101.41 Problem number 7352

\[ \int \frac {-16 x^2-68 x^3-32 x^4-4 x^5+e^2 \left (64 x^3+32 x^4+4 x^5\right )+\left (4 x^3+e^2 \left (16 x^2+4 x^3\right )\right ) \log (x)+\left (64 x^3+32 x^4+4 x^5+\left (16 x^2+4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\left (-32 x-120 x^2+4 x^3+24 x^4+4 x^5+e^2 \left (128 x^2-24 x^4-4 x^5\right )+\left (8 x^2-4 x^3+e^2 \left (32 x-8 x^2-4 x^3\right )\right ) \log (x)+\left (128 x^2-24 x^4-4 x^5+\left (32 x-8 x^2-4 x^3\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log \left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-64 x^2-32 x^3-4 x^4\right )+e^2 \left (-16 x-4 x^2\right ) \log (x)+\left (-64 x^2-32 x^3-4 x^4+\left (-16 x-4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^2\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )+\left (e^2 \left (-128 x+24 x^3+4 x^4\right )+e^2 \left (-32+8 x+4 x^2\right ) \log (x)+\left (-128 x+24 x^3+4 x^4+\left (-32+8 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )}{\left (e^2 \left (16 x^4+8 x^5+x^6\right )+e^2 \left (4 x^3+x^4\right ) \log (x)+\left (16 x^4+8 x^5+x^6+\left (4 x^3+x^4\right ) \log (x)\right ) \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )\right ) \log ^3\left (\frac {e^4+2 e^2 \log \left (\frac {4 x+x^2+\log (x)}{4+x}\right )+\log ^2\left (\frac {4 x+x^2+\log (x)}{4+x}\right )}{x^2}\right )} \, dx \]

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

integrate(((((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)

Sympy 1.10.1 under Python 3.10.4 output

\[ \frac {x + \left (4 - 2 x\right ) \log {\left (\frac {\log {\left (\frac {x^{2} + 4 x + \log {\left (x \right )}}{x + 4} \right )}^{2} + 2 e^{2} \log {\left (\frac {x^{2} + 4 x + \log {\left (x \right )}}{x + 4} \right )} + e^{4}}{x^{2}} \right )}}{x \log {\left (\frac {\log {\left (\frac {x^{2} + 4 x + \log {\left (x \right )}}{x + 4} \right )}^{2} + 2 e^{2} \log {\left (\frac {x^{2} + 4 x + \log {\left (x \right )}}{x + 4} \right )} + e^{4}}{x^{2}} \right )}^{2}} + \frac {4 - 4 x}{x^{2}} \]

Sympy 1.8 under Python 3.8.8 output

\[ \text {Timed out} \]________________________________________________________________________________________