\[ \int \frac {x+(4+8 x) \log \left (\frac {2+4 x}{x}\right )}{(16+32 x) \log \left (\frac {2+4 x}{x}\right )+\left (8 x+16 x^2\right ) \log \left (\frac {2+4 x}{x}\right ) \log \left (\log \left (\frac {2+4 x}{x}\right )\right )+\left (x^2+2 x^3\right ) \log \left (\frac {2+4 x}{x}\right ) \log ^2\left (\log \left (\frac {2+4 x}{x}\right )\right )} \, dx \]
Optimal antiderivative \[ -1+\frac {x}{4+\ln \left (\ln \left (\frac {2}{x}+4\right )\right ) x} \]
command
int(((8*x+4)*ln((4*x+2)/x)+x)/((2*x^3+x^2)*ln((4*x+2)/x)*ln(ln((4*x+2)/x))^2+(16*x^2+8*x)*ln((4*x+2)/x)*ln(ln((4*x+2)/x))+(32*x+16)*ln((4*x+2)/x)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\frac {1}{-8+\frac {8 x +4}{x}+\ln \left (\ln \left (2\right )+\ln \left (\frac {2 x +1}{x}\right )\right )}\) | \(29\) |
Maple 2021.1 output
\[\int \frac {\left (8 x +4\right ) \ln \left (\frac {4 x +2}{x}\right )+x}{\left (2 x^{3}+x^{2}\right ) \ln \left (\frac {4 x +2}{x}\right ) \ln \left (\ln \left (\frac {4 x +2}{x}\right )\right )^{2}+\left (16 x^{2}+8 x \right ) \ln \left (\frac {4 x +2}{x}\right ) \ln \left (\ln \left (\frac {4 x +2}{x}\right )\right )+\left (32 x +16\right ) \ln \left (\frac {4 x +2}{x}\right )}\, dx\]________________________________________________________________________________________