Integrand size = 219, antiderivative size = 23 \[ \int \frac {e^{\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{4 x+4 x^2} \, dx=e^{\frac {1}{16} \left (-1+2 x+\log \left (\frac {x^2}{1+x}\right )\right )^4} \]
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(23)=46\).
Time = 0.39 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.78 \[ \int \frac {e^{\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{4 x+4 x^2} \, dx=e^{\frac {1}{16} \left ((1-2 x)^4+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (\frac {x^2}{1+x}\right )^{\frac {1}{4} (-1+2 x)^3} \]
Integrate[(E^((1 - 8*x + 24*x^2 - 32*x^3 + 16*x^4 + (-4 + 24*x - 48*x^2 + 32*x^3)*Log[x^2/(1 + x)] + (6 - 24*x + 24*x^2)*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(-2 + 9*x - 8*x^2 - 8*x^ 3 + 16*x^5 + (6 - 15*x - 6*x^2 + 12*x^3 + 24*x^4)*Log[x^2/(1 + x)] + (-6 + 3*x + 12*x^2 + 12*x^3)*Log[x^2/(1 + x)]^2 + (2 + 3*x + 2*x^2)*Log[x^2/(1 + x)]^3))/(4*x + 4*x^2),x]
E^(((1 - 2*x)^4 + 6*(1 - 2*x)^2*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*Log[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(x^2/(1 + x))^((-1 + 2*x)^3/4)
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.
| \(\displaystyle \int \frac {\left (16 x^5-8 x^3-8 x^2+\left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right )+\left (12 x^3+12 x^2+3 x-6\right ) \log ^2\left (\frac {x^2}{x+1}\right )+\left (24 x^4+12 x^3-6 x^2-15 x+6\right ) \log \left (\frac {x^2}{x+1}\right )+9 x-2\right ) \exp \left (\frac {1}{16} \left (16 x^4-32 x^3+24 x^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )+\left (24 x^2-24 x+6\right ) \log ^2\left (\frac {x^2}{x+1}\right )+\left (32 x^3-48 x^2+24 x-4\right ) \log \left (\frac {x^2}{x+1}\right )-8 x+1\right )\right )}{4 x^2+4 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (16 x^5-8 x^3-8 x^2+\left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right )+\left (12 x^3+12 x^2+3 x-6\right ) \log ^2\left (\frac {x^2}{x+1}\right )+\left (24 x^4+12 x^3-6 x^2-15 x+6\right ) \log \left (\frac {x^2}{x+1}\right )+9 x-2\right ) \exp \left (\frac {1}{16} \left (16 x^4-32 x^3+24 x^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )+\left (24 x^2-24 x+6\right ) \log ^2\left (\frac {x^2}{x+1}\right )+\left (32 x^3-48 x^2+24 x-4\right ) \log \left (\frac {x^2}{x+1}\right )-8 x+1\right )\right )}{x (4 x+4)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (-2 x^2-3 x-2\right ) \left (-\log \left (\frac {x^2}{x+1}\right )-2 x+1\right )^3 \exp \left (\frac {1}{16} \left (\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )+6 (1-2 x)^2 \log ^2\left (\frac {x^2}{x+1}\right )+(1-2 