Integrand size = 197, antiderivative size = 31 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5}{x+x \left (e^{1+\frac {(-4+x)^2}{x^2}}+x^2\right ) \left (x+\log ^2(3)\right )} \]
Time = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5 e^{8/x}}{x \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \]
Integrate[(-5*x^2 - 20*x^5 - 15*x^4*Log[3]^2 + E^((16 - 8*x + 2*x^2)/x^2)* (160*x - 40*x^2 - 10*x^3 + (160 - 40*x - 5*x^2)*Log[3]^2))/(x^4 + 2*x^7 + x^10 + (2*x^6 + 2*x^9)*Log[3]^2 + x^8*Log[3]^4 + E^((2*(16 - 8*x + 2*x^2)) /x^2)*(x^6 + 2*x^5*Log[3]^2 + x^4*Log[3]^4) + E^((16 - 8*x + 2*x^2)/x^2)*( 2*x^5 + 2*x^8 + (2*x^4 + 4*x^7)*Log[3]^2 + 2*x^6*Log[3]^4)),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-20 x^5-15 x^4 \log ^2(3)-5 x^2+e^{\frac {2 x^2-8 x+16}{x^2}} \left (-10 x^3-40 x^2+\left (-5 x^2-40 x+160\right ) \log ^2(3)+160 x\right )}{x^{10}+x^8 \log ^4(3)+2 x^7+x^4+\left (2 x^9+2 x^6\right ) \log ^2(3)+e^{\frac {2 \left (2 x^2-8 x+16\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {2 x^2-8 x+16}{x^2}} \left (2 x^8+2 x^6 \log ^4(3)+2 x^5+\left (4 x^7+2 x^4\right ) \log ^2(3)\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{16/x} \left (-20 x^5-15 x^4 \log ^2(3)-5 x^2+e^{\frac {2 x^2-8 x+16}{x^2}} \left (-10 x^3-40 x^2+\left (-5 x^2-40 x+160\right ) \log ^2(3)+160 x\right )\right )}{x^4 \left (e^{8/x} x^3+e^{\frac {16}{x^2}+2} x+e^{8/x} x^2 \log ^2(3)+e^{\frac {16}{x^2}+2} \log ^2(3)+e^{8/x}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{-\frac {16}{x^2}+\frac {8}{x}-2} \left (-2 x^3-x^2 \left (8+\log ^2(3)\right )+8 x \left (4-\log ^2(3)\right )+32 \log ^2(3)\right )}{x^4 \left (x+\log ^2(3)\right )^2}+\frac {5 e^{-\frac {16}{x^2}+\frac {16}{x}-2} \left (x^3+x^2 \log ^2(3)+1\right ) \left (2 x^3+x^2 \left (8+\log ^2(3)\right )-8 x \left (4-\log ^2(3)\right )-32 \log ^2(3)\right )}{x^4 \left (x+\log ^2(3)\right )^2 \left (e^{8/x} x^3+e^{\frac {16}{x^2}+2} x+e^{8/x} x^2 \log ^2(3)+e^{\frac {16}{x^2}+2} \log ^2(3)+e^{8/x}\right )}+\frac {5 e^{16/x} \left (-2 x^6+8 x^5 \left (1-\frac {\log ^2(3)}{2}\right )-32 x^4 \left (1+\frac {1}{16} \log ^2(3) \left (\log ^2(3)-8\right )\right )+x^3 \left (1+8 \log ^2(3) \left (\log ^2(3)-8\right )\right )+8 x^2 \left (1-4 \log ^4(3)\right )-32 x \left (1-\frac {\log ^2(3)}{4}\right )-32 \log ^2(3)\right )}{x^4 \left (x+\log ^2(3)\right ) \left (e^{8/x} x^3+e^{\frac {16}{x^2}+2} x+e^{8/x} x^2 \log ^2(3)+e^{\frac {16}{x^2}+2} \log ^2(3)+e^{8/x}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 e^{8/x} \left (-e^{\frac {16}{x^2}+2} \left (2 x^3+x^2 \left (8+\log ^2(3)\right )+8 x \left (\log ^2(3)-4\right )-32 \log ^2(3)\right )-e^{8/x} \left (4 x^5+3 x^4 \log ^2(3)+x^2\right )\right )}{x^4 \left (e^{\frac {16}{x^2}+2} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+x^2 \log ^2(3)+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int -\frac {e^{8/x} \left (e^{8/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{2+\frac {16}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2-8 \left (4-\log ^2(3)\right ) x-32 \log ^2(3)\right )\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 \int \frac {e^{8/x} \left (e^{8/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{2+\frac {16}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2-8 \left (4-\log ^2(3)\right ) x-32 \log ^2(3)\right )\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {e^{8/x} \left (4 x^3+3 \log ^2(3) x^2+1\right )}{x^2 \left (x^3+\log ^2(3) x^2+1\right ) \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{2+\frac {8}{x}+\frac {16}{x^2}} \left (-2 x^6+8 \left (1-\frac {\log ^2(3)}{2}\right ) x^5-32 \left (1+\frac {1}{16} \log ^2(3) \left (-8+\log ^2(3)\right )\right ) x^4+\left (1+8 \log ^2(3) \left (-8+\log ^2(3)\right )\right ) x^3+8 \left (1-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right ) \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -5 \int \frac {e^{16/x} \left (4 x^5+3 \log ^2(3) x^4+x^2\right )+e^{\frac {2 \left (x^2+4 x+8\right )}{x^2}} \left (2 x^3+\left (8+\log ^2(3)\right ) x^2+8 \left (-4+\log ^2(3)\right ) x-32 \log ^2(3)\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (x^3+\log ^2(3) x^2+1\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -5 \int \left (\frac {4 x^3+3 \log ^2(3) x^2+1}{x^2 \left (x^3+\log ^2(3) x^2+1\right )^2}+\frac {e^{2+\frac {16}{x^2}} \left (-6 x^6+8 \left (1-\frac {11 \log ^2(3)}{8}\right ) x^5-32 \left (1+\frac {1}{32} \log ^2(3) \left (-16+5 \log ^2(3)\right )\right ) x^4-64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x^3+8 \left (1-\frac {\log ^2(3)}{8}-4 \log ^4(3)\right ) x^2-32 \left (1-\frac {\log ^2(3)}{4}\right ) x-32 \log ^2(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )}+\frac {e^{4+\frac {32}{x^2}} \left (2 x^7-8 \left (1-\frac {3 \log ^2(3)}{4}\right ) x^6+32 \left (1+\frac {3}{16} \log ^2(3) \left (-4+\log ^2(3)\right )\right ) x^5-\left (1-2 \log ^2(3) \left (48-12 \log ^2(3)+\log ^4(3)\right )\right ) x^4-8 \left (1+\frac {\log ^2(3)}{8}-12 \log ^4(3)+\log ^6(3)\right ) x^3+32 \left (1-\frac {\log ^2(3)}{2}+\log ^6(3)\right ) x^2+64 \log ^2(3) \left (1-\frac {\log ^2(3)}{8}\right ) x+32 \log ^4(3)\right )}{x^4 \left (x^3+\log ^2(3) x^2+1\right )^2 \left (e^{8/x} x^3+e^{8/x} \log ^2(3) x^2+e^{2+\frac {16}{x^2}} x+e^{8/x}+e^{2+\frac {16}{x^2}} \log ^2(3)\right )^2}\right )dx\) |
Int[(-5*x^2 - 20*x^5 - 15*x^4*Log[3]^2 + E^((16 - 8*x + 2*x^2)/x^2)*(160*x - 40*x^2 - 10*x^3 + (160 - 40*x - 5*x^2)*Log[3]^2))/(x^4 + 2*x^7 + x^10 + (2*x^6 + 2*x^9)*Log[3]^2 + x^8*Log[3]^4 + E^((2*(16 - 8*x + 2*x^2))/x^2)* (x^6 + 2*x^5*Log[3]^2 + x^4*Log[3]^4) + E^((16 - 8*x + 2*x^2)/x^2)*(2*x^5 + 2*x^8 + (2*x^4 + 4*x^7)*Log[3]^2 + 2*x^6*Log[3]^4)),x]
3.6.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81
| method | result | size |
