2*I*x^4-I*ln(5)*exp(exp(5))+I*x*ln(5)-I*exp(exp (5))*x+I*x^2+Pi*exp(exp(5))-Pi*x)
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {8 x+32 x^3+e^{e^5} \left (-4-24 x^2\right )+4 (i \pi +\log (5))}{x^4+4 x^6+4 x^8+\left (2 x^3+4 x^5\right ) (i \pi +\log (5))+x^2 (i \pi +\log (5))^2+e^{2 e^5} \left (x^2+4 x^4+4 x^6+\left (2 x+4 x^3\right ) (i \pi +\log (5))+(i \pi +\log (5))^2\right )+e^{e^5} \left (-2 x^3-8 x^5-8 x^7+\left (-4 x^2-8 x^4\right ) (i \pi +\log (5))-2 x (i \pi +\log (5))^2\right )} \, dx=-\frac {4}{2 \, x^{4} + i \, \pi x + x^{2} - {\left (i \, \pi + 2 \, x^{3} + x + \log \left (5\right )\right )} e^{\left (e^{5}\right )} + x \log \left (5\right )} \]
integrate(((-24*x^2-4)*exp(exp(5))+4*log(5)+4*I*pi+32*x^3+8*x)/(((log(5)+I *pi)^2+(4*x^3+2*x)*(log(5)+I*pi)+4*x^6+4*x^4+x^2)*exp(exp(5))^2+(-2*x*(log (5)+I*pi)^2+(-8*x^4-4*x^2)*(log(5)+I*pi)-8*x^7-8*x^5-2*x^3)*exp(exp(5))+x^ 2*(log(5)+I*pi)^2+(4*x^5+2*x^3)*(log(5)+I*pi)+4*x^8+4*x^6+x^4),x, algorith m=\
Timed out. \[ \int \frac {8 x+32 x^3+e^{e^5} \left (-4-24 x^2\right )+4 (i \pi +\log (5))}{x^4+4 x^6+4 x^8+\left (2 x^3+4 x^5\right ) (i \pi +\log (5))+x^2 (i \pi +\log (5))^2+e^{2 e^5} \left (x^2+4 x^4+4 x^6+\left (2 x+4 x^3\right ) (i \pi +\log (5))+(i \pi +\log (5))^2\right )+e^{e^5} \left (-2 x^3-8 x^5-8 x^7+\left (-4 x^2-8 x^4\right ) (i \pi +\log (5))-2 x (i \pi +\log (5))^2\right )} \, dx=\text {Timed out} \]
integrate(((-24*x**2-4)*exp(exp(5))+4*ln(5)+4*I*pi+32*x**3+8*x)/(((ln(5)+I *pi)**2+(4*x**3+2*x)*(ln(5)+I*pi)+4*x**6+4*x**4+x**2)*exp(exp(5))**2+(-2*x *(ln(5)+I*pi)**2+(-8*x**4-4*x**2)*(ln(5)+I*pi)-8*x**7-8*x**5-2*x**3)*exp(e xp(5))+x**2*(ln(5)+I*pi)**2+(4*x**5+2*x**3)*(ln(5)+I*pi)+4*x**8+4*x**6+x** 4),x)
Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {8 x+32 x^3+e^{e^5} \left (-4-24 x^2\right )+4 (i \pi +\log (5))}{x^4+4 x^6+4 x^8+\left (2 x^3+4 x^5\right ) (i \pi +\log (5))+x^2 (i \pi +\log (5))^2+e^{2 e^5} \left (x^2+4 x^4+4 x^6+\left (2 x+4 x^3\right ) (i \pi +\log (5))+(i \pi +\log (5))^2\right )+e^{e^5} \left (-2 x^3-8 x^5-8 x^7+\left (-4 x^2-8 x^4\right ) (i \pi +\log (5))-2 x (i \pi +\log (5))^2\right )} \, dx=-\frac {4}{2 \, x^{4} - 2 \, x^{3} e^{\left (e^{5}\right )} + {\left (i \, \pi - e^{\left (e^{5}\right )} + \log \left (5\right )\right )} x + x^{2} - i \, \pi e^{\left (e^{5}\right )} - e^{\left (e^{5}\right )} \log \left (5\right )} \]