x)^4\right )\right )}{4 x^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int -\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2-4 \log ^3\left (\frac {x^2}{x+1}\right ) (1-2 x)+\log ^4\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2-4 \log ^3\left (\frac {x^2}{x+1}\right ) (1-2 x)+\log ^4\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2-4 \log ^3\left (\frac {x^2}{x+1}\right ) (1-2 x)+\log ^4\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2-4 \log ^3\left (\frac {x^2}{x+1}\right ) (1-2 x)+\log ^4\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2-4 \log ^3\left (\frac {x^2}{x+1}\right ) (1-2 x)+\log ^4\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2-4 \log ^3\left (\frac {x^2}{x+1}\right ) (1-2 x)+\log ^4\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (2 x^2+3 x+2\right ) \log ^3\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1) \left (2 x^2+3 x+2\right ) \log ^2\left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^3 \left (2 x^2+3 x+2\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}-\frac {3 \exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) (2 x-1)^2 \left (2 x^2+3 x+2\right ) \log \left (\frac {x^2}{x+1}\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{4} \int \frac {\exp \left (\frac {1}{16} \left ((1-2 x)^4+6 \log ^2\left (\frac {x^2}{x+1}\right ) (1-2 x)^2+\log ^4\left (\frac {x^2}{x+1}\right )+(8 x-4) \log ^3\left (\frac {x^2}{x+1}\right )\right )\right ) \left (\frac {x^2}{x+1}\right )^{2 x^3-3 x^2+\frac {3 x}{2}+\frac {3}{4}} \left (2 x^2+3 x+2\right ) \left (-2 x-\log \left (\frac {x^2}{x+1}\right )+1\right )^3}{x^3}dx\) |
Int[(E^((1 - 8*x + 24*x^2 - 32*x^3 + 16*x^4 + (-4 + 24*x - 48*x^2 + 32*x^3 )*Log[x^2/(1 + x)] + (6 - 24*x + 24*x^2)*Log[x^2/(1 + x)]^2 + (-4 + 8*x)*L og[x^2/(1 + x)]^3 + Log[x^2/(1 + x)]^4)/16)*(-2 + 9*x - 8*x^2 - 8*x^3 + 16 *x^5 + (6 - 15*x - 6*x^2 + 12*x^3 + 24*x^4)*Log[x^2/(1 + x)] + (-6 + 3*x + 12*x^2 + 12*x^3)*Log[x^2/(1 + x)]^2 + (2 + 3*x + 2*x^2)*Log[x^2/(1 + x)]^ 3))/(4*x + 4*x^2),x]
3.19.7.3.1 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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(37)=74\).
Time = 0.52 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\ln \left (\frac {x^{2}}{1+x}\right )^{4}}{16}+\frac {\left (8 x -4\right ) \ln \left (\frac {x^{2}}{1+x}\right )^{3}}{16}+\frac {\left (24 x^{2}-24 x +6\right ) \ln \left (\frac {x^{2}}{1+x}\right )^{2}}{16}+\frac {\left (32 x^{3}-48 x^{2}+24 x -4\right ) \ln \left (\frac {x^{2}}{1+x}\right )}{16}+x^{4}-2 x^{3}+\frac {3 x^{2}}{2}-\frac {x}{2}+\frac {1}{16}}\) | \(104\) |