| risch | \(\frac {5}{x \left (x^{2} \ln \left (3\right )^{2}+\ln \left (3\right )^{2} {\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+x^{3}+x \,{\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+1\right )}\) | \(56\) |
| parallelrisch | \(\frac {5}{x \left (x^{2} \ln \left (3\right )^{2}+\ln \left (3\right )^{2} {\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+x^{3}+x \,{\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+1\right )}\) | \(56\) |
| norman | \(\frac {5}{x \left (x^{2} \ln \left (3\right )^{2}+\ln \left (3\right )^{2} {\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+x^{3}+x \,{\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+1\right )}\) | \(58\) |
int((((-5*x^2-40*x+160)*ln(3)^2-10*x^3-40*x^2+160*x)*exp((2*x^2-8*x+16)/x^ 2)-15*x^4*ln(3)^2-20*x^5-5*x^2)/((x^4*ln(3)^4+2*x^5*ln(3)^2+x^6)*exp((2*x^ 2-8*x+16)/x^2)^2+(2*x^6*ln(3)^4+(4*x^7+2*x^4)*ln(3)^2+2*x^8+2*x^5)*exp((2* x^2-8*x+16)/x^2)+x^8*ln(3)^4+(2*x^9+2*x^6)*ln(3)^2+x^10+2*x^7+x^4),x,metho d=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5}{x^{3} \log \left (3\right )^{2} + x^{4} + {\left (x \log \left (3\right )^{2} + x^{2}\right )} e^{\left (\frac {2 \, {\left (x^{2} - 4 \, x + 8\right )}}{x^{2}}\right )} + x} \]
integrate((((-5*x^2-40*x+160)*log(3)^2-10*x^3-40*x^2+160*x)*exp((2*x^2-8*x +16)/x^2)-15*x^4*log(3)^2-20*x^5-5*x^2)/((x^4*log(3)^4+2*x^5*log(3)^2+x^6) *exp((2*x^2-8*x+16)/x^2)^2+(2*x^6*log(3)^4+(4*x^7+2*x^4)*log(3)^2+2*x^8+2* x^5)*exp((2*x^2-8*x+16)/x^2)+x^8*log(3)^4+(2*x^9+2*x^6)*log(3)^2+x^10+2*x^ 7+x^4),x, algorithm=\
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5}{x^{4} + x^{3} \log {\left (3 \right )}^{2} + x + \left (x^{2} + x \log {\left (3 \right )}^{2}\right ) e^{\frac {2 x^{2} - 8 x + 16}{x^{2}}}} \]
integrate((((-5*x**2-40*x+160)*ln(3)**2-10*x**3-40*x**2+160*x)*exp((2*x**2 -8*x+16)/x**2)-15*x**4*ln(3)**2-20*x**5-5*x**2)/((x**4*ln(3)**4+2*x**5*ln( 3)**2+x**6)*exp((2*x**2-8*x+16)/x**2)**2+(2*x**6*ln(3)**4+(4*x**7+2*x**4)* ln(3)**2+2*x**8+2*x**5)*exp((2*x**2-8*x+16)/x**2)+x**8*ln(3)**4+(2*x**9+2* x**6)*ln(3)**2+x**10+2*x**7+x**4),x)
Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5 \, e^{\frac {8}{x}}}{{\left (x^{3} \log \left (3\right )^{2} + x^{4} + x\right )} e^{\frac {8}{x}} + {\left (x e^{2} \log \left (3\right )^{2} + x^{2} e^{2}\right )} e^{\left (\frac {16}{x^{2}}\right )}} \]
integrate((((-5*x^2-40*x+160)*log(3)^2-10*x^3-40*x^2+160*x)*exp((2*x^2-8*x +16)/x^2)-15*x^4*log(3)^2-20*x^5-5*x^2)/((x^4*log(3)^4+2*x^5*log(3)^2+x^6) *exp((2*x^2-8*x+16)/x^2)^2+(2*x^6*log(3)^4+(4*x^7+2*x^4)*log(3)^2+2*x^8+2* x^5)*exp((2*x^2-8*x+16)/x^2)+x^8*log(3)^4+(2*x^9+2*x^6)*log(3)^2+x^10+2*x^ 7+x^4),x, algorithm=\
Time = 0.76 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5}{x^{3} \log \left (3\right )^{2} + x^{4} + x e^{\left (\frac {2 \, {\left (x^{2} - 4 \, x + 8\right )}}{x^{2}}\right )} \log \left (3\right )^{2} + x^{2} e^{\left (\frac {2 \, {\left (x^{2} - 4 \, x + 8\right )}}{x^{2}}\right )} + x} \]
