\[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (\frac {x}{\ln \left (\frac {\left ({\mathrm e}^{2}+\ln \left (\frac {\ln \left (x \right )}{4+x}+x \right )\right )^{2}}{x^{2}}\right )}-x +2\right )^{2}}{x^{2}} \]
command
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+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)
Maple 2022.1 output
\[\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}}\, dx\]
Maple 2021.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(6785\) |