Integrand size = 166, antiderivative size = 40 \[ \int \frac {e^{\frac {64 x^2+32 x^3+16 x^4+\left (32 x+16 x^2+8 x^3\right ) \log (x)+\left (4+2 x+x^2\right ) \log ^2(x)}{\left (32 x+16 x^2\right ) (i \pi +\log (5-\log (5)))^2}} \left (32 x+96 x^2+80 x^3+68 x^4+16 x^5+\left (8+8 x+4 x^2+17 x^3+4 x^4\right ) \log (x)+(-4-4 x) \log ^2(x)\right )}{\left (32 x^2+32 x^3+8 x^4\right ) (i \pi +\log (5-\log (5)))^2} \, dx=e^{\frac {\left (x+\frac {4}{2+x}\right ) \left (x+\frac {\log (x)}{4}\right )^2}{x (i \pi +\log (5-\log (5)))^2}} \] Output:
exp((4/(2+x)+x)/ln(ln(5)-5)^2*(x+1/4*ln(x))^2/x)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(40)=80\).
Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.05 \[ \int \frac {e^{\frac {64 x^2+32 x^3+16 x^4+\left (32 x+16 x^2+8 x^3\right ) \log (x)+\left (4+2 x+x^2\right ) \log ^2(x)}{\left (32 x+16 x^2\right ) (i \pi +\log (5-\log (5)))^2}} \left (32 x+96 x^2+80 x^3+68 x^4+16 x^5+\left (8+8 x+4 x^2+17 x^3+4 x^4\right ) \log (x)+(-4-4 x) \log ^2(x)\right )}{\left (32 x^2+32 x^3+8 x^4\right ) (i \pi +\log (5-\log (5)))^2} \, dx=e^{-\frac {\left (4+2 x+x^2\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (2+x) (\pi -i \log (5-\log (5)))^2}} x^{-\frac {4+2 x+x^2}{2 (2+x) (\pi -i \log (5-\log (5)))^2}} \] Input:
Integrate[(E^((64*x^2 + 32*x^3 + 16*x^4 + (32*x + 16*x^2 + 8*x^3)*Log[x] + (4 + 2*x + x^2)*Log[x]^2)/((32*x + 16*x^2)*(I*Pi + Log[5 - Log[5]])^2))*( 32*x + 96*x^2 + 80*x^3 + 68*x^4 + 16*x^5 + (8 + 8*x + 4*x^2 + 17*x^3 + 4*x ^4)*Log[x] + (-4 - 4*x)*Log[x]^2))/((32*x^2 + 32*x^3 + 8*x^4)*(I*Pi + Log[ 5 - Log[5]])^2),x]
Output:
1/(E^(((4 + 2*x + x^2)*(16*x^2 + Log[x]^2))/(16*x*(2 + x)*(Pi - I*Log[5 - Log[5]])^2))*x^((4 + 2*x + x^2)/(2*(2 + x)*(Pi - I*Log[5 - Log[5]])^2)))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (16 x^5+68 x^4+80 x^3+96 x^2+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+32 x+(-4 x-4) \log ^2(x)\right ) \exp \left (\frac {16 x^4+32 x^3+64 x^2+\left (x^2+2 x+4\right ) \log ^2(x)+\left (8 x^3+16 x^2+32 x\right ) \log (x)}{\left (16 x^2+32 x\right ) (\log (5-\log (5))+i \pi )^2}\right )}{\left (8 x^4+32 x^3+32 x^2\right ) (\log (5-\log (5))+i \pi )^2} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {16 x^4+32 x^3+64 x^2+\left (x^2+2 x+4\right ) \log ^2(x)}{16 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {8 x^3+16 x^2+32 x}{\left (16 x^2+32 x\right ) (i \pi +\log (5-\log (5)))^2}} \left (16 x^5+68 x^4+80 x^3+96 x^2+32 x-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)\right )}{8 \left (x^4+4 x^3+4 x^2\right )}dx}{(\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {16 x^4+32 x^3+64 x^2+\left (x^2+2 x+4\right ) \log ^2(x)}{16 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}} \left (16 x^5+68 x^4+80 x^3+96 x^2+32 x-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)\right )}{x^4+4 x^3+4 x^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {16 x^4+32 x^3+64 x^2+\left (x^2+2 x+4\right ) \log ^2(x)}{16 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}-2} \left (16 x^5+68 x^4+80 x^3+96 x^2+32 x-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)\right )}{x^2+4 