Integrand size = 679, antiderivative size = 37 \[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx=\frac {\left (2-x+\frac {x}{\log \left (\frac {\left (e^2+\log \left (x+\frac {\log (x)}{4+x}\right )\right )^2}{x^2}\right )}\right )^2}{x^2} \] Output:
(x/ln((exp(2)+ln(ln(x)/(4+x)+x))^2/x^2)-x+2)^2/x^2
Time = 0.56 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.95 \[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx=4 \left (\frac {1-x}{x^2}+\frac {1}{4 \log ^2\left (\frac {\left (e^2+\log \left (x+\frac {\log (x)}{4+x}\right )\right )^2}{x^2}\right )}+\frac {-\frac {1}{2}+\frac {1}{x}}{\log \left (\frac {\left (e^2+\log \left (x+\frac {\log (x)}{4+x}\right )\right )^2}{x^2}\right )}\right ) \] Input:
Integrate[(-16*x^2 - 68*x^3 - 32*x^4 - 4*x^5 + E^2*(64*x^3 + 32*x^4 + 4*x^ 5) + (4*x^3 + E^2*(16*x^2 + 4*x^3))*Log[x] + (64*x^3 + 32*x^4 + 4*x^5 + (1 6*x^2 + 4*x^3)*Log[x])*Log[(4*x + x^2 + Log[x])/(4 + x)] + (-32*x - 120*x^ 2 + 4*x^3 + 24*x^4 + 4*x^5 + E^2*(128*x^2 - 24*x^4 - 4*x^5) + (8*x^2 - 4*x ^3 + E^2*(32*x - 8*x^2 - 4*x^3))*Log[x] + (128*x^2 - 24*x^4 - 4*x^5 + (32* x - 8*x^2 - 4*x^3)*Log[x])*Log[(4*x + x^2 + Log[x])/(4 + x)])*Log[(E^4 + 2 *E^2*Log[(4*x + x^2 + Log[x])/(4 + x)] + Log[(4*x + x^2 + Log[x])/(4 + x)] ^2)/x^2] + (E^2*(-64*x^2 - 32*x^3 - 4*x^4) + E^2*(-16*x - 4*x^2)*Log[x] + (-64*x^2 - 32*x^3 - 4*x^4 + (-16*x - 4*x^2)*Log[x])*Log[(4*x + x^2 + Log[x ])/(4 + x)])*Log[(E^4 + 2*E^2*Log[(4*x + x^2 + Log[x])/(4 + x)] + Log[(4*x + x^2 + Log[x])/(4 + x)]^2)/x^2]^2 + (E^2*(-128*x + 24*x^3 + 4*x^4) + E^2 *(-32 + 8*x + 4*x^2)*Log[x] + (-128*x + 24*x^3 + 4*x^4 + (-32 + 8*x + 4*x^ 2)*Log[x])*Log[(4*x + x^2 + Log[x])/(4 + x)])*Log[(E^4 + 2*E^2*Log[(4*x + x^2 + Log[x])/(4 + x)] + Log[(4*x + x^2 + Log[x])/(4 + x)]^2)/x^2]^3)/((E^ 2*(16*x^4 + 8*x^5 + x^6) + E^2*(4*x^3 + x^4)*Log[x] + (16*x^4 + 8*x^5 + x^ 6 + (4*x^3 + x^4)*Log[x])*Log[(4*x + x^2 + Log[x])/(4 + x)])*Log[(E^4 + 2* E^2*Log[(4*x + x^2 + Log[x])/(4 + x)] + Log[(4*x + x^2 + Log[x])/(4 + x)]^ 2)/x^2]^3),x]
Output:
4*((1 - x)/x^2 + 1/(4*Log[(E^2 + Log[x + Log[x]/(4 + x)])^2/x^2]^2) + (-1/ 2 + x^(-1))/Log[(E^2 + Log[x + Log[x]/(4 + x)])^2/x^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.
| \(\displaystyle \int \frac {-4 x^5-32 x^4-68 x^3-16 x^2+\left (4 x^3+e^2 \left (4 x^3+16 x^2\right )\right ) \log (x)+e^2 \left (4 x^5+32 x^4+64 x^3\right )+\left (e^2 \left (-4 x^2-16 x\right ) \log (x)+e^2 \left (-4 x^4-32 x^3-64 x^2\right )+\left (-4 x^4-32 x^3-64 x^2+\left (-4 x^2-16 x\right ) \log (x)\right ) \log \left (\frac {x^2+4 x+\log (x)}{x+4}\right )\right ) \log ^2\left (\frac {\log ^2\left (\frac {x^2+4 x+\log (x)}{x+4}\right )+2 e^2 \log \left (\frac {x^2+4 x+\log (x)}{x+4}\right )+e^4}{x^2}\right )+\left (e^2 \left (4 x^2+8 x-32\right ) \log (x)+e^2 \left (4 x^4+24 x^3-128 x\right )+\left (4 x^4+24 x^3+\left (4 x^2+8 x-32\right ) \log (x)-128 x\right ) \log \left (\frac {x^2+4 x+\log (x)}{x+4}\right )\right ) \log ^3\left (\frac {\log ^2\left (\frac {x^2+4 x+\log (x)}{x+4}\right )+2 e^2 \log \left (\frac {x^2+4 x+\log (x)}{x+4}\right )+e^4}{x^2}\right )+\left (4 x^5+24 x^4+4 x^3-120 x^2+\left (-4 x^3+8 x^2+e^2 \left (-4 x^3-8 x^2+32 x\right )\right ) \log (x)+e^2 \left (-4 x^5-24 x^4+128 x^2\right )+\left (-4 x^5-24 x^4+128 x^2+\left (-4 x^3-8 x^2+32 x\right ) \log (x)\right ) \log \left (\frac {x^2+4 x+\log (x)}{x+4}\right )-32 x\right ) \log \left (\frac {\log ^2\left (\frac {x^2+4 x+\log (x)}{x+4}\right )+2 e^2 \log \left (\frac {x^2+4 