\[ \int \frac {-1500 e^{15-3 x} x^5+e^{20-4 x} \left (-2500 x^5+2500 x^6\right )+e^x \left (e^{20-4 x} \left (2500 x^3-2500 x^4\right )+e^{15-3 x} \left (-1500 x^3+1500 x^4\right )\right )+\left (-2700 e^{10-2 x} x^5+e^{15-3 x} \left (-6000 x^5+4500 x^6\right )+e^x \left (e^{15-3 x} \left (6000 x^3-4500 x^4\right )+e^{10-2 x} \left (-2700 x^3+2700 x^4\right )\right )\right ) \log \left (\frac {-e^x+x^2}{2 x}\right )+\left (-1620 e^{5-x} x^5+e^{10-2 x} \left (-5400 x^5+2700 x^6\right )+e^x \left (e^{10-2 x} \left (5400 x^3-2700 x^4\right )+e^{5-x} \left (-1620 x^3+1620 x^4\right )\right )\right ) \log ^2\left (\frac {-e^x+x^2}{2 x}\right )+\left (-324 x^5+e^{5-x} \left (-2160 x^5+540 x^6\right )+e^x \left (-324 x^3+324 x^4+e^{5-x} \left (2160 x^3-540 x^4\right )\right )\right ) \log ^3\left (\frac {-e^x+x^2}{2 x}\right )+\left (324 e^x x^3-324 x^5\right ) \log ^4\left (\frac {-e^x+x^2}{2 x}\right )}{81 e^x-81 x^2} \, dx \]
Optimal antiderivative \[ \left (x \ln \! \left (\frac {x}{2}-\frac {{\mathrm e}^{x}}{2 x}\right )+\frac {5 x \,{\mathrm e}^{5-x}}{3}\right )^{4} \]
command
integrate(((324*exp(x)*x^3-324*x^5)*log(1/2*(-exp(x)+x^2)/x)^4+(((-540*x^4+2160*x^3)*exp(5-x)+324*x^4-324*x^3)*exp(x)+(540*x^6-2160*x^5)*exp(5-x)-324*x^5)*log(1/2*(-exp(x)+x^2)/x)^3+(((-2700*x^4+5400*x^3)*exp(5-x)^2+(1620*x^4-1620*x^3)*exp(5-x))*exp(x)+(2700*x^6-5400*x^5)*exp(5-x)^2-1620*x^5*exp(5-x))*log(1/2*(-exp(x)+x^2)/x)^2+(((-4500*x^4+6000*x^3)*exp(5-x)^3+(2700*x^4-2700*x^3)*exp(5-x)^2)*exp(x)+(4500*x^6-6000*x^5)*exp(5-x)^3-2700*x^5*exp(5-x)^2)*log(1/2*(-exp(x)+x^2)/x)+((-2500*x^4+2500*x^3)*exp(5-x)^4+(1500*x^4-1500*x^3)*exp(5-x)^3)*exp(x)+(2500*x^6-2500*x^5)*exp(5-x)^4-1500*x^5*exp(5-x)^3)/(81*exp(x)-81*x^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Timed out} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {1}{81} \, {\left (81 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right )^{4} - 324 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right )^{3} \log \left (x^{2} - e^{x}\right ) + 486 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right )^{2} \log \left (x^{2} - e^{x}\right )^{2} - 324 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right ) \log \left (x^{2} - e^{x}\right )^{3} + 81 \, x^{4} e^{\left (4 \, x\right )} \log \left (x^{2} - e^{x}\right )^{4} + 324 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right )^{3} \log \left (x\right ) - 972 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right )^{2} \log \left (x^{2} - e^{x}\right ) \log \left (x\right ) + 972 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right ) \log \left (x^{2} - e^{x}\right )^{2} \log \left (x\right ) - 324 \, x^{4} e^{\left (4 \, x\right )} \log \left (x^{2} - e^{x}\right )^{3} \log \left (x\right ) + 486 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right )^{2} \log \left (x\right )^{2} - 972 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right ) \log \left (x^{2} - e^{x}\right ) \log \left (x\right )^{2} + 486 \, x^{4} e^{\left (4 \, x\right )} \log \left (x^{2} - e^{x}\right )^{2} \log \left (x\right )^{2} + 324 \, x^{4} e^{\left (4 \, x\right )} \log \left (2\right ) \log \left (x\right )^{3} - 324 \, x^{4} e^{\left (4 \, x\right )} \log \left (x^{2} - e^{x}\right ) \log \left (x\right )^{3} + 81 \, x^{4} e^{\left (4 \, x\right )} \log \left (x\right )^{4} - 540 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (2\right )^{3} + 1620 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (2\right )^{2} \log \left (x^{2} - e^{x}\right ) - 1620 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (2\right ) \log \left (x^{2} - e^{x}\right )^{2} + 540 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (x^{2} - e^{x}\right )^{3} - 1620 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (2\right )^{2} \log \left (x\right ) + 3240 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (2\right ) \log \left (x^{2} - e^{x}\right ) \log \left (x\right ) - 1620 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (x^{2} - e^{x}\right )^{2} \log \left (x\right ) - 1620 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (2\right ) \log \left (x\right )^{2} + 1620 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (x^{2} - e^{x}\right ) \log \left (x\right )^{2} - 540 \, x^{4} e^{\left (3 \, x + 5\right )} \log \left (x\right )^{3} + 1350 \, x^{4} e^{\left (2 \, x + 10\right )} \log \left (2\right )^{2} - 2700 \, x^{4} e^{\left (2 \, x + 10\right )} \log \left (2\right ) \log \left (x^{2} - e^{x}\right ) + 1350 \, x^{4} e^{\left (2 \, x + 10\right )} \log \left (x^{2} - e^{x}\right )^{2} + 2700 \, x^{4} e^{\left (2 \, x + 10\right )} \log \left (2\right ) \log \left (x\right ) - 2700 \, x^{4} e^{\left (2 \, x + 10\right )} \log \left (x^{2} - e^{x}\right ) \log \left (x\right ) + 1350 \, x^{4} e^{\left (2 \, x + 10\right )} \log \left (x\right )^{2} - 1500 \, x^{4} e^{\left (x + 15\right )} \log \left (2\right ) + 1500 \, x^{4} e^{\left (x + 15\right )} \log \left (x^{2} - e^{x}\right ) - 1500 \, x^{4} e^{\left (x + 15\right )} \log \left (x\right ) + 625 \, x^{4} e^{20}\right )} e^{\left (-4 \, x\right )} \]