Integrand size = 159, antiderivative size = 21 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=4+\log \left (\log \left (4+\frac {x}{\left (1+2 e^x\right )^2}+x \log (x)\right )\right ) \] Output:
ln(ln(x*ln(x)+4+x/(1+2*exp(x))^2))+4
Time = 0.13 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log \left (\log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (1+2 e^x\right )^2 x \log (x)}{\left (1+2 e^x\right )^2}\right )\right ) \] Input:
Integrate[(2 + 12*E^(2*x) + 8*E^(3*x) + E^x*(8 - 4*x) + (1 + 6*E^x + 12*E^ (2*x) + 8*E^(3*x))*Log[x])/((4 + 48*E^(2*x) + 32*E^(3*x) + x + E^x*(24 + 2 *x) + (x + 6*E^x*x + 12*E^(2*x)*x + 8*E^(3*x)*x)*Log[x])*Log[(4 + 16*E^x + 16*E^(2*x) + x + (x + 4*E^x*x + 4*E^(2*x)*x)*Log[x])/(1 + 4*E^x + 4*E^(2* x))]),x]
Output:
Log[Log[(4 + 16*E^x + 16*E^(2*x) + x + (1 + 2*E^x)^2*x*Log[x])/(1 + 2*E^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.
| \(\displaystyle \int \frac {e^x (8-4 x)+12 e^{2 x}+8 e^{3 x}+\left (6 e^x+12 e^{2 x}+8 e^{3 x}+1\right ) \log (x)+2}{\left (x+48 e^{2 x}+32 e^{3 x}+e^x (2 x+24)+\left (6 e^x x+12 e^{2 x} x+8 e^{3 x} x+x\right ) \log (x)+4\right ) \log \left (\frac {x+16 e^x+16 e^{2 x}+\left (4 e^x x+4 e^{2 x} x+x\right ) \log (x)+4}{4 e^x+4 e^{2 x}+1}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^x (8-4 x)+12 e^{2 x}+8 e^{3 x}+\left (6 e^x+12 e^{2 x}+8 e^{3 x}+1\right ) \log (x)+2}{\left (2 e^x+1\right ) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {x+16 e^x+16 e^{2 x}+\left (4 e^x x+4 e^{2 x} x+x\right ) \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {12 e^{2 x}+8 e^{3 x}-4 e^x (x-2)+\left (2 e^x+1\right )^3 \log (x)+2}{\left (2 e^x+1\right ) \left (16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {4 e^x x^2 \log ^2(x)+2 x^2 \log ^2(x)+2 x^2 \log (x)+9 x+64 e^x+32 e^x x \log (x)+16 x \log (x)+28}{(x \log (x)+4) \left (x+16 e^x+16 e^{2 x}+4 e^x x \log (x)+4 e^{2 x} x \log (x)+x \log (x)+4\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {\log (x)+1}{(x \log (x)+4) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}+\frac {2}{\left (2 e^x+1\right ) \log \left (\frac {16 e^x+16 e^{2 x}+x+x \left (2 e^x+1\right )^2 \log (x)+4}{\left (2 e^x+1\right )^2}\right )}\right )dx\) |
Input:
Int[(2 + 12*E^(2*x) + 8*E^(3*x) + E^x*(8 - 4*x) + (1 + 6*E^x + 12*E^(2*x) + 8*E^(3*x))*Log[x])/((4 + 48*E^(2*x) + 32*E^(3*x) + x + E^x*(24 + 2*x) + (x + 6*E^x*x + 12*E^(2*x)*x + 8*E^(3*x)*x)*Log[x])*Log[(4 + 16*E^x + 16*E^ (2*x) + x + (x + 4*E^x*x + 4*E^(2*x)*x)*Log[x])/(1 + 4*E^x + 4*E^(2*x))]), x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(20)=40\).