| risch | \(\left (\frac {x^{2}}{1+x}\right )^{\frac {\left (-1+2 x \right )^{3}}{4}} {\mathrm e}^{\frac {\ln \left (\frac {x^{2}}{1+x}\right )^{4}}{16}+\frac {\ln \left (\frac {x^{2}}{1+x}\right )^{3} x}{2}-\frac {\ln \left (\frac {x^{2}}{1+x}\right )^{3}}{4}+\frac {3 \ln \left (\frac {x^{2}}{1+x}\right )^{2} x^{2}}{2}-\frac {3 \ln \left (\frac {x^{2}}{1+x}\right )^{2} x}{2}+\frac {3 \ln \left (\frac {x^{2}}{1+x}\right )^{2}}{8}+\frac {1}{16}+x^{4}-2 x^{3}+\frac {3 x^{2}}{2}-\frac {x}{2}}\) | \(129\) |
int(((2*x^2+3*x+2)*ln(x^2/(1+x))^3+(12*x^3+12*x^2+3*x-6)*ln(x^2/(1+x))^2+( 24*x^4+12*x^3-6*x^2-15*x+6)*ln(x^2/(1+x))+16*x^5-8*x^3-8*x^2+9*x-2)*exp(1/ 16*ln(x^2/(1+x))^4+1/16*(8*x-4)*ln(x^2/(1+x))^3+1/16*(24*x^2-24*x+6)*ln(x^ 2/(1+x))^2+1/16*(32*x^3-48*x^2+24*x-4)*ln(x^2/(1+x))+x^4-2*x^3+3/2*x^2-1/2 *x+1/16)/(4*x^2+4*x),x,method=_RETURNVERBOSE)
exp(1/16*ln(x^2/(1+x))^4+1/16*(8*x-4)*ln(x^2/(1+x))^3+1/16*(24*x^2-24*x+6) *ln(x^2/(1+x))^2+1/16*(32*x^3-48*x^2+24*x-4)*ln(x^2/(1+x))+x^4-2*x^3+3/2*x ^2-1/2*x+1/16)
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 4.48 \[ \int \frac {e^{\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{4 x+4 x^2} \, dx=e^{\left (x^{4} + \frac {1}{4} \, {\left (2 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {1}{16} \, \log \left (\frac {x^{2}}{x + 1}\right )^{4} - 2 \, x^{3} + \frac {3}{8} \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{2} + \frac {3}{2} \, x^{2} + \frac {1}{4} \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right ) - \frac {1}{2} \, x + \frac {1}{16}\right )} \]
integrate(((2*x^2+3*x+2)*log(x^2/(1+x))^3+(12*x^3+12*x^2+3*x-6)*log(x^2/(1 +x))^2+(24*x^4+12*x^3-6*x^2-15*x+6)*log(x^2/(1+x))+16*x^5-8*x^3-8*x^2+9*x- 2)*exp(1/16*log(x^2/(1+x))^4+1/16*(8*x-4)*log(x^2/(1+x))^3+1/16*(24*x^2-24 *x+6)*log(x^2/(1+x))^2+1/16*(32*x^3-48*x^2+24*x-4)*log(x^2/(1+x))+x^4-2*x^ 3+3/2*x^2-1/2*x+1/16)/(4*x^2+4*x),x, algorithm=\
e^(x^4 + 1/4*(2*x - 1)*log(x^2/(x + 1))^3 + 1/16*log(x^2/(x + 1))^4 - 2*x^ 3 + 3/8*(4*x^2 - 4*x + 1)*log(x^2/(x + 1))^2 + 3/2*x^2 + 1/4*(8*x^3 - 12*x ^2 + 6*x - 1)*log(x^2/(x + 1)) - 1/2*x + 1/16)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (34) = 68\).
Time = 0.46 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int \frac {e^{\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{4 x+4 x^2} \, dx=e^{x^{4} - 2 x^{3} + \frac {3 x^{2}}{2} - \frac {x}{2} + \left (\frac {x}{2} - \frac {1}{4}\right ) \log {\left (\frac {x^{2}}{x + 1} \right )}^{3} + \left (\frac {3 x^{2}}{2} - \frac {3 x}{2} + \frac {3}{8}\right ) \log {\left (\frac {x^{2}}{x + 1} \right )}^{2} + \left (2 x^{3} - 3 x^{2} + \frac {3 x}{2} - \frac {1}{4}\right ) \log {\left (\frac {x^{2}}{x + 1} \right )} + \frac {\log {\left (\frac {x^{2}}{x + 1} \right )}^{4}}{16} + \frac {1}{16}} \]