x+\log (x)}{x+4}\right )+e^4}{x^2}\right )+\left (4 x^5+32 x^4+64 x^3+\left (4 x^3+16 x^2\right ) \log (x)\right ) \log \left (\frac {x^2+4 x+\log (x)}{x+4}\right )}{\left (e^2 \left (x^4+4 x^3\right ) \log (x)+e^2 \left (x^6+8 x^5+16 x^4\right )+\left (x^6+8 x^5+16 x^4+\left (x^4+4 x^3\right ) \log (x)\right ) \log \left (\frac {x^2+4 x+\log (x)}{x+4}\right )\right ) \log ^3\left (\frac {\log ^2\left (\frac {x^2+4 x+\log (x)}{x+4}\right )+2 e^2 \log \left (\frac {x^2+4 x+\log (x)}{x+4}\right )+e^4}{x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 \left (-x^5-8 x^4+e^2 (x+4)^2 x^3-17 x^3-4 x^2+\left (x^2+2 x-8\right ) (x (x+4)+\log (x)) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^3\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )-(x+4) x (x (x+4)+\log (x)) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^2\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )+\left (x+e^2 (x+4)\right ) x^2 \log (x)+(x+4) x^2 (x (x+4)+\log (x)) \log \left (x+\frac {\log (x)}{x+4}\right )+x \left (x^4+6 x^3+x^2-\left (x^2+2 x-8\right ) (x (x+4)+\log (x)) \log \left (x+\frac {\log (x)}{x+4}\right )-e^2 (x-2) (x+4)^2 x-30 x-(x-2) \left (x+e^2 (x+4)\right ) \log (x)-8\right ) \log \left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )\right )}{x^3 (x+4) (x (x+4)+\log (x)) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^3\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {x^5+8 x^4-e^2 (x+4)^2 x^3+17 x^3-\left (x+e^2 (x+4)\right ) \log (x) x^2-(x+4) (x (x+4)+\log (x)) \log \left (x+\frac {\log (x)}{x+4}\right ) x^2+4 x^2+(x+4) (x (x+4)+\log (x)) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^2\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right ) x+\left (-x^4-6 x^3-x^2-e^2 (2-x) (x+4)^2 x+30 x-(2-x) \left (x+e^2 (x+4)\right ) \log (x)-\left (-x^2-2 x+8\right ) (x (x+4)+\log (x)) \log \left (x+\frac {\log (x)}{x+4}\right )+8\right ) \log \left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right ) x+\left (-x^2-2 x+8\right ) (x (x+4)+\log (x)) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^3\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}{x^3 (x+4) (x (x+4)+\log (x)) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^3\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {x^5+8 x^4-e^2 (x+4)^2 x^3+17 x^3-\left (x+e^2 (x+4)\right ) \log (x) x^2-(x+4) (x (x+4)+\log (x)) \log \left (x+\frac {\log (x)}{x+4}\right ) x^2+4 x^2+(x+4) (x (x+4)+\log (x)) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^2\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right ) x+\left (-x^4-6 x^3-x^2-e^2 (2-x) (x+4)^2 x+30 x-(2-x) \left (x+e^2 (x+4)\right ) \log (x)-\left (-x^2-2 x+8\right ) (x (x+4)+\log (x)) \log \left (x+\frac {\log (x)}{x+4}\right )+8\right ) \log \left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right ) x+\left (-x^2-2 x+8\right ) (x (x+4)+\log (x)) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^3\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}{x^3 (x+4) (x (x+4)+\log (x)) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^3\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {\left (-\log \left (x+\frac {\log (x)}{x+4}\right ) x^3+\left (1-e^2\right ) x^3-8 \log \left (x+\frac {\log (x)}{x+4}\right ) x^2+8 \left (1-e^2\right ) x^2-\left (1+e^2\right ) \log (x) x-\log (x) \log \left (x+\frac {\log (x)}{x+4}\right ) x-16 \log \left (x+\frac {\log (x)}{x+4}\right ) x+17 \left (1-\frac {16 e^2}{17}\right ) x-4 e^2 \log (x)-4 \log (x) \log \left (x+\frac {\log (x)}{x+4}\right )+4\right ) (2-x)}{x^2 (x+4) \left (x^2+4 x+\log (x)\right ) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^2\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}+\frac {2-x}{x^3}+\frac {1}{x^2 \log \left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}+\frac {-\log \left (x+\frac {\log (x)}{x+4}\right ) x^3+\left (1-e^2\right ) x^3-8 \log \left (x+\frac {\log (x)}{x+4}\right ) x^2+8 \left (1-e^2\right ) x^2-\left (1+e^2\right ) \log (x) x-\log (x) \log \left (x+\frac {\log (x)}{x+4}\right ) x-16 \log \left (x+\frac {\log (x)}{x+4}\right ) x+17 \left (1-\frac {16 e^2}{17}\right ) x-4 e^2 \log (x)-4 \log (x) \log \left (x+\frac {\log (x)}{x+4}\right )+4}{x (x+4) \left (x^2+4 x+\log (x)\right ) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^3\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -4 \int \left (\frac {\left (-\log \left (x+\frac {\log (x)}{x+4}\right ) x^3+\left (1-e^2\right ) x^3-8 \log \left (x+\frac {\log (x)}{x+4}\right ) x^2+8 \left (1-e^2\right ) x^2-\left (1+e^2\right ) \log (x) x-\log (x) \log \left (x+\frac {\log (x)}{x+4}\right ) x-16 \log \left (x+\frac {\log (x)}{x+4}\right ) x+17 \left (1-\frac {16 e^2}{17}\right ) x-4 e^2 \log (x)-4 \log (x) \log \left (x+\frac {\log (x)}{x+4}\right )+4\right ) (2-x)}{x^2 (x+4) \left (x^2+4 x+\log (x)\right ) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^2\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}+\frac {2-x}{x^3}+\frac {1}{x^2 \log \left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}+\frac {-\log \left (x+\frac {\log (x)}{x+4}\right ) x^3+\left (1-e^2\right ) x^3-8 \log \left (x+\frac {\log (x)}{x+4}\right ) x^2+8 \left (1-e^2\right ) x^2-\left (1+e^2\right ) \log (x) x-\log (x) \log \left (x+\frac {\log (x)}{x+4}\right ) x-16 \log \left (x+\frac {\log (x)}{x+4}\right ) x+17 \left (1-\frac {16 e^2}{17}\right ) x-4 e^2 \log (x)-4 \log (x) \log \left (x+\frac {\log (x)}{x+4}\right )+4}{x (x+4) \left (x^2+4 x+\log (x)\right ) \left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right ) \log ^3\left (\frac {\left (\log \left (x+\frac {\log (x)}{x+4}\right )+e^2\right )^2}{x^2}\right )}\right )dx\) |
Input:
Int[(-16*x^2 - 68*x^3 - 32*x^4 - 4*x^5 + E^2*(64*x^3 + 32*x^4 + 4*x^5) + ( 4*x^3 + E^2*(16*x^2 + 4*x^3))*Log[x] + (64*x^3 + 32*x^4 + 4*x^5 + (16*x^2 + 4*x^3)*Log[x])*Log[(4*x + x^2 + Log[x])/(4 + x)] + (-32*x - 120*x^2 + 4* x^3 + 24*x^4 + 4*x^5 + E^2*(128*x^2 - 24*x^4 - 4*x^5) + (8*x^2 - 4*x^3 + E ^2*(32*x - 8*x^2 - 4*x^3))*Log[x] + (128*x^2 - 24*x^4 - 4*x^5 + (32*x - 8* x^2 - 4*x^3)*Log[x])*Log[(4*x + x^2 + Log[x])/(4 + x)])*Log[(E^4 + 2*E^2*L og[(4*x + x^2 + Log[x])/(4 + x)] + Log[(4*x + x^2 + Log[x])/(4 + x)]^2)/x^ 2] + (E^2*(-64*x^2 - 32*x^3 - 4*x^4) + E^2*(-16*x - 4*x^2)*Log[x] + (-64*x ^2 - 32*x^3 - 4*x^4 + (-16*x - 4*x^2)*Log[x])*Log[(4*x + x^2 + Log[x])/(4 + x)])*Log[(E^4 + 2*E^2*Log[(4*x + x^2 + Log[x])/(4 + x)] + Log[(4*x + x^2 + Log[x])/(4 + x)]^2)/x^2]^2 + (E^2*(-128*x + 24*x^3 + 4*x^4) + E^2*(-32 + 8*x + 4*x^2)*Log[x] + (-128*x + 24*x^3 + 4*x^4 + (-32 + 8*x + 4*x^2)*Log [x])*Log[(4*x + x^2 + Log[x])/(4 + x)])*Log[(E^4 + 2*E^2*Log[(4*x + x^2 + Log[x])/(4 + x)] + Log[(4*x + x^2 + Log[x])/(4 + x)]^2)/x^2]^3)/((E^2*(16* x^4 + 8*x^5 + x^6) + E^2*(4*x^3 + x^4)*Log[x] + (16*x^4 + 8*x^5 + x^6 + (4 *x^3 + x^4)*Log[x])*Log[(4*x + x^2 + Log[x])/(4 + x)])*Log[(E^4 + 2*E^2*Lo g[(4*x + x^2 + Log[x])/(4 + x)] + Log[(4*x + x^2 + Log[x])/(4 + x)]^2)/x^2 ]^3),x]
Output:
$Aborted
Timed out.