Time = 113.84 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29
| method | result | size |
| parallelrisch | \(\ln \left (\ln \left (\frac {\left (4 x \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} x +x \right ) \ln \left (x \right )+16 \,{\mathrm e}^{2 x}+16 \,{\mathrm e}^{x}+4+x}{4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1}\right )\right )\) | \(48\) |
| risch | \(\ln \left (\ln \left (\left ({\mathrm e}^{2 x} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )-\frac {i \left (\pi \,\operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}+\frac {1}{2}\right )^{2}}\right ) \operatorname {csgn}\left (i \left (\left ({\mathrm e}^{2 x} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{2 x} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )}{\left ({\mathrm e}^{x}+\frac {1}{2}\right )^{2}}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}+\frac {1}{2}\right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{2 x} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )}{\left ({\mathrm e}^{x}+\frac {1}{2}\right )^{2}}\right )}^{2}-\pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+\frac {1}{2}\right )\right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+\frac {1}{2}\right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+\frac {1}{2}\right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+\frac {1}{2}\right )^{2}\right )}^{2}-\pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+\frac {1}{2}\right )^{2}\right )}^{3}-\pi \,\operatorname {csgn}\left (i \left (\left ({\mathrm e}^{2 x} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{2 x} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )}{\left ({\mathrm e}^{x}+\frac {1}{2}\right )^{2}}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{2 x} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+\frac {\ln \left (x \right )}{4}+\frac {1}{4}\right ) x +4 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x}+1\right )}{\left ({\mathrm e}^{x}+\frac {1}{2}\right )^{2}}\right )}^{3}-4 i \ln \left ({\mathrm e}^{x}+\frac {1}{2}\right )\right )}{2}\right )\) | \(385\) |
Input:
int(((8*exp(x)^3+12*exp(x)^2+6*exp(x)+1)*ln(x)+8*exp(x)^3+12*exp(x)^2+(-4* x+8)*exp(x)+2)/((8*x*exp(x)^3+12*x*exp(x)^2+6*exp(x)*x+x)*ln(x)+32*exp(x)^ 3+48*exp(x)^2+(2*x+24)*exp(x)+4+x)/ln(((4*x*exp(x)^2+4*exp(x)*x+x)*ln(x)+1 6*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)),x,method=_RETURNVERBOSE )
Output:
ln(ln(((4*x*exp(x)^2+4*exp(x)*x+x)*ln(x)+16*exp(x)^2+16*exp(x)+4+x)/(4*exp (x)^2+4*exp(x)+1)))
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log \left (\log \left (\frac {{\left (4 \, x e^{\left (2 \, x\right )} + 4 \, x e^{x} + x\right )} \log \left (x\right ) + x + 16 \, e^{\left (2 \, x\right )} + 16 \, e^{x} + 4}{4 \, e^{\left (2 \, x\right )} + 4 \, e^{x} + 1}\right )\right ) \] Input:
integrate(((8*exp(x)^3+12*exp(x)^2+6*exp(x)+1)*log(x)+8*exp(x)^3+12*exp(x) ^2+(-4*x+8)*exp(x)+2)/((8*x*exp(x)^3+12*x*exp(x)^2+6*exp(x)*x+x)*log(x)+32 *exp(x)^3+48*exp(x)^2+(2*x+24)*exp(x)+4+x)/log(((4*x*exp(x)^2+4*exp(x)*x+x )*log(x)+16*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)),x, algorithm= "fricas")
Output:
log(log(((4*x*e^(2*x) + 4*x*e^x + x)*log(x) + x + 16*e^(2*x) + 16*e^x + 4) /(4*e^(2*x) + 4*e^x + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).
Time = 3.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log {\left (\log {\left (\frac {x + \left (4 x e^{2 x} + 4 x e^{x} + x\right ) \log {\left (x \right )} + 16 e^{2 x} + 16 e^{x} + 4}{4 e^{2 x} + 4 e^{x} + 1} \right )} \right )} \] Input:
integrate(((8*exp(x)**3+12*exp(x)**2+6*exp(x)+1)*ln(x)+8*exp(x)**3+12*exp( x)**2+(-4*x+8)*exp(x)+2)/((8*x*exp(x)**3+12*x*exp(x)**2+6*exp(x)*x+x)*ln(x )+32*exp(x)**3+48*exp(x)**2+(2*x+24)*exp(x)+4+x)/ln(((4*x*exp(x)**2+4*exp( x)*x+x)*ln(x)+16*exp(x)**2+16*exp(x)+4+x)/(4*exp(x)**2+4*exp(x)+1)),x)
Output:
log(log((x + (4*x*exp(2*x) + 4*x*exp(x) + x)*log(x) + 16*exp(2*x) + 16*exp (x) + 4)/(4*exp(2*x) + 4*exp(x) + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 6.75 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log \left (\log \left ({\left (4 \, x e^{\left (2 \, x\right )} + 4 \, x e^{x} + x\right )} \log \left (x\right ) + x + 16 \, e^{\left (2 \, x\right )} + 16 \, e^{x} + 4\right ) - 2 \, \log \left (2 \, e^{x} + 1\right )\right ) \] Input:
integrate(((8*exp(x)^3+12*exp(x)^2+6*exp(x)+1)*log(x)+8*exp(x)^3+12*exp(x) ^2+(-4*x+8)*exp(x)+2)/((8*x*exp(x)^3+12*x*exp(x)^2+6*exp(x)*x+x)*log(x)+32 *exp(x)^3+48*exp(x)^2+(2*x+24)*exp(x)+4+x)/log(((4*x*exp(x)^2+4*exp(x)*x+x )*log(x)+16*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)),x, algorithm= "maxima")
Output:
log(log((4*x*e^(2*x) + 4*x*e^x + x)*log(x) + x + 16*e^(2*x) + 16*e^x + 4) - 2*log(2*e^x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\log \left (-\log \left (4 \, x e^{\left (2 \, x\right )} \log \left (x\right ) + 4 \, x e^{x} \log \left (x\right ) + x \log \left (x\right ) + x + 16 \, e^{\left (2 \, x\right )} + 16 \, e^{x} + 4\right ) + \log \left (4 \, e^{\left (2 \, x\right )} + 4 \, e^{x} + 1\right )\right ) \] Input:
integrate(((8*exp(x)^3+12*exp(x)^2+6*exp(x)+1)*log(x)+8*exp(x)^3+12*exp(x) ^2+(-4*x+8)*exp(x)+2)/((8*x*exp(x)^3+12*x*exp(x)^2+6*exp(x)*x+x)*log(x)+32 *exp(x)^3+48*exp(x)^2+(2*x+24)*exp(x)+4+x)/log(((4*x*exp(x)^2+4*exp(x)*x+x )*log(x)+16*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)),x, algorithm= "giac")
Output:
log(-log(4*x*e^(2*x)*log(x) + 4*x*e^x*log(x) + x*log(x) + x + 16*e^(2*x) + 16*e^x + 4) + log(4*e^(2*x) + 4*e^x + 1))
Time = 4.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\ln \left (\ln \left (x+16\,{\mathrm {e}}^{2\,x}+16\,{\mathrm {e}}^x+x\,\ln \left (x\right )+4\,x\,{\mathrm {e}}^x\,\ln \left (x\right )+4\,x\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )+4\right )-\ln \left (4\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^x+1\right )\right ) \] Input:
int((12*exp(2*x) + 8*exp(3*x) + log(x)*(12*exp(2*x) + 8*exp(3*x) + 6*exp(x ) + 1) - exp(x)*(4*x - 8) + 2)/(log((x + 16*exp(2*x) + 16*exp(x) + log(x)* (x + 4*x*exp(2*x) + 4*x*exp(x)) + 4)/(4*exp(2*x) + 4*exp(x) + 1))*(x + 48* exp(2*x) + 32*exp(3*x) + exp(x)*(2*x + 24) + log(x)*(x + 12*x*exp(2*x) + 8 *x*exp(3*x) + 6*x*exp(x)) + 4)),x)
Output:
log(log(x + 16*exp(2*x) + 16*exp(x) + x*log(x) + 4*x*exp(x)*log(x) + 4*x*e xp(2*x)*log(x) + 4) - log(4*exp(2*x) + 4*exp(x) + 1))
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.67 \[ \int \frac {2+12 e^{2 x}+8 e^{3 x}+e^x (8-4 x)+\left (1+6 e^x+12 e^{2 x}+8 e^{3 x}\right ) \log (x)}{\left (4+48 e^{2 x}+32 e^{3 x}+x+e^x (24+2 x)+\left (x+6 e^x x+12 e^{2 x} x+8 e^{3 x} x\right ) \log (x)\right ) \log \left (\frac {4+16 e^x+16 e^{2 x}+x+\left (x+4 e^x x+4 e^{2 x} x\right ) \log (x)}{1+4 e^x+4 e^{2 x}}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\frac {4 e^{2 x} \mathrm {log}\left (x \right ) x +16 e^{2 x}+4 e^{x} \mathrm {log}\left (x \right ) x +16 e^{x}+\mathrm {log}\left (x \right ) x +x +4}{4 e^{2 x}+4 e^{x}+1}\right )\right ) \] Input:
int(((8*exp(x)^3+12*exp(x)^2+6*exp(x)+1)*log(x)+8*exp(x)^3+12*exp(x)^2+(-4 *x+8)*exp(x)+2)/((8*x*exp(x)^3+12*x*exp(x)^2+6*exp(x)*x+x)*log(x)+32*exp(x )^3+48*exp(x)^2+(2*x+24)*exp(x)+4+x)/log(((4*x*exp(x)^2+4*exp(x)*x+x)*log( x)+16*exp(x)^2+16*exp(x)+4+x)/(4*exp(x)^2+4*exp(x)+1)),x)
Output:
log(log((4*e**(2*x)*log(x)*x + 16*e**(2*x) + 4*e**x*log(x)*x + 16*e**x + l og(x)*x + x + 4)/(4*e**(2*x) + 4*e**x + 1)))