\[ \int \frac {-1-x+e^{5 x} (1+x)+\left (2-2 e^{5 x}\right ) \log (x)+\left (-x+e^{5 x} x+\left (1-e^{5 x}\right ) \log (x)\right ) \log (-x+\log (x))+\left (2 x+e^{5 x} \left (-2 x-10 x^2\right )+\left (-2+e^{5 x} (2+10 x)\right ) \log (x)+\left (x+e^{5 x} \left (-x-5 x^2\right )+\left (-1+e^{5 x} (1+5 x)\right ) \log (x)\right ) \log (-x+\log (x))\right ) \log \left (-\frac {x}{2+\log (-x+\log (x))}\right )}{(-2 x+2 \log (x)+(-x+\log (x)) \log (-x+\log (x))) \log ^2\left (-\frac {x}{2+\log (-x+\log (x))}\right )} \, dx \]
Optimal antiderivative \[ 1+\frac {x \left ({\mathrm e}^{5 x}-1\right )}{\ln \! \left (\frac {x}{-2-\ln \left (\ln \left (x \right )-x \right )}\right )} \]
command
int((((((1+5*x)*exp(5*x)-1)*ln(x)+(-5*x^2-x)*exp(5*x)+x)*ln(ln(x)-x)+((10*x+2)*exp(5*x)-2)*ln(x)+(-10*x^2-2*x)*exp(5*x)+2*x)*ln(-x/(ln(ln(x)-x)+2))+((-exp(5*x)+1)*ln(x)+x*exp(5*x)-x)*ln(ln(x)-x)+(-2*exp(5*x)+2)*ln(x)+(x+1)*exp(5*x)-x-1)/((ln(x)-x)*ln(ln(x)-x)+2*ln(x)-2*x)/ln(-x/(ln(ln(x)-x)+2))^2,x)
Maple 2022.1 output
\[\int \frac {\left (\left (\left (\left (1+5 x \right ) {\mathrm e}^{5 x}-1\right ) \ln \left (x \right )+\left (-5 x^{2}-x \right ) {\mathrm e}^{5 x}+x \right ) \ln \left (\ln \left (x \right )-x \right )+\left (\left (10 x +2\right ) {\mathrm e}^{5 x}-2\right ) \ln \left (x \right )+\left (-10 x^{2}-2 x \right ) {\mathrm e}^{5 x}+2 x \right ) \ln \left (-\frac {x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )+\left (\left (-{\mathrm e}^{5 x}+1\right ) \ln \left (x \right )+x \,{\mathrm e}^{5 x}-x \right ) \ln \left (\ln \left (x \right )-x \right )+\left (-2 \,{\mathrm e}^{5 x}+2\right ) \ln \left (x \right )+\left (x +1\right ) {\mathrm e}^{5 x}-x -1}{\left (\left (\ln \left (x \right )-x \right ) \ln \left (\ln \left (x \right )-x \right )+2 \ln \left (x \right )-2 x \right ) \ln \left (-\frac {x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{2}}\, dx\]
Maple 2021.1 output
| method | result | size |
| risch | \(-\frac {2 i x \left ({\mathrm e}^{5 x}-1\right )}{\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )-x \right )+2}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )-x \right )+2}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right ) \mathrm {csgn}\left (i x \right )+\pi \mathrm {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{3}-2 \pi \mathrm {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i x}{\ln \left (\ln \left (x \right )-x \right )+2}\right )^{2} \mathrm {csgn}\left (i x \right )+2 \pi -2 i \ln \left (x \right )+2 i \ln \left (\ln \left (\ln \left (x \right )-x \right )+2\right )}\) | \(175\) |