\[\int \frac {\left (\left (\left (4 x^{2}+8 x -32\right ) \ln \left (x \right )+4 x^{4}+24 x^{3}-128 x \right ) \ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )+\left (4 x^{2}+8 x -32\right ) {\mathrm e}^{2} \ln \left (x \right )+\left (4 x^{4}+24 x^{3}-128 x \right ) {\mathrm e}^{2}\right ) {\ln \left (\frac {\ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )^{2}+2 \,{\mathrm e}^{2} \ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )+{\mathrm e}^{4}}{x^{2}}\right )}^{3}+\left (\left (\left (-4 x^{2}-16 x \right ) \ln \left (x \right )-4 x^{4}-32 x^{3}-64 x^{2}\right ) \ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )+\left (-4 x^{2}-16 x \right ) {\mathrm e}^{2} \ln \left (x \right )+\left (-4 x^{4}-32 x^{3}-64 x^{2}\right ) {\mathrm e}^{2}\right ) {\ln \left (\frac {\ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )^{2}+2 \,{\mathrm e}^{2} \ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )+{\mathrm e}^{4}}{x^{2}}\right )}^{2}+\left (\left (\left (-4 x^{3}-8 x^{2}+32 x \right ) \ln \left (x \right )-4 x^{5}-24 x^{4}+128 x^{2}\right ) \ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )+\left (\left (-4 x^{3}-8 x^{2}+32 x \right ) {\mathrm e}^{2}-4 x^{3}+8 x^{2}\right ) \ln \left (x \right )+\left (-4 x^{5}-24 x^{4}+128 x^{2}\right ) {\mathrm e}^{2}+4 x^{5}+24 x^{4}+4 x^{3}-120 x^{2}-32 x \right ) \ln \left (\frac {\ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )^{2}+2 \,{\mathrm e}^{2} \ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )+{\mathrm e}^{4}}{x^{2}}\right )+\left (\left (4 x^{3}+16 x^{2}\right ) \ln \left (x \right )+4 x^{5}+32 x^{4}+64 x^{3}\right ) \ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )+\left (\left (4 x^{3}+16 x^{2}\right ) {\mathrm e}^{2}+4 x^{3}\right ) \ln \left (x \right )+\left (4 x^{5}+32 x^{4}+64 x^{3}\right ) {\mathrm e}^{2}-4 x^{5}-32 x^{4}-68 x^{3}-16 x^{2}}{\left (\left (\left (x^{4}+4 x^{3}\right ) \ln \left (x \right )+x^{6}+8 x^{5}+16 x^{4}\right ) \ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )+\left (x^{4}+4 x^{3}\right ) {\mathrm e}^{2} \ln \left (x \right )+\left (x^{6}+8 x^{5}+16 x^{4}\right ) {\mathrm e}^{2}\right ) {\ln \left (\frac {\ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )^{2}+2 \,{\mathrm e}^{2} \ln \left (\frac {\ln \left (x \right )+x^{2}+4 x}{4+x}\right )+{\mathrm e}^{4}}{x^{2}}\right )}^{3}}d x\]
Input:
int(((((4*x^2+8*x-32)*ln(x)+4*x^4+24*x^3-128*x)*ln((ln(x)+x^2+4*x)/(4+x))+ (4*x^2+8*x-32)*exp(2)*ln(x)+(4*x^4+24*x^3-128*x)*exp(2))*ln((ln((ln(x)+x^2 +4*x)/(4+x))^2+2*exp(2)*ln((ln(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^3+(((-4*x ^2-16*x)*ln(x)-4*x^4-32*x^3-64*x^2)*ln((ln(x)+x^2+4*x)/(4+x))+(-4*x^2-16*x )*exp(2)*ln(x)+(-4*x^4-32*x^3-64*x^2)*exp(2))*ln((ln((ln(x)+x^2+4*x)/(4+x) )^2+2*exp(2)*ln((ln(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^2+(((-4*x^3-8*x^2+32 *x)*ln(x)-4*x^5-24*x^4+128*x^2)*ln((ln(x)+x^2+4*x)/(4+x))+((-4*x^3-8*x^2+3 2*x)*exp(2)-4*x^3+8*x^2)*ln(x)+(-4*x^5-24*x^4+128*x^2)*exp(2)+4*x^5+24*x^4 +4*x^3-120*x^2-32*x)*ln((ln((ln(x)+x^2+4*x)/(4+x))^2+2*exp(2)*ln((ln(x)+x^ 2+4*x)/(4+x))+exp(2)^2)/x^2)+((4*x^3+16*x^2)*ln(x)+4*x^5+32*x^4+64*x^3)*ln ((ln(x)+x^2+4*x)/(4+x))+((4*x^3+16*x^2)*exp(2)+4*x^3)*ln(x)+(4*x^5+32*x^4+ 64*x^3)*exp(2)-4*x^5-32*x^4-68*x^3-16*x^2)/(((x^4+4*x^3)*ln(x)+x^6+8*x^5+1 6*x^4)*ln((ln(x)+x^2+4*x)/(4+x))+(x^4+4*x^3)*exp(2)*ln(x)+(x^6+8*x^5+16*x^ 4)*exp(2))/ln((ln((ln(x)+x^2+4*x)/(4+x))^2+2*exp(2)*ln((ln(x)+x^2+4*x)/(4+ x))+exp(2)^2)/x^2)^3,x)
Output:
int(((((4*x^2+8*x-32)*ln(x)+4*x^4+24*x^3-128*x)*ln((ln(x)+x^2+4*x)/(4+x))+ (4*x^2+8*x-32)*exp(2)*ln(x)+(4*x^4+24*x^3-128*x)*exp(2))*ln((ln((ln(x)+x^2 +4*x)/(4+x))^2+2*exp(2)*ln((ln(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^3+(((-4*x ^2-16*x)*ln(x)-4*x^4-32*x^3-64*x^2)*ln((ln(x)+x^2+4*x)/(4+x))+(-4*x^2-16*x )*exp(2)*ln(x)+(-4*x^4-32*x^3-64*x^2)*exp(2))*ln((ln((ln(x)+x^2+4*x)/(4+x) )^2+2*exp(2)*ln((ln(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^2+(((-4*x^3-8*x^2+32 *x)*ln(x)-4*x^5-24*x^4+128*x^2)*ln((ln(x)+x^2+4*x)/(4+x))+((-4*x^3-8*x^2+3 2*x)*exp(2)-4*x^3+8*x^2)*ln(x)+(-4*x^5-24*x^4+128*x^2)*exp(2)+4*x^5+24*x^4 +4*x^3-120*x^2-32*x)*ln((ln((ln(x)+x^2+4*x)/(4+x))^2+2*exp(2)*ln((ln(x)+x^ 2+4*x)/(4+x))+exp(2)^2)/x^2)+((4*x^3+16*x^2)*ln(x)+4*x^5+32*x^4+64*x^3)*ln ((ln(x)+x^2+4*x)/(4+x))+((4*x^3+16*x^2)*exp(2)+4*x^3)*ln(x)+(4*x^5+32*x^4+ 64*x^3)*exp(2)-4*x^5-32*x^4-68*x^3-16*x^2)/(((x^4+4*x^3)*ln(x)+x^6+8*x^5+1 6*x^4)*ln((ln(x)+x^2+4*x)/(4+x))+(x^4+4*x^3)*exp(2)*ln(x)+(x^6+8*x^5+16*x^ 4)*exp(2))/ln((ln((ln(x)+x^2+4*x)/(4+x))^2+2*exp(2)*ln((ln(x)+x^2+4*x)/(4+ x))+exp(2)^2)/x^2)^3,x)
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (35) = 70\).
Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 4.51 \[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx=-\frac {4 \, {\left (x - 1\right )} \log \left (\frac {2 \, e^{2} \log \left (\frac {x^{2} + 4 \, x + \log \left (x\right )}{x + 4}\right ) + \log \left (\frac {x^{2} + 4 \, x + \log \left (x\right )}{x + 4}\right )^{2} + e^{4}}{x^{2}}\right )^{2} - x^{2} + 2 \, {\left (x^{2} - 2 \, x\right )} \log \left (\frac {2 \, e^{2} \log \left (\frac {x^{2} + 4 \, x + \log \left (x\right )}{x + 4}\right ) + \log \left (\frac {x^{2} + 4 \, x + \log \left (x\right )}{x + 4}\right )^{2} + e^{4}}{x^{2}}\right )}{x^{2} \log \left (\frac {2 \, e^{2} \log \left (\frac {x^{2} + 4 \, x + \log \left (x\right )}{x + 4}\right ) + \log \left (\frac {x^{2} + 4 \, x + \log \left (x\right )}{x + 4}\right )^{2} + e^{4}}{x^{2}}\right )^{2}} \] Input:
integrate(((((4*x^2+8*x-32)*log(x)+4*x^4+24*x^3-128*x)*log((log(x)+x^2+4*x )/(4+x))+(4*x^2+8*x-32)*exp(2)*log(x)+(4*x^4+24*x^3-128*x)*exp(2))*log((lo g((log(x)+x^2+4*x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2) /x^2)^3+(((-4*x^2-16*x)*log(x)-4*x^4-32*x^3-64*x^2)*log((log(x)+x^2+4*x)/( 4+x))+(-4*x^2-16*x)*exp(2)*log(x)+(-4*x^4-32*x^3-64*x^2)*exp(2))*log((log( (log(x)+x^2+4*x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x ^2)^2+(((-4*x^3-8*x^2+32*x)*log(x)-4*x^5-24*x^4+128*x^2)*log((log(x)+x^2+4 *x)/(4+x))+((-4*x^3-8*x^2+32*x)*exp(2)-4*x^3+8*x^2)*log(x)+(-4*x^5-24*x^4+ 128*x^2)*exp(2)+4*x^5+24*x^4+4*x^3-120*x^2-32*x)*log((log((log(x)+x^2+4*x) /(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)+((4*x^3+16*x ^2)*log(x)+4*x^5+32*x^4+64*x^3)*log((log(x)+x^2+4*x)/(4+x))+((4*x^3+16*x^2 )*exp(2)+4*x^3)*log(x)+(4*x^5+32*x^4+64*x^3)*exp(2)-4*x^5-32*x^4-68*x^3-16 *x^2)/(((x^4+4*x^3)*log(x)+x^6+8*x^5+16*x^4)*log((log(x)+x^2+4*x)/(4+x))+( x^4+4*x^3)*exp(2)*log(x)+(x^6+8*x^5+16*x^4)*exp(2))/log((log((log(x)+x^2+4 *x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^3,x, algo rithm="fricas")
Output:
-(4*(x - 1)*log((2*e^2*log((x^2 + 4*x + log(x))/(x + 4)) + log((x^2 + 4*x + log(x))/(x + 4))^2 + e^4)/x^2)^2 - x^2 + 2*(x^2 - 2*x)*log((2*e^2*log((x ^2 + 4*x + log(x))/(x + 4)) + log((x^2 + 4*x + log(x))/(x + 4))^2 + e^4)/x ^2))/(x^2*log((2*e^2*log((x^2 + 4*x + log(x))/(x + 4)) + log((x^2 + 4*x + log(x))/(x + 4))^2 + e^4)/x^2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (29) = 58\).