integrate(((2*x**2+3*x+2)*ln(x**2/(1+x))**3+(12*x**3+12*x**2+3*x-6)*ln(x** 2/(1+x))**2+(24*x**4+12*x**3-6*x**2-15*x+6)*ln(x**2/(1+x))+16*x**5-8*x**3- 8*x**2+9*x-2)*exp(1/16*ln(x**2/(1+x))**4+1/16*(8*x-4)*ln(x**2/(1+x))**3+1/ 16*(24*x**2-24*x+6)*ln(x**2/(1+x))**2+1/16*(32*x**3-48*x**2+24*x-4)*ln(x** 2/(1+x))+x**4-2*x**3+3/2*x**2-1/2*x+1/16)/(4*x**2+4*x),x)
exp(x**4 - 2*x**3 + 3*x**2/2 - x/2 + (x/2 - 1/4)*log(x**2/(x + 1))**3 + (3 *x**2/2 - 3*x/2 + 3/8)*log(x**2/(x + 1))**2 + (2*x**3 - 3*x**2 + 3*x/2 - 1 /4)*log(x**2/(x + 1)) + log(x**2/(x + 1))**4/16 + 1/16)
\[ \int \frac {e^{\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{4 x+4 x^2} \, dx=\int { \frac {{\left (16 \, x^{5} + {\left (2 \, x^{2} + 3 \, x + 2\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{3} - 8 \, x^{3} + 3 \, {\left (4 \, x^{3} + 4 \, x^{2} + x - 2\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{2} - 8 \, x^{2} + 3 \, {\left (8 \, x^{4} + 4 \, x^{3} - 2 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {x^{2}}{x + 1}\right ) + 9 \, x - 2\right )} e^{\left (x^{4} + \frac {1}{4} \, {\left (2 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {1}{16} \, \log \left (\frac {x^{2}}{x + 1}\right )^{4} - 2 \, x^{3} + \frac {3}{8} \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {x^{2}}{x + 1}\right )^{2} + \frac {3}{2} \, x^{2} + \frac {1}{4} \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )} \log \left (\frac {x^{2}}{x + 1}\right ) - \frac {1}{2} \, x + \frac {1}{16}\right )}}{4 \, {\left (x^{2} + x\right )}} \,d x } \]
integrate(((2*x^2+3*x+2)*log(x^2/(1+x))^3+(12*x^3+12*x^2+3*x-6)*log(x^2/(1 +x))^2+(24*x^4+12*x^3-6*x^2-15*x+6)*log(x^2/(1+x))+16*x^5-8*x^3-8*x^2+9*x- 2)*exp(1/16*log(x^2/(1+x))^4+1/16*(8*x-4)*log(x^2/(1+x))^3+1/16*(24*x^2-24 *x+6)*log(x^2/(1+x))^2+1/16*(32*x^3-48*x^2+24*x-4)*log(x^2/(1+x))+x^4-2*x^ 3+3/2*x^2-1/2*x+1/16)/(4*x^2+4*x),x, algorithm=\
1/4*integrate((16*x^5 + (2*x^2 + 3*x + 2)*log(x^2/(x + 1))^3 - 8*x^3 + 3*( 4*x^3 + 4*x^2 + x - 2)*log(x^2/(x + 1))^2 - 8*x^2 + 3*(8*x^4 + 4*x^3 - 2*x ^2 - 5*x + 2)*log(x^2/(x + 1)) + 9*x - 2)*e^(x^4 + 1/4*(2*x - 1)*log(x^2/( x + 1))^3 + 1/16*log(x^2/(x + 1))^4 - 2*x^3 + 3/8*(4*x^2 - 4*x + 1)*log(x^ 2/(x + 1))^2 + 3/2*x^2 + 1/4*(8*x^3 - 12*x^2 + 6*x - 1)*log(x^2/(x + 1)) - 1/2*x + 1/16)/(x^2 + x), x)
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (20) = 40\).