x+4}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {16 x^4+32 x^3+64 x^2+\left (x^2+2 x+4\right ) \log ^2(x)}{16 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}-2} \left (16 x^5+68 x^4+80 x^3+96 x^2+32 x-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}-2} \left (16 x^5+68 x^4+80 x^3+96 x^2+32 x-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {96 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}}}{(x+2)^2}-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}-2}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}-2}}{(x+2)^2}+\frac {32 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}-1}}{(x+2)^2}+\frac {80 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}+1}}{(x+2)^2}+\frac {68 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}+2}}{(x+2)^2}+\frac {16 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{\frac {x^3+2 x^2+4 x}{2 \left (x^2+2 x\right ) (i \pi +\log (5-\log (5)))^2}+3}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {\int \frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}} \left (-4 (x+1) \log ^2(x)+\left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x)+4 x \left (4 x^4+17 x^3+20 x^2+24 x+8\right )\right )}{(x+2)^2}dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) (x+1) \log ^2(x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {\exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+4 x^2+8 x+8\right ) \log (x) x^{-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}+\frac {4 \exp \left (\frac {\left (x^2+2 x+4\right ) \left (16 x^2+\log ^2(x)\right )}{16 x (x+2) (i \pi +\log (5-\log (5)))^2}\right ) \left (4 x^4+17 x^3+20 x^2+24 x+8\right ) x^{1-\frac {x^2+2 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right ) x+4 \left (1+2 \pi ^2-4 i \pi \log (5-\log (5))-2 \log ^2(5-\log (5))\right )}{2 (x+2) (\pi -i \log (5-\log (5)))^2}}}{(x+2)^2}\right )dx}{8 (\log (5-\log (5))+i \pi )^2}\) |
Input:
Int[(E^((64*x^2 + 32*x^3 + 16*x^4 + (32*x + 16*x^2 + 8*x^3)*Log[x] + (4 + 2*x + x^2)*Log[x]^2)/((32*x + 16*x^2)*(I*Pi + Log[5 - Log[5]])^2))*(32*x + 96*x^2 + 80*x^3 + 68*x^4 + 16*x^5 + (8 + 8*x + 4*x^2 + 17*x^3 + 4*x^4)*Lo g[x] + (-4 - 4*x)*Log[x]^2))/((32*x^2 + 32*x^3 + 8*x^4)*(I*Pi + Log[5 - Lo g[5]])^2),x]
Output:
$Aborted
Time = 4.51 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \({\mathrm e}^{\frac {\left (x^{2}+2 x +4\right ) \left (4 x +\ln \left (x \right )\right )^{2}}{16 x \left (2+x \right ) \ln \left (\ln \left (5\right )-5\right )^{2}}}\) | \(35\) |
| parallelrisch | \({\mathrm e}^{\frac {\left (x^{2}+2 x +4\right ) \ln \left (x \right )^{2}+\left (8 x^{3}+16 x^{2}+32 x \right ) \ln \left (x \right )+16 x^{4}+32 x^{3}+64 x^{2}}{16 x \left (2+x \right ) \ln \left (\ln \left (5\right )-5\right )^{2}}}\) | \(65\) |
Input:
int(((-4-4*x)*ln(x)^2+(4*x^4+17*x^3+4*x^2+8*x+8)*ln(x)+16*x^5+68*x^4+80*x^ 3+96*x^2+32*x)*exp(((x^2+2*x+4)*ln(x)^2+(8*x^3+16*x^2+32*x)*ln(x)+16*x^4+3 2*x^3+64*x^2)/(16*x^2+32*x)/ln(ln(5)-5)^2)/(8*x^4+32*x^3+32*x^2)/ln(ln(5)- 5)^2,x,method=_RETURNVERBOSE)
Output:
exp(1/16*(x^2+2*x+4)*(4*x+ln(x))^2/x/(2+x)/ln(ln(5)-5)^2)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (30) = 60\).
