\[ \int \frac {-12 x^2-4 x^3-2 x^4+e^3 \left (5+12 x+4 x^2+2 x^3\right )+\left (-6 x^2-2 x^3-x^4+e^3 \left (6 x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )}{-2 x^2-4 x^3-2 x^4+e^3 \left (2 x+4 x^2+2 x^3\right )+\left (-x^2-2 x^3-x^4+e^3 \left (x+2 x^2+x^3\right )\right ) \log \left (\frac {-e^3+x}{x}\right )+\left (4 x^2+4 x^3+e^3 \left (-4 x-4 x^2\right )+\left (2 x^2+2 x^3+e^3 \left (-2 x-2 x^2\right )\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log \left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )+\left (2 e^3 x-2 x^2+\left (e^3 x-x^2\right ) \log \left (\frac {-e^3+x}{x}\right )\right ) \log ^2\left (8+4 \log \left (\frac {-e^3+x}{x}\right )\right )} \, dx \]
Optimal antiderivative \[ -1+\frac {5}{\ln \left (4 \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+8\right )-1-x}+x \]
command
int((((x*exp(3)-x^2)*ln((x-exp(3))/x)+2*x*exp(3)-2*x^2)*ln(4*ln((x-exp(3))/x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*ln((x-exp(3))/x)+(-4*x^2-4*x)*exp(3)+4*x^3+4*x^2)*ln(4*ln((x-exp(3))/x)+8)+((x^3+2*x^2+6*x)*exp(3)-x^4-2*x^3-6*x^2)*ln((x-exp(3))/x)+(2*x^3+4*x^2+12*x+5)*exp(3)-2*x^4-4*x^3-12*x^2)/(((x*exp(3)-x^2)*ln((x-exp(3))/x)+2*x*exp(3)-2*x^2)*ln(4*ln((x-exp(3))/x)+8)^2+(((-2*x^2-2*x)*exp(3)+2*x^3+2*x^2)*ln((x-exp(3))/x)+(-4*x^2-4*x)*exp(3)+4*x^3+4*x^2)*ln(4*ln((x-exp(3))/x)+8)+((x^3+2*x^2+x)*exp(3)-x^4-2*x^3-x^2)*ln((x-exp(3))/x)+(2*x^3+4*x^2+2*x)*exp(3)-2*x^4-4*x^3-2*x^2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(-\frac {{\mathrm e}^{3}}{-\frac {-x +{\mathrm e}^{3}}{x}-1}+\frac {-\frac {5 \left (-x +{\mathrm e}^{3}\right )}{x}-5}{-\frac {2 \ln \left (2\right ) \left (-x +{\mathrm e}^{3}\right )}{x}-\frac {\ln \left (\ln \left (-\frac {-x +{\mathrm e}^{3}}{x}\right )+2\right ) \left (-x +{\mathrm e}^{3}\right )}{x}+{\mathrm e}^{3}-2 \ln \left (2\right )+\frac {-x +{\mathrm e}^{3}}{x}-\ln \left (\ln \left (-\frac {-x +{\mathrm e}^{3}}{x}\right )+2\right )+1}\) | \(112\) |
Maple 2021.1 output
\[\int \frac {\left (\left (x \,{\mathrm e}^{3}-x^{2}\right ) \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+2 x \,{\mathrm e}^{3}-2 x^{2}\right ) \ln \left (4 \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+8\right )^{2}+\left (\left (\left (-2 x^{2}-2 x \right ) {\mathrm e}^{3}+2 x^{3}+2 x^{2}\right ) \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+\left (-4 x^{2}-4 x \right ) {\mathrm e}^{3}+4 x^{3}+4 x^{2}\right ) \ln \left (4 \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+8\right )+\left (\left (x^{3}+2 x^{2}+6 x \right ) {\mathrm e}^{3}-x^{4}-2 x^{3}-6 x^{2}\right ) \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+\left (2 x^{3}+4 x^{2}+12 x +5\right ) {\mathrm e}^{3}-2 x^{4}-4 x^{3}-12 x^{2}}{\left (\left (x \,{\mathrm e}^{3}-x^{2}\right ) \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+2 x \,{\mathrm e}^{3}-2 x^{2}\right ) \ln \left (4 \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+8\right )^{2}+\left (\left (\left (-2 x^{2}-2 x \right ) {\mathrm e}^{3}+2 x^{3}+2 x^{2}\right ) \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+\left (-4 x^{2}-4 x \right ) {\mathrm e}^{3}+4 x^{3}+4 x^{2}\right ) \ln \left (4 \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+8\right )+\left (\left (x^{3}+2 x^{2}+x \right ) {\mathrm e}^{3}-x^{4}-2 x^{3}-x^{2}\right ) \ln \left (\frac {x -{\mathrm e}^{3}}{x}\right )+\left (2 x^{3}+4 x^{2}+2 x \right ) {\mathrm e}^{3}-2 x^{4}-4 x^{3}-2 x^{2}}\, dx\]________________________________________________________________________________________