Time = 2.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 7.09 \[ \int \frac {e^{\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{4 x+4 x^2} \, dx=e^{\left (x^{4} + 2 \, x^{3} \log \left (\frac {x^{2}}{x + 1}\right ) + \frac {3}{2} \, x^{2} \log \left (\frac {x^{2}}{x + 1}\right )^{2} + \frac {1}{2} \, x \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {1}{16} \, \log \left (\frac {x^{2}}{x + 1}\right )^{4} - 2 \, x^{3} - 3 \, x^{2} \log \left (\frac {x^{2}}{x + 1}\right ) - \frac {3}{2} \, x \log \left (\frac {x^{2}}{x + 1}\right )^{2} - \frac {1}{4} \, \log \left (\frac {x^{2}}{x + 1}\right )^{3} + \frac {3}{2} \, x^{2} + \frac {3}{2} \, x \log \left (\frac {x^{2}}{x + 1}\right ) + \frac {3}{8} \, \log \left (\frac {x^{2}}{x + 1}\right )^{2} - \frac {1}{2} \, x - \frac {1}{4} \, \log \left (\frac {x^{2}}{x + 1}\right ) + \frac {1}{16}\right )} \]
integrate(((2*x^2+3*x+2)*log(x^2/(1+x))^3+(12*x^3+12*x^2+3*x-6)*log(x^2/(1 +x))^2+(24*x^4+12*x^3-6*x^2-15*x+6)*log(x^2/(1+x))+16*x^5-8*x^3-8*x^2+9*x- 2)*exp(1/16*log(x^2/(1+x))^4+1/16*(8*x-4)*log(x^2/(1+x))^3+1/16*(24*x^2-24 *x+6)*log(x^2/(1+x))^2+1/16*(32*x^3-48*x^2+24*x-4)*log(x^2/(1+x))+x^4-2*x^ 3+3/2*x^2-1/2*x+1/16)/(4*x^2+4*x),x, algorithm=\
e^(x^4 + 2*x^3*log(x^2/(x + 1)) + 3/2*x^2*log(x^2/(x + 1))^2 + 1/2*x*log(x ^2/(x + 1))^3 + 1/16*log(x^2/(x + 1))^4 - 2*x^3 - 3*x^2*log(x^2/(x + 1)) - 3/2*x*log(x^2/(x + 1))^2 - 1/4*log(x^2/(x + 1))^3 + 3/2*x^2 + 3/2*x*log(x ^2/(x + 1)) + 3/8*log(x^2/(x + 1))^2 - 1/2*x - 1/4*log(x^2/(x + 1)) + 1/16 )
Time = 10.98 (sec) , antiderivative size = 338, normalized size of antiderivative = 14.70 \[ \int \frac {e^{\frac {1}{16} \left (1-8 x+24 x^2-32 x^3+16 x^4+\left (-4+24 x-48 x^2+32 x^3\right ) \log \left (\frac {x^2}{1+x}\right )+\left (6-24 x+24 x^2\right ) \log ^2\left (\frac {x^2}{1+x}\right )+(-4+8 x) \log ^3\left (\frac {x^2}{1+x}\right )+\log ^4\left (\frac {x^2}{1+x}\right )\right )} \left (-2+9 x-8 x^2-8 x^3+16 x^5+\left (6-15 x-6 x^2+12 x^3+24 x^4\right ) \log \left (\frac {x^2}{1+x}\right )+\left (-6+3 x+12 x^2+12 x^3\right ) \log ^2\left (\frac {x^2}{1+x}\right )+\left (2+3 x+2 x^2\right ) \log ^3\left (\frac {x^2}{1+x}\right )\right )}{4 x+4 x^2} \, dx={\mathrm {e}}^{\frac {\ln \left (x^2\right )\,{\ln \left (\frac {1}{x+1}\right )}^3}{4}}\,{\mathrm {e}}^{\frac {{\ln \left (x^2\right )}^3\,\ln \left (\frac {1}{x+1}\right )}{4}}\,{\mathrm {e}}^{-\frac {3\,\ln \left (x^2\right )\,{\ln \left (\frac {1}{x+1}\right )}^2}{4}}\,{\mathrm {e}}^{-\frac {3\,{\ln \left (x^2\right )}^2\,\ln \left (\frac {1}{x+1}\right )}{4}}\,{\mathrm {e}}^{-\frac {x}{2}}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{\frac {x\,{\ln \left (\frac {1}{x+1}\right )}^3}{2}}\,{\mathrm {e}}^{-\frac {3\,x\,{\ln \left (\frac {1}{x+1}\right )}^2}{2}}\,{\mathrm {e}}^{\frac {3\,x^2\,{\ln \left (x^2\right )}^2}{2}}\,{\mathrm {e}}^{1/16}\,{\mathrm {e}}^{-\frac {{\ln \left (x^2\right )}^3}{4}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (x^2\right )}^2}{8}}\,{\mathrm {e}}^{\frac {{\ln \left (x^2\right )}^4}{16}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (x^2\right )}^2\,{\ln \left (\frac {1}{x+1}\right )}^2}{8}}\,{\mathrm {e}}^{\frac {3\,x\,\ln \left (x^2\right )\,{\ln \left (\frac {1}{x+1}\right )}^2}{2}}\,{\mathrm {e}}^{\frac {3\,x\,{\ln \left (x^2\right )}^2\,\ln \left (\frac {1}{x+1}\right )}{2}}\,{\mathrm {e}}^{-2\,x^3}\,{\mathrm {e}}^{\frac {3\,x^2}{2}}\,{\mathrm {e}}^{\frac {3\,x^2\,{\ln \left (\frac {1}{x+1}\right )}^2}{2}}\,{\mathrm {e}}^{\frac {3\,\ln \left (x^2\right )\,\ln \left (\frac {1}{x+1}\right )}{4}}\,{\mathrm {e}}^{-\frac {{\ln \left (\frac {1}{x+1}\right )}^3}{4}}\,{\mathrm {e}}^{\frac {3\,{\ln \left (\frac {1}{x+1}\right )}^2}{8}}\,{\mathrm {e}}^{\frac {{\ln \left (\frac {1}{x+1}\right )}^4}{16}}\,{\mathrm {e}}^{\frac {3\,x\,\ln \left (\frac {1}{x+1}\right )}{2}}\,{\mathrm {e}}^{\frac {x\,{\ln \left (x^2\right )}^3}{2}}\,{\mathrm {e}}^{-\frac {3\,x\,{\ln \left (x^2\right )}^2}{2}}\,{\left (\frac {1}{x+1}\right )}^{3\,x^2\,\ln \left (x^2\right )-3\,x\,\ln \left (x^2\right )-3\,x^2+2\,x^3-\frac {1}{4}}\,{\left (x^2\right )}^{2\,x^3-3\,x^2+\frac {3\,x}{2}-\frac {1}{4}} \]
int((exp((log(x^2/(x + 1))^2*(24*x^2 - 24*x + 6))/16 - x/2 + log(x^2/(x + 1))^4/16 + (log(x^2/(x + 1))^3*(8*x - 4))/16 + (3*x^2)/2 - 2*x^3 + x^4 + ( log(x^2/(x + 1))*(24*x - 48*x^2 + 32*x^3 - 4))/16 + 1/16)*(9*x + log(x^2/( x + 1))^3*(3*x + 2*x^2 + 2) + log(x^2/(x + 1))*(12*x^3 - 6*x^2 - 15*x + 24 *x^4 + 6) + log(x^2/(x + 1))^2*(3*x + 12*x^2 + 12*x^3 - 6) - 8*x^2 - 8*x^3 + 16*x^5 - 2))/(4*x + 4*x^2),x)
exp((log(x^2)*log(1/(x + 1))^3)/4)*exp((log(x^2)^3*log(1/(x + 1)))/4)*exp( -(3*log(x^2)*log(1/(x + 1))^2)/4)*exp(-(3*log(x^2)^2*log(1/(x + 1)))/4)*ex p(-x/2)*exp(x^4)*exp((x*log(1/(x + 1))^3)/2)*exp(-(3*x*log(1/(x + 1))^2)/2 )*exp((3*x^2*log(x^2)^2)/2)*exp(1/16)*exp(-log(x^2)^3/4)*exp((3*log(x^2)^2 )/8)*exp(log(x^2)^4/16)*exp((3*log(x^2)^2*log(1/(x + 1))^2)/8)*exp((3*x*lo g(x^2)*log(1/(x + 1))^2)/2)*exp((3*x*log(x^2)^2*log(1/(x + 1)))/2)*exp(-2* x^3)*exp((3*x^2)/2)*exp((3*x^2*log(1/(x + 1))^2)/2)*exp((3*log(x^2)*log(1/ (x + 1)))/4)*exp(-log(1/(x + 1))^3/4)*exp((3*log(1/(x + 1))^2)/8)*exp(log( 1/(x + 1))^4/16)*exp((3*x*log(1/(x + 1)))/2)*exp((x*log(x^2)^3)/2)*exp(-(3 *x*log(x^2)^2)/2)*(1/(x + 1))^(3*x^2*log(x^2) - 3*x*log(x^2) - 3*x^2 + 2*x ^3 - 1/4)*(x^2)^((3*x)/2 - 3*x^2 + 2*x^3 - 1/4)