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.60 \[ \int \frac {e^{\frac {64 x^2+32 x^3+16 x^4+\left (32 x+16 x^2+8 x^3\right ) \log (x)+\left (4+2 x+x^2\right ) \log ^2(x)}{\left (32 x+16 x^2\right ) (i \pi +\log (5-\log (5)))^2}} \left (32 x+96 x^2+80 x^3+68 x^4+16 x^5+\left (8+8 x+4 x^2+17 x^3+4 x^4\right ) \log (x)+(-4-4 x) \log ^2(x)\right )}{\left (32 x^2+32 x^3+8 x^4\right ) (i \pi +\log (5-\log (5)))^2} \, dx=e^{\left (\frac {16 \, x^{4} + 32 \, x^{3} + {\left (x^{2} + 2 \, x + 4\right )} \log \left (x\right )^{2} + 64 \, x^{2} + 8 \, {\left (x^{3} + 2 \, x^{2} + 4 \, x\right )} \log \left (x\right )}{16 \, {\left (x^{2} + 2 \, x\right )} \log \left (\log \left (5\right ) - 5\right )^{2}}\right )} \] Input:
integrate(((-4-4*x)*log(x)^2+(4*x^4+17*x^3+4*x^2+8*x+8)*log(x)+16*x^5+68*x ^4+80*x^3+96*x^2+32*x)*exp(((x^2+2*x+4)*log(x)^2+(8*x^3+16*x^2+32*x)*log(x )+16*x^4+32*x^3+64*x^2)/(16*x^2+32*x)/log(log(5)-5)^2)/(8*x^4+32*x^3+32*x^ 2)/log(log(5)-5)^2,x, algorithm="fricas")
Output:
e^(1/16*(16*x^4 + 32*x^3 + (x^2 + 2*x + 4)*log(x)^2 + 64*x^2 + 8*(x^3 + 2* x^2 + 4*x)*log(x))/((x^2 + 2*x)*log(log(5) - 5)^2))
Timed out. \[ \int \frac {e^{\frac {64 x^2+32 x^3+16 x^4+\left (32 x+16 x^2+8 x^3\right ) \log (x)+\left (4+2 x+x^2\right ) \log ^2(x)}{\left (32 x+16 x^2\right ) (i \pi +\log (5-\log (5)))^2}} \left (32 x+96 x^2+80 x^3+68 x^4+16 x^5+\left (8+8 x+4 x^2+17 x^3+4 x^4\right ) \log (x)+(-4-4 x) \log ^2(x)\right )}{\left (32 x^2+32 x^3+8 x^4\right ) (i \pi +\log (5-\log (5)))^2} \, dx=\text {Timed out} \] Input:
integrate(((-4-4*x)*ln(x)**2+(4*x**4+17*x**3+4*x**2+8*x+8)*ln(x)+16*x**5+6 8*x**4+80*x**3+96*x**2+32*x)*exp(((x**2+2*x+4)*ln(x)**2+(8*x**3+16*x**2+32 *x)*ln(x)+16*x**4+32*x**3+64*x**2)/(16*x**2+32*x)/ln(ln(5)-5)**2)/(8*x**4+ 32*x**3+32*x**2)/ln(ln(5)-5)**2,x)
Output:
Timed out
\[ \int \frac {e^{\frac {64 x^2+32 x^3+16 x^4+\left (32 x+16 x^2+8 x^3\right ) \log (x)+\left (4+2 x+x^2\right ) \log ^2(x)}{\left (32 x+16 x^2\right ) (i \pi +\log (5-\log (5)))^2}} \left (32 x+96 x^2+80 x^3+68 x^4+16 x^5+\left (8+8 x+4 x^2+17 x^3+4 x^4\right ) \log (x)+(-4-4 x) \log ^2(x)\right )}{\left (32 x^2+32 x^3+8 x^4\right ) (i \pi +\log (5-\log (5)))^2} \, dx=\int { \frac {{\left (16 \, x^{5} + 68 \, x^{4} + 80 \, x^{3} - 4 \, {\left (x + 1\right )} \log \left (x\right )^{2} + 96 \, x^{2} + {\left (4 \, x^{4} + 17 \, x^{3} + 4 \, x^{2} + 8 \, x + 8\right )} \log \left (x\right ) + 32 \, x\right )} e^{\left (\frac {16 \, x^{4} + 32 \, x^{3} + {\left (x^{2} + 2 \, x + 4\right )} \log \left (x\right )^{2} + 64 \, x^{2} + 8 \, {\left (x^{3} + 2 \, x^{2} + 4 \, x\right )} \log \left (x\right )}{16 \, {\left (x^{2} + 2 \, x\right )} \log \left (\log \left (5\right ) - 5\right )^{2}}\right )}}{8 \, {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )} \log \left (\log \left (5\right ) - 5\right )^{2}} \,d x } \] Input:
integrate(((-4-4*x)*log(x)^2+(4*x^4+17*x^3+4*x^2+8*x+8)*log(x)+16*x^5+68*x ^4+80*x^3+96*x^2+32*x)*exp(((x^2+2*x+4)*log(x)^2+(8*x^3+16*x^2+32*x)*log(x )+16*x^4+32*x^3+64*x^2)/(16*x^2+32*x)/log(log(5)-5)^2)/(8*x^4+32*x^3+32*x^ 2)/log(log(5)-5)^2,x, algorithm="maxima")
Output:
1/8*integrate((16*x^5 + 68*x^4 + 80*x^3 - 4*(x + 1)*log(x)^2 + 96*x^2 + (4 *x^4 + 17*x^3 + 4*x^2 + 8*x + 8)*log(x) + 32*x)*e^(1/16*(16*x^4 + 32*x^3 + (x^2 + 2*x + 4)*log(x)^2 + 64*x^2 + 8*(x^3 + 2*x^2 + 4*x)*log(x))/((x^2 + 2*x)*log(log(5) - 5)^2))/(x^4 + 4*x^3 + 4*x^2), x)/log(log(5) - 5)^2
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (30) = 60\).
Time = 0.35 (sec) , antiderivative size = 272, normalized size of antiderivative = 6.80 \[ \int \frac {e^{\frac {64 x^2+32 x^3+16 x^4+\left (32 x+16 x^2+8 x^3\right ) \log (x)+\left (4+2 x+x^2\right ) \log ^2(x)}{\left (32 x+16 x^2\right ) (i \pi +\log (5-\log (5)))^2}} \left (32 x+96 x^2+80 x^3+68 x^4+16 x^5+\left (8+8 x+4 x^2+17 x^3+4 x^4\right ) \log (x)+(-4-4 x) \log ^2(x)\right )}{\left (32 x^2+32 x^3+8 x^4\right ) (i \pi +\log (5-\log (5)))^2} \, dx=e^{\left (\frac {x^{4}}{x^{2} \log \left (\log \left (5\right ) - 5\right )^{2} + 2 \, x \log \left (\log \left (5\right ) - 5\right )^{2}} + \frac {x^{3} \log \left (x\right )}{2 \, {\left (x^{2} \log \left (\log \left (5\right ) - 5\right )^{2} + 2 \, x \log \left (\log \left (5\right ) - 5\right )^{2}\right )}} + \frac {x^{2} \log \left (x\right )^{2}}{16 \, {\left (x^{2} \log \left (\log \left (5\right ) - 5\right )^{2} + 2 \, x \log \left (\log \left (5\right ) - 5\right )^{2}\right )}} + \frac {2 \, x^{3}}{x^{2} \log \left (\log \left (5\right ) - 5\right )^{2} + 2 \, x \log \left (\log \left (5\right ) - 5\right )^{2}} + \frac {x^{2} \log \left (x\right )}{x^{2} \log \left (\log \left (5\right ) - 5\right )^{2} + 2 \, x \log \left (\log \left (5\right ) - 5\right )^{2}} + \frac {x \log \left (x\right )^{2}}{8 \, {\left (x^{2} \log \left (\log \left (5\right ) - 5\right )^{2} + 2 \, x \log \left (\log \left (5\right ) - 5\right )^{2}\right )}} + \frac {4 \, x^{2}}{x^{2} \log \left (\log \left (5\right ) - 5\right )^{2} + 2 \, x \log \left (\log \left (5\right ) - 5\right )^{2}} + \frac {2 \, x \log \left (x\right )}{x^{2} \log \left (\log \left (5\right ) - 5\right )^{2} + 2 \, x \log \left (\log \left (5\right ) - 5\right )^{2}} + \frac {\log \left (x\right )^{2}}{4 \, {\left (x^{2} \log \left (\log \left (5\right ) - 5\right )^{2} + 2 \, x \log \left (\log \left (5\right ) - 5\right )^{2}\right )}}\right )} \] Input:
integrate(((-4-4*x)*log(x)^2+(4*x^4+17*x^3+4*x^2+8*x+8)*log(x)+16*x^5+68*x ^4+80*x^3+96*x^2+32*x)*exp(((x^2+2*x+4)*log(x)^2+(8*x^3+16*x^2+32*x)*log(x )+16*x^4+32*x^3+64*x^2)/(16*x^2+32*x)/log(log(5)-5)^2)/(8*x^4+32*x^3+32*x^ 2)/log(log(5)-5)^2,x, algorithm="giac")
Output:
e^(x^4/(x^2*log(log(5) - 5)^2 + 2*x*log(log(5) - 5)^2) + 1/2*x^3*log(x)/(x ^2*log(log(5) - 5)^2 + 2*x*log(log(5) - 5)^2) + 1/16*x^2*log(x)^2/(x^2*log (log(5) - 5)^2 + 2*x*log(log(5) - 5)^2) + 2*x^3/(x^2*log(log(5) - 5)^2 + 2 *x*log(log(5) - 5)^2) + x^2*log(x)/(x^2*log(log(5) - 5)^2 + 2*x*log(log(5) - 5)^2) + 1/8*x*log(x)^2/(x^2*log(log(5) - 5)^2 + 2*x*log(log(5) - 5)^2) + 4*x^2/(x^2*log(log(5) - 5)^2 + 2*x*log(log(5) - 5)^2) + 2*x*log(x)/(x^2* log(log(5) - 5)^2 + 2*x*log(log(5) - 5)^2) + 1/4*log(x)^2/(x^2*log(log(5) - 5)^2 + 2*x*log(log(5) - 5)^2))
Time = 3.23 (sec) , antiderivative size = 254, normalized size of antiderivative = 6.35 \[ \int \frac {e^{\frac {64 x^2+32 x^3+16 x^4+\left (32 x+16 x^2+8 x^3\right ) \log (x)+\left (4+2 x+x^2\right ) \log ^2(x)}{\left (32 x+16 x^2\right ) (i \pi +\log (5-\log (5)))^2}} \left (32 x+96 x^2+80 x^3+68 x^4+16 x^5+\left (8+8 x+4 x^2+17 x^3+4 x^4\right ) \log (x)+(-4-4 x) \log ^2(x)\right )}{\left (32 x^2+32 x^3+8 x^4\right ) (i \pi +\log (5-\log (5)))^2} \, dx=x^{\frac {16\,x^2}{16\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x^2+32\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x}+\frac {x^2+4}{2\,\left (x\,{\ln \left (\ln \left (5\right )-5\right )}^2+2\,{\ln \left (\ln \left (5\right )-5\right )}^2\right )}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (x\right )}^2}{16\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x^2+32\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x}}\,{\mathrm {e}}^{\frac {2\,x\,{\ln \left (x\right )}^2}{16\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x^2+32\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x}}\,{\mathrm {e}}^{\frac {16\,x^4}{16\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x^2+32\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x}}\,{\mathrm {e}}^{\frac {32\,x^3}{16\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x^2+32\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x}}\,{\mathrm {e}}^{\frac {64\,x^2}{16\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x^2+32\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x}}\,{\mathrm {e}}^{\frac {x^2\,{\ln \left (x\right )}^2}{16\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x^2+32\,{\ln \left (\ln \left (5\right )-5\right )}^2\,x}} \] Input:
int((exp((log(x)^2*(2*x + x^2 + 4) + 64*x^2 + 32*x^3 + 16*x^4 + log(x)*(32 *x + 16*x^2 + 8*x^3))/(log(log(5) - 5)^2*(32*x + 16*x^2)))*(32*x + log(x)* (8*x + 4*x^2 + 17*x^3 + 4*x^4 + 8) + 96*x^2 + 80*x^3 + 68*x^4 + 16*x^5 - l og(x)^2*(4*x + 4)))/(log(log(5) - 5)^2*(32*x^2 + 32*x^3 + 8*x^4)),x)
Output:
x^((16*x^2)/(32*x*log(log(5) - 5)^2 + 16*x^2*log(log(5) - 5)^2) + (x^2 + 4 )/(2*(x*log(log(5) - 5)^2 + 2*log(log(5) - 5)^2)))*exp((4*log(x)^2)/(32*x* log(log(5) - 5)^2 + 16*x^2*log(log(5) - 5)^2))*exp((2*x*log(x)^2)/(32*x*lo g(log(5) - 5)^2 + 16*x^2*log(log(5) - 5)^2))*exp((16*x^4)/(32*x*log(log(5) - 5)^2 + 16*x^2*log(log(5) - 5)^2))*exp((32*x^3)/(32*x*log(log(5) - 5)^2 + 16*x^2*log(log(5) - 5)^2))*exp((64*x^2)/(32*x*log(log(5) - 5)^2 + 16*x^2 *log(log(5) - 5)^2))*exp((x^2*log(x)^2)/(32*x*log(log(5) - 5)^2 + 16*x^2*l og(log(5) - 5)^2))
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.10 \[ \int \frac {e^{\frac {64 x^2+32 x^3+16 x^4+\left (32 x+16 x^2+8 x^3\right ) \log (x)+\left (4+2 x+x^2\right ) \log ^2(x)}{\left (32 x+16 x^2\right ) (i \pi +\log (5-\log (5)))^2}} \left (32 x+96 x^2+80 x^3+68 x^4+16 x^5+\left (8+8 x+4 x^2+17 x^3+4 x^4\right ) \log (x)+(-4-4 x) \log ^2(x)\right )}{\left (32 x^2+32 x^3+8 x^4\right ) (i \pi +\log (5-\log (5)))^2} \, dx=e^{\frac {\mathrm {log}\left (x \right )^{2} x^{2}+2 \mathrm {log}\left (x \right )^{2} x +4 \mathrm {log}\left (x \right )^{2}+8 \,\mathrm {log}\left (x \right ) x^{3}+16 \,\mathrm {log}\left (x \right ) x^{2}+32 \,\mathrm {log}\left (x \right ) x +16 x^{4}+32 x^{3}+64 x^{2}}{16 \mathrm {log}\left (\mathrm {log}\left (5\right )-5\right )^{2} x^{2}+32 \mathrm {log}\left (\mathrm {log}\left (5\right )-5\right )^{2} x}} \] Input:
int(((-4-4*x)*log(x)^2+(4*x^4+17*x^3+4*x^2+8*x+8)*log(x)+16*x^5+68*x^4+80* x^3+96*x^2+32*x)*exp(((x^2+2*x+4)*log(x)^2+(8*x^3+16*x^2+32*x)*log(x)+16*x ^4+32*x^3+64*x^2)/(16*x^2+32*x)/log(log(5)-5)^2)/(8*x^4+32*x^3+32*x^2)/log (log(5)-5)^2,x)
Output:
e**((log(x)**2*x**2 + 2*log(x)**2*x + 4*log(x)**2 + 8*log(x)*x**3 + 16*log (x)*x**2 + 32*log(x)*x + 16*x**4 + 32*x**3 + 64*x**2)/(16*log(log(5) - 5)* *2*x**2 + 32*log(log(5) - 5)**2*x))