\[ \int \frac {e^{-\frac {6}{e^{e^{5 x}}-x^4-2 x^3 \log (5)-x^2 \log ^2(5)+\left (-2 x^3-2 x^2 \log (5)\right ) \log (x)-x^2 \log ^2(x)}} \left (-30 e^{e^{5 x}+5 x}+12 x^2+24 x^3+\left (12 x+36 x^2\right ) \log (5)+12 x \log ^2(5)+\left (12 x+36 x^2+24 x \log (5)\right ) \log (x)+12 x \log ^2(x)\right )}{e^{2 e^{5 x}}+x^8+4 x^7 \log (5)+6 x^6 \log ^2(5)+4 x^5 \log ^3(5)+x^4 \log ^4(5)+\left (4 x^7+12 x^6 \log (5)+12 x^5 \log ^2(5)+4 x^4 \log ^3(5)\right ) \log (x)+\left (6 x^6+12 x^5 \log (5)+6 x^4 \log ^2(5)\right ) \log ^2(x)+\left (4 x^5+4 x^4 \log (5)\right ) \log ^3(x)+x^4 \log ^4(x)+e^{e^{5 x}} \left (-2 x^4-4 x^3 \log (5)-2 x^2 \log ^2(5)+\left (-4 x^3-4 x^2 \log (5)\right ) \log (x)-2 x^2 \log ^2(x)\right )} \, dx \]
Optimal antiderivative \[ 4-{\mathrm e}^{\frac {6}{x^{2} \left (\ln \left (5 x \right )+x \right )^{2}-{\mathrm e}^{{\mathrm e}^{5 x}}}} \]
command
integrate((-30*exp(5*x)*exp(exp(5*x))+12*x*log(x)^2+(24*x*log(5)+36*x^2+12*x)*log(x)+12*x*log(5)^2+(36*x^2+12*x)*log(5)+24*x^3+12*x^2)*exp(-3/(exp(exp(5*x))-x^2*log(x)^2+(-2*x^2*log(5)-2*x^3)*log(x)-x^2*log(5)^2-2*x^3*log(5)-x^4))^2/(exp(exp(5*x))^2+(-2*x^2*log(x)^2+(-4*x^2*log(5)-4*x^3)*log(x)-2*x^2*log(5)^2-4*x^3*log(5)-2*x^4)*exp(exp(5*x))+x^4*log(x)^4+(4*x^4*log(5)+4*x^5)*log(x)^3+(6*x^4*log(5)^2+12*x^5*log(5)+6*x^6)*log(x)^2+(4*x^4*log(5)^3+12*x^5*log(5)^2+12*x^6*log(5)+4*x^7)*log(x)+x^4*log(5)^4+4*x^5*log(5)^3+6*x^6*log(5)^2+4*x^7*log(5)+x^8),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -e^{\left (\frac {6}{x^{4} + 2 \, x^{3} \log \left (5\right ) + x^{2} \log \left (5\right )^{2} + 2 \, x^{3} \log \left (x\right ) + 2 \, x^{2} \log \left (5\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - e^{\left (e^{\left (5 \, x\right )}\right )}}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {6 \, {\left (4 \, x^{3} + 2 \, x \log \left (5\right )^{2} + 2 \, x \log \left (x\right )^{2} + 2 \, x^{2} + 2 \, {\left (3 \, x^{2} + x\right )} \log \left (5\right ) + 2 \, {\left (3 \, x^{2} + 2 \, x \log \left (5\right ) + x\right )} \log \left (x\right ) - 5 \, e^{\left (5 \, x + e^{\left (5 \, x\right )}\right )}\right )} e^{\left (\frac {6}{x^{4} + 2 \, x^{3} \log \left (5\right ) + x^{2} \log \left (5\right )^{2} + x^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{3} + x^{2} \log \left (5\right )\right )} \log \left (x\right ) - e^{\left (e^{\left (5 \, x\right )}\right )}}\right )}}{x^{8} + 4 \, x^{7} \log \left (5\right ) + 6 \, x^{6} \log \left (5\right )^{2} + 4 \, x^{5} \log \left (5\right )^{3} + x^{4} \log \left (5\right )^{4} + x^{4} \log \left (x\right )^{4} + 4 \, {\left (x^{5} + x^{4} \log \left (5\right )\right )} \log \left (x\right )^{3} + 6 \, {\left (x^{6} + 2 \, x^{5} \log \left (5\right ) + x^{4} \log \left (5\right )^{2}\right )} \log \left (x\right )^{2} - 2 \, {\left (x^{4} + 2 \, x^{3} \log \left (5\right ) + x^{2} \log \left (5\right )^{2} + x^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{3} + x^{2} \log \left (5\right )\right )} \log \left (x\right )\right )} e^{\left (e^{\left (5 \, x\right )}\right )} + 4 \, {\left (x^{7} + 3 \, x^{6} \log \left (5\right ) + 3 \, x^{5} \log \left (5\right )^{2} + x^{4} \log \left (5\right )^{3}\right )} \log \left (x\right ) + e^{\left (2 \, e^{\left (5 \, x\right )}\right )}}\,{d x} \]________________________________________________________________________________________