Time = 103.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.95 \[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx=\frac {x + \left (4 - 2 x\right ) \log {\left (\frac {\log {\left (\frac {x^{2} + 4 x + \log {\left (x \right )}}{x + 4} \right )}^{2} + 2 e^{2} \log {\left (\frac {x^{2} + 4 x + \log {\left (x \right )}}{x + 4} \right )} + e^{4}}{x^{2}} \right )}}{x \log {\left (\frac {\log {\left (\frac {x^{2} + 4 x + \log {\left (x \right )}}{x + 4} \right )}^{2} + 2 e^{2} \log {\left (\frac {x^{2} + 4 x + \log {\left (x \right )}}{x + 4} \right )} + e^{4}}{x^{2}} \right )}^{2}} + \frac {4 - 4 x}{x^{2}} \] Input:
integrate(((((4*x**2+8*x-32)*ln(x)+4*x**4+24*x**3-128*x)*ln((ln(x)+x**2+4* x)/(4+x))+(4*x**2+8*x-32)*exp(2)*ln(x)+(4*x**4+24*x**3-128*x)*exp(2))*ln(( ln((ln(x)+x**2+4*x)/(4+x))**2+2*exp(2)*ln((ln(x)+x**2+4*x)/(4+x))+exp(2)** 2)/x**2)**3+(((-4*x**2-16*x)*ln(x)-4*x**4-32*x**3-64*x**2)*ln((ln(x)+x**2+ 4*x)/(4+x))+(-4*x**2-16*x)*exp(2)*ln(x)+(-4*x**4-32*x**3-64*x**2)*exp(2))* ln((ln((ln(x)+x**2+4*x)/(4+x))**2+2*exp(2)*ln((ln(x)+x**2+4*x)/(4+x))+exp( 2)**2)/x**2)**2+(((-4*x**3-8*x**2+32*x)*ln(x)-4*x**5-24*x**4+128*x**2)*ln( (ln(x)+x**2+4*x)/(4+x))+((-4*x**3-8*x**2+32*x)*exp(2)-4*x**3+8*x**2)*ln(x) +(-4*x**5-24*x**4+128*x**2)*exp(2)+4*x**5+24*x**4+4*x**3-120*x**2-32*x)*ln ((ln((ln(x)+x**2+4*x)/(4+x))**2+2*exp(2)*ln((ln(x)+x**2+4*x)/(4+x))+exp(2) **2)/x**2)+((4*x**3+16*x**2)*ln(x)+4*x**5+32*x**4+64*x**3)*ln((ln(x)+x**2+ 4*x)/(4+x))+((4*x**3+16*x**2)*exp(2)+4*x**3)*ln(x)+(4*x**5+32*x**4+64*x**3 )*exp(2)-4*x**5-32*x**4-68*x**3-16*x**2)/(((x**4+4*x**3)*ln(x)+x**6+8*x**5 +16*x**4)*ln((ln(x)+x**2+4*x)/(4+x))+(x**4+4*x**3)*exp(2)*ln(x)+(x**6+8*x* *5+16*x**4)*exp(2))/ln((ln((ln(x)+x**2+4*x)/(4+x))**2+2*exp(2)*ln((ln(x)+x **2+4*x)/(4+x))+exp(2)**2)/x**2)**3,x)
Output:
(x + (4 - 2*x)*log((log((x**2 + 4*x + log(x))/(x + 4))**2 + 2*exp(2)*log(( x**2 + 4*x + log(x))/(x + 4)) + exp(4))/x**2))/(x*log((log((x**2 + 4*x + l og(x))/(x + 4))**2 + 2*exp(2)*log((x**2 + 4*x + log(x))/(x + 4)) + exp(4)) /x**2)**2) + (4 - 4*x)/x**2
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (35) = 70\).
Time = 2.35 (sec) , antiderivative size = 155, normalized size of antiderivative = 4.19 \[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx=-\frac {16 \, {\left (x - 1\right )} \log \left (x\right )^{2} + 16 \, {\left (x - 1\right )} \log \left (e^{2} + \log \left (x^{2} + 4 \, x + \log \left (x\right )\right ) - \log \left (x + 4\right )\right )^{2} - x^{2} - 4 \, {\left (x^{2} - 2 \, x\right )} \log \left (x\right ) + 4 \, {\left (x^{2} - 8 \, {\left (x - 1\right )} \log \left (x\right ) - 2 \, x\right )} \log \left (e^{2} + \log \left (x^{2} + 4 \, x + \log \left (x\right )\right ) - \log \left (x + 4\right )\right )}{4 \, {\left (x^{2} \log \left (x\right )^{2} - 2 \, x^{2} \log \left (x\right ) \log \left (e^{2} + \log \left (x^{2} + 4 \, x + \log \left (x\right )\right ) - \log \left (x + 4\right )\right ) + x^{2} \log \left (e^{2} + \log \left (x^{2} + 4 \, x + \log \left (x\right )\right ) - \log \left (x + 4\right )\right )^{2}\right )}} \] Input:
integrate(((((4*x^2+8*x-32)*log(x)+4*x^4+24*x^3-128*x)*log((log(x)+x^2+4*x )/(4+x))+(4*x^2+8*x-32)*exp(2)*log(x)+(4*x^4+24*x^3-128*x)*exp(2))*log((lo g((log(x)+x^2+4*x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2) /x^2)^3+(((-4*x^2-16*x)*log(x)-4*x^4-32*x^3-64*x^2)*log((log(x)+x^2+4*x)/( 4+x))+(-4*x^2-16*x)*exp(2)*log(x)+(-4*x^4-32*x^3-64*x^2)*exp(2))*log((log( (log(x)+x^2+4*x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x ^2)^2+(((-4*x^3-8*x^2+32*x)*log(x)-4*x^5-24*x^4+128*x^2)*log((log(x)+x^2+4 *x)/(4+x))+((-4*x^3-8*x^2+32*x)*exp(2)-4*x^3+8*x^2)*log(x)+(-4*x^5-24*x^4+ 128*x^2)*exp(2)+4*x^5+24*x^4+4*x^3-120*x^2-32*x)*log((log((log(x)+x^2+4*x) /(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)+((4*x^3+16*x ^2)*log(x)+4*x^5+32*x^4+64*x^3)*log((log(x)+x^2+4*x)/(4+x))+((4*x^3+16*x^2 )*exp(2)+4*x^3)*log(x)+(4*x^5+32*x^4+64*x^3)*exp(2)-4*x^5-32*x^4-68*x^3-16 *x^2)/(((x^4+4*x^3)*log(x)+x^6+8*x^5+16*x^4)*log((log(x)+x^2+4*x)/(4+x))+( x^4+4*x^3)*exp(2)*log(x)+(x^6+8*x^5+16*x^4)*exp(2))/log((log((log(x)+x^2+4 *x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^3,x, algo rithm="maxima")