integrate((((-5*x^2-40*x+160)*log(3)^2-10*x^3-40*x^2+160*x)*exp((2*x^2-8*x +16)/x^2)-15*x^4*log(3)^2-20*x^5-5*x^2)/((x^4*log(3)^4+2*x^5*log(3)^2+x^6) *exp((2*x^2-8*x+16)/x^2)^2+(2*x^6*log(3)^4+(4*x^7+2*x^4)*log(3)^2+2*x^8+2* x^5)*exp((2*x^2-8*x+16)/x^2)+x^8*log(3)^4+(2*x^9+2*x^6)*log(3)^2+x^10+2*x^ 7+x^4),x, algorithm=\
5/(x^3*log(3)^2 + x^4 + x*e^(2*(x^2 - 4*x + 8)/x^2)*log(3)^2 + x^2*e^(2*(x ^2 - 4*x + 8)/x^2) + x)
Time = 12.54 (sec) , antiderivative size = 383, normalized size of antiderivative = 12.35 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5\,{\left (x^5+2\,{\ln \left (3\right )}^2\,x^4+{\ln \left (3\right )}^4\,x^3\right )}^2\,\left (32\,x+64\,x^3\,{\ln \left (3\right )}^2+32\,x^2\,{\ln \left (3\right )}^4-16\,x^4\,{\ln \left (3\right )}^2-8\,x^3\,{\ln \left (3\right )}^4+4\,x^5\,{\ln \left (3\right )}^2+2\,x^4\,{\ln \left (3\right )}^4-8\,x\,{\ln \left (3\right )}^2+32\,{\ln \left (3\right )}^2-8\,x^2-x^3+32\,x^4-8\,x^5+2\,x^6\right )}{x^4\,\left ({\mathrm {e}}^{\frac {16}{x^2}-\frac {8}{x}+2}+\frac {x^3+{\ln \left (3\right )}^2\,x^2+1}{x+{\ln \left (3\right )}^2}\right )\,{\left (x+{\ln \left (3\right )}^2\right )}^3\,\left (96\,x^5\,{\ln \left (3\right )}^2+96\,x^4\,{\ln \left (3\right )}^4-24\,x^6\,{\ln \left (3\right )}^2+32\,x^3\,{\ln \left (3\right )}^6-24\,x^5\,{\ln \left (3\right )}^4-2\,x^7\,{\ln \left (3\right )}^2-8\,x^4\,{\ln \left (3\right )}^6-x^6\,{\ln \left (3\right )}^4+128\,x^8\,{\ln \left (3\right )}^2+192\,x^7\,{\ln \left (3\right )}^4-32\,x^9\,{\ln \left (3\right )}^2+128\,x^6\,{\ln \left (3\right )}^6-48\,x^8\,{\ln \left (3\right )}^4+8\,x^{10}\,{\ln \left (3\right )}^2+32\,x^5\,{\ln \left (3\right )}^8-32\,x^7\,{\ln \left (3\right )}^6+12\,x^9\,{\ln \left (3\right )}^4-8\,x^6\,{\ln \left (3\right )}^8+8\,x^8\,{\ln \left (3\right )}^6+2\,x^7\,{\ln \left (3\right )}^8+32\,x^6-8\,x^7-x^8+32\,x^9-8\,x^{10}+2\,x^{11}\right )} \]
int(-(15*x^4*log(3)^2 + exp((2*x^2 - 8*x + 16)/x^2)*(log(3)^2*(40*x + 5*x^ 2 - 160) - 160*x + 40*x^2 + 10*x^3) + 5*x^2 + 20*x^5)/(x^8*log(3)^4 + exp( (2*x^2 - 8*x + 16)/x^2)*(2*x^6*log(3)^4 + 2*x^5 + 2*x^8 + log(3)^2*(2*x^4 + 4*x^7)) + exp((2*(2*x^2 - 8*x + 16))/x^2)*(2*x^5*log(3)^2 + x^4*log(3)^4 + x^6) + x^4 + 2*x^7 + x^10 + log(3)^2*(2*x^6 + 2*x^9)),x)
(5*(2*x^4*log(3)^2 + x^3*log(3)^4 + x^5)^2*(32*x + 64*x^3*log(3)^2 + 32*x^ 2*log(3)^4 - 16*x^4*log(3)^2 - 8*x^3*log(3)^4 + 4*x^5*log(3)^2 + 2*x^4*log (3)^4 - 8*x*log(3)^2 + 32*log(3)^2 - 8*x^2 - x^3 + 32*x^4 - 8*x^5 + 2*x^6) )/(x^4*(exp(16/x^2 - 8/x + 2) + (x^2*log(3)^2 + x^3 + 1)/(x + log(3)^2))*( x + log(3)^2)^3*(96*x^5*log(3)^2 + 96*x^4*log(3)^4 - 24*x^6*log(3)^2 + 32* x^3*log(3)^6 - 24*x^5*log(3)^4 - 2*x^7*log(3)^2 - 8*x^4*log(3)^6 - x^6*log (3)^4 + 128*x^8*log(3)^2 + 192*x^7*log(3)^4 - 32*x^9*log(3)^2 + 128*x^6*lo g(3)^6 - 48*x^8*log(3)^4 + 8*x^10*log(3)^2 + 32*x^5*log(3)^8 - 32*x^7*log( 3)^6 + 12*x^9*log(3)^4 - 8*x^6*log(3)^8 + 8*x^8*log(3)^6 + 2*x^7*log(3)^8 + 32*x^6 - 8*x^7 - x^8 + 32*x^9 - 8*x^10 + 2*x^11))