integrate(((-24*x^2-4)*exp(exp(5))+4*log(5)+4*I*pi+32*x^3+8*x)/(((log(5)+I *pi)^2+(4*x^3+2*x)*(log(5)+I*pi)+4*x^6+4*x^4+x^2)*exp(exp(5))^2+(-2*x*(log (5)+I*pi)^2+(-8*x^4-4*x^2)*(log(5)+I*pi)-8*x^7-8*x^5-2*x^3)*exp(exp(5))+x^ 2*(log(5)+I*pi)^2+(4*x^5+2*x^3)*(log(5)+I*pi)+4*x^8+4*x^6+x^4),x, algorith m=\
-4/(2*x^4 - 2*x^3*e^(e^5) + (I*pi - e^(e^5) + log(5))*x + x^2 - I*pi*e^(e^ 5) - e^(e^5)*log(5))
Timed out. \[ \int \frac {8 x+32 x^3+e^{e^5} \left (-4-24 x^2\right )+4 (i \pi +\log (5))}{x^4+4 x^6+4 x^8+\left (2 x^3+4 x^5\right ) (i \pi +\log (5))+x^2 (i \pi +\log (5))^2+e^{2 e^5} \left (x^2+4 x^4+4 x^6+\left (2 x+4 x^3\right ) (i \pi +\log (5))+(i \pi +\log (5))^2\right )+e^{e^5} \left (-2 x^3-8 x^5-8 x^7+\left (-4 x^2-8 x^4\right ) (i \pi +\log (5))-2 x (i \pi +\log (5))^2\right )} \, dx=\text {Timed out} \]
integrate(((-24*x^2-4)*exp(exp(5))+4*log(5)+4*I*pi+32*x^3+8*x)/(((log(5)+I *pi)^2+(4*x^3+2*x)*(log(5)+I*pi)+4*x^6+4*x^4+x^2)*exp(exp(5))^2+(-2*x*(log (5)+I*pi)^2+(-8*x^4-4*x^2)*(log(5)+I*pi)-8*x^7-8*x^5-2*x^3)*exp(exp(5))+x^ 2*(log(5)+I*pi)^2+(4*x^5+2*x^3)*(log(5)+I*pi)+4*x^8+4*x^6+x^4),x, algorith m=\
Time = 18.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {8 x+32 x^3+e^{e^5} \left (-4-24 x^2\right )+4 (i \pi +\log (5))}{x^4+4 x^6+4 x^8+\left (2 x^3+4 x^5\right ) (i \pi +\log (5))+x^2 (i \pi +\log (5))^2+e^{2 e^5} \left (x^2+4 x^4+4 x^6+\left (2 x+4 x^3\right ) (i \pi +\log (5))+(i \pi +\log (5))^2\right )+e^{e^5} \left (-2 x^3-8 x^5-8 x^7+\left (-4 x^2-8 x^4\right ) (i \pi +\log (5))-2 x (i \pi +\log (5))^2\right )} \, dx=\frac {2}{-x^4+{\mathrm {e}}^{{\mathrm {e}}^5}\,x^3-\frac {x^2}{2}+\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^5}}{2}-\frac {\ln \left (5\right )}{2}-\frac {\Pi \,1{}\mathrm {i}}{2}\right )\,x+\frac {\Pi \,{\mathrm {e}}^{{\mathrm {e}}^5}\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{{\mathrm {e}}^5}\,\ln \left (5\right )}{2}} \]
int((Pi*4i + 8*x + 4*log(5) - exp(exp(5))*(24*x^2 + 4) + 32*x^3)/(x^2*(Pi* 1i + log(5))^2 + exp(2*exp(5))*((2*x + 4*x^3)*(Pi*1i + log(5)) + x^2 + 4*x ^4 + 4*x^6 + (Pi*1i + log(5))^2) - exp(exp(5))*((Pi*1i + log(5))*(4*x^2 + 8*x^4) + 2*x*(Pi*1i + log(5))^2 + 2*x^3 + 8*x^5 + 8*x^7) + (Pi*1i + log(5) )*(2*x^3 + 4*x^5) + x^4 + 4*x^6 + 4*x^8),x)