Output:
-1/4*(16*(x - 1)*log(x)^2 + 16*(x - 1)*log(e^2 + log(x^2 + 4*x + log(x)) - log(x + 4))^2 - x^2 - 4*(x^2 - 2*x)*log(x) + 4*(x^2 - 8*(x - 1)*log(x) - 2*x)*log(e^2 + log(x^2 + 4*x + log(x)) - log(x + 4)))/(x^2*log(x)^2 - 2*x^ 2*log(x)*log(e^2 + log(x^2 + 4*x + log(x)) - log(x + 4)) + x^2*log(e^2 + l og(x^2 + 4*x + log(x)) - log(x + 4))^2)
Timed out. \[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx=\text {Timed out} \] Input:
integrate(((((4*x^2+8*x-32)*log(x)+4*x^4+24*x^3-128*x)*log((log(x)+x^2+4*x )/(4+x))+(4*x^2+8*x-32)*exp(2)*log(x)+(4*x^4+24*x^3-128*x)*exp(2))*log((lo g((log(x)+x^2+4*x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2) /x^2)^3+(((-4*x^2-16*x)*log(x)-4*x^4-32*x^3-64*x^2)*log((log(x)+x^2+4*x)/( 4+x))+(-4*x^2-16*x)*exp(2)*log(x)+(-4*x^4-32*x^3-64*x^2)*exp(2))*log((log( (log(x)+x^2+4*x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x ^2)^2+(((-4*x^3-8*x^2+32*x)*log(x)-4*x^5-24*x^4+128*x^2)*log((log(x)+x^2+4 *x)/(4+x))+((-4*x^3-8*x^2+32*x)*exp(2)-4*x^3+8*x^2)*log(x)+(-4*x^5-24*x^4+ 128*x^2)*exp(2)+4*x^5+24*x^4+4*x^3-120*x^2-32*x)*log((log((log(x)+x^2+4*x) /(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)+((4*x^3+16*x ^2)*log(x)+4*x^5+32*x^4+64*x^3)*log((log(x)+x^2+4*x)/(4+x))+((4*x^3+16*x^2 )*exp(2)+4*x^3)*log(x)+(4*x^5+32*x^4+64*x^3)*exp(2)-4*x^5-32*x^4-68*x^3-16 *x^2)/(((x^4+4*x^3)*log(x)+x^6+8*x^5+16*x^4)*log((log(x)+x^2+4*x)/(4+x))+( x^4+4*x^3)*exp(2)*log(x)+(x^6+8*x^5+16*x^4)*exp(2))/log((log((log(x)+x^2+4 *x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^3,x, algo rithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx =\text {Too large to display} \] Input:
int(-(log((exp(4) + log((4*x + log(x) + x^2)/(x + 4))^2 + 2*exp(2)*log((4* x + log(x) + x^2)/(x + 4)))/x^2)*(32*x + log((4*x + log(x) + x^2)/(x + 4)) *(24*x^4 - 128*x^2 + 4*x^5 + log(x)*(8*x^2 - 32*x + 4*x^3)) + log(x)*(exp( 2)*(8*x^2 - 32*x + 4*x^3) - 8*x^2 + 4*x^3) + exp(2)*(24*x^4 - 128*x^2 + 4* x^5) + 120*x^2 - 4*x^3 - 24*x^4 - 4*x^5) - log(x)*(exp(2)*(16*x^2 + 4*x^3) + 4*x^3) - log((4*x + log(x) + x^2)/(x + 4))*(log(x)*(16*x^2 + 4*x^3) + 6 4*x^3 + 32*x^4 + 4*x^5) + log((exp(4) + log((4*x + log(x) + x^2)/(x + 4))^ 2 + 2*exp(2)*log((4*x + log(x) + x^2)/(x + 4)))/x^2)^2*(log((4*x + log(x) + x^2)/(x + 4))*(log(x)*(16*x + 4*x^2) + 64*x^2 + 32*x^3 + 4*x^4) + exp(2) *(64*x^2 + 32*x^3 + 4*x^4) + exp(2)*log(x)*(16*x + 4*x^2)) - log((exp(4) + log((4*x + log(x) + x^2)/(x + 4))^2 + 2*exp(2)*log((4*x + log(x) + x^2)/( x + 4)))/x^2)^3*(exp(2)*(24*x^3 - 128*x + 4*x^4) + log((4*x + log(x) + x^2 )/(x + 4))*(log(x)*(8*x + 4*x^2 - 32) - 128*x + 24*x^3 + 4*x^4) + exp(2)*l og(x)*(8*x + 4*x^2 - 32)) - exp(2)*(64*x^3 + 32*x^4 + 4*x^5) + 16*x^2 + 68 *x^3 + 32*x^4 + 4*x^5)/(log((exp(4) + log((4*x + log(x) + x^2)/(x + 4))^2 + 2*exp(2)*log((4*x + log(x) + x^2)/(x + 4)))/x^2)^3*(exp(2)*(16*x^4 + 8*x ^5 + x^6) + log((4*x + log(x) + x^2)/(x + 4))*(log(x)*(4*x^3 + x^4) + 16*x ^4 + 8*x^5 + x^6) + exp(2)*log(x)*(4*x^3 + x^4))),x)
Output:
int(-(log((exp(4) + log((4*x + log(x) + x^2)/(x + 4))^2 + 2*exp(2)*log((4* x + log(x) + x^2)/(x + 4)))/x^2)*(32*x + log((4*x + log(x) + x^2)/(x + 4)) *(24*x^4 - 128*x^2 + 4*x^5 + log(x)*(8*x^2 - 32*x + 4*x^3)) + log(x)*(exp( 2)*(8*x^2 - 32*x + 4*x^3) - 8*x^2 + 4*x^3) + exp(2)*(24*x^4 - 128*x^2 + 4* x^5) + 120*x^2 - 4*x^3 - 24*x^4 - 4*x^5) - log(x)*(exp(2)*(16*x^2 + 4*x^3) + 4*x^3) - log((4*x + log(x) + x^2)/(x + 4))*(log(x)*(16*x^2 + 4*x^3) + 6 4*x^3 + 32*x^4 + 4*x^5) + log((exp(4) + log((4*x + log(x) + x^2)/(x + 4))^ 2 + 2*exp(2)*log((4*x + log(x) + x^2)/(x + 4)))/x^2)^2*(log((4*x + log(x) + x^2)/(x + 4))*(log(x)*(16*x + 4*x^2) + 64*x^2 + 32*x^3 + 4*x^4) + exp(2) *(64*x^2 + 32*x^3 + 4*x^4) + exp(2)*log(x)*(16*x + 4*x^2)) - log((exp(4) + log((4*x + log(x) + x^2)/(x + 4))^2 + 2*exp(2)*log((4*x + log(x) + x^2)/( x + 4)))/x^2)^3*(exp(2)*(24*x^3 - 128*x + 4*x^4) + log((4*x + log(x) + x^2 )/(x + 4))*(log(x)*(8*x + 4*x^2 - 32) - 128*x + 24*x^3 + 4*x^4) + exp(2)*l og(x)*(8*x + 4*x^2 - 32)) - exp(2)*(64*x^3 + 32*x^4 + 4*x^5) + 16*x^2 + 68 *x^3 + 32*x^4 + 4*x^5)/(log((exp(4) + log((4*x + log(x) + x^2)/(x + 4))^2 + 2*exp(2)*log((4*x + log(x) + x^2)/(x + 4)))/x^2)^3*(exp(2)*(16*x^4 + 8*x ^5 + x^6) + log((4*x + log(x) + x^2)/(x + 4))*(log(x)*(4*x^3 + x^4) + 16*x ^4 + 8*x^5 + x^6) + exp(2)*log(x)*(4*x^3 + x^4))), x)
Time = 0.26 (sec) , antiderivative size = 267, normalized size of antiderivative = 7.22 \[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx=\frac {-4 {\mathrm {log}\left (\frac {\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right )^{2}+2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right ) e^{2}+e^{4}}{x^{2}}\right )}^{2} x +4 {\mathrm {log}\left (\frac {\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right )^{2}+2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right ) e^{2}+e^{4}}{x^{2}}\right )}^{2}-2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right )^{2}+2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right ) e^{2}+e^{4}}{x^{2}}\right ) x^{2}+4 \,\mathrm {log}\left (\frac {\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right )^{2}+2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right ) e^{2}+e^{4}}{x^{2}}\right ) x +x^{2}}{{\mathrm {log}\left (\frac {\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right )^{2}+2 \,\mathrm {log}\left (\frac {\mathrm {log}\left (x \right )+x^{2}+4 x}{x +4}\right ) e^{2}+e^{4}}{x^{2}}\right )}^{2} x^{2}} \] Input:
int(((((4*x^2+8*x-32)*log(x)+4*x^4+24*x^3-128*x)*log((log(x)+x^2+4*x)/(4+x ))+(4*x^2+8*x-32)*exp(2)*log(x)+(4*x^4+24*x^3-128*x)*exp(2))*log((log((log (x)+x^2+4*x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^ 3+(((-4*x^2-16*x)*log(x)-4*x^4-32*x^3-64*x^2)*log((log(x)+x^2+4*x)/(4+x))+ (-4*x^2-16*x)*exp(2)*log(x)+(-4*x^4-32*x^3-64*x^2)*exp(2))*log((log((log(x )+x^2+4*x)/(4+x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^2+ (((-4*x^3-8*x^2+32*x)*log(x)-4*x^5-24*x^4+128*x^2)*log((log(x)+x^2+4*x)/(4 +x))+((-4*x^3-8*x^2+32*x)*exp(2)-4*x^3+8*x^2)*log(x)+(-4*x^5-24*x^4+128*x^ 2)*exp(2)+4*x^5+24*x^4+4*x^3-120*x^2-32*x)*log((log((log(x)+x^2+4*x)/(4+x) )^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)+((4*x^3+16*x^2)*lo g(x)+4*x^5+32*x^4+64*x^3)*log((log(x)+x^2+4*x)/(4+x))+((4*x^3+16*x^2)*exp( 2)+4*x^3)*log(x)+(4*x^5+32*x^4+64*x^3)*exp(2)-4*x^5-32*x^4-68*x^3-16*x^2)/ (((x^4+4*x^3)*log(x)+x^6+8*x^5+16*x^4)*log((log(x)+x^2+4*x)/(4+x))+(x^4+4* x^3)*exp(2)*log(x)+(x^6+8*x^5+16*x^4)*exp(2))/log((log((log(x)+x^2+4*x)/(4 +x))^2+2*exp(2)*log((log(x)+x^2+4*x)/(4+x))+exp(2)^2)/x^2)^3,x)
Output:
( - 4*log((log((log(x) + x**2 + 4*x)/(x + 4))**2 + 2*log((log(x) + x**2 + 4*x)/(x + 4))*e**2 + e**4)/x**2)**2*x + 4*log((log((log(x) + x**2 + 4*x)/( x + 4))**2 + 2*log((log(x) + x**2 + 4*x)/(x + 4))*e**2 + e**4)/x**2)**2 - 2*log((log((log(x) + x**2 + 4*x)/(x + 4))**2 + 2*log((log(x) + x**2 + 4*x) /(x + 4))*e**2 + e**4)/x**2)*x**2 + 4*log((log((log(x) + x**2 + 4*x)/(x + 4))**2 + 2*log((log(x) + x**2 + 4*x)/(x + 4))*e**2 + e**4)/x**2)*x + x**2) /(log((log((log(x) + x**2 + 4*x)/(x + 4))**2 + 2*log((log(x) + x**2 + 4*x) /(x + 4))*e**2 + e**4)/x**2)**2*x**2)