\[ \int \frac {8000+4800 x+960 x^2-56 x^3-12 x^4+\left (6000+2400 x+240 x^2-35 x^3\right ) \log (x)+(1500+300 x) \log ^2(x)+125 \log ^3(x)}{\left (8000 x+4800 x^2+960 x^3+44 x^4-4 x^5+\left (6000 x+2400 x^2+240 x^3-5 x^4\right ) \log (x)+\left (1500 x+300 x^2\right ) \log ^2(x)+125 x \log ^3(x)\right ) \log \left (\frac {160000 x+128000 x^2+38400 x^3+4320 x^4-64 x^5-32 x^6+x^7+\left (160000 x+96000 x^2+19200 x^3+880 x^4-80 x^5\right ) \log (x)+\left (60000 x+24000 x^2+2400 x^3-50 x^4\right ) \log ^2(x)+\left (10000 x+2000 x^2\right ) \log ^3(x)+625 x \log ^4(x)}{160000+128000 x+38400 x^2+5120 x^3+256 x^4+\left (160000+96000 x+19200 x^2+1280 x^3\right ) \log (x)+\left (60000+24000 x+2400 x^2\right ) \log ^2(x)+(10000+2000 x) \log ^3(x)+625 \log ^4(x)}\right )} \, dx \]
Optimal antiderivative \[ \ln \! \left (\ln \! \left (\frac {\left (x -\frac {x^{4}}{\left (4 x +20+5 \ln \left (x \right )\right )^{2}}\right )^{2}}{x}\right )\right ) \]
command
integrate((125*log(x)^3+(300*x+1500)*log(x)^2+(-35*x^3+240*x^2+2400*x+6000)*log(x)-12*x^4-56*x^3+960*x^2+4800*x+8000)/(125*x*log(x)^3+(300*x^2+1500*x)*log(x)^2+(-5*x^4+240*x^3+2400*x^2+6000*x)*log(x)-4*x^5+44*x^4+960*x^3+4800*x^2+8000*x)/log((625*x*log(x)^4+(2000*x^2+10000*x)*log(x)^3+(-50*x^4+2400*x^3+24000*x^2+60000*x)*log(x)^2+(-80*x^5+880*x^4+19200*x^3+96000*x^2+160000*x)*log(x)+x^7-32*x^6-64*x^5+4320*x^4+38400*x^3+128000*x^2+160000*x)/(625*log(x)^4+(2000*x+10000)*log(x)^3+(2400*x^2+24000*x+60000)*log(x)^2+(1280*x^3+19200*x^2+96000*x+160000)*log(x)+256*x^4+5120*x^3+38400*x^2+128000*x+160000)),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
\[ \log \left (-\log \left (x^{6} - 32 \, x^{5} - 80 \, x^{4} \log \left (x\right ) - 50 \, x^{3} \log \left (x\right )^{2} - 64 \, x^{4} + 880 \, x^{3} \log \left (x\right ) + 2400 \, x^{2} \log \left (x\right )^{2} + 2000 \, x \log \left (x\right )^{3} + 625 \, \log \left (x\right )^{4} + 4320 \, x^{3} + 19200 \, x^{2} \log \left (x\right ) + 24000 \, x \log \left (x\right )^{2} + 10000 \, \log \left (x\right )^{3} + 38400 \, x^{2} + 96000 \, x \log \left (x\right ) + 60000 \, \log \left (x\right )^{2} + 128000 \, x + 160000 \, \log \left (x\right ) + 160000\right ) + \log \left (256 \, x^{4} + 1280 \, x^{3} \log \left (x\right ) + 2400 \, x^{2} \log \left (x\right )^{2} + 2000 \, x \log \left (x\right )^{3} + 625 \, \log \left (x\right )^{4} + 5120 \, x^{3} + 19200 \, x^{2} \log \left (x\right ) + 24000 \, x \log \left (x\right )^{2} + 10000 \, \log \left (x\right )^{3} + 38400 \, x^{2} + 96000 \, x \log \left (x\right ) + 60000 \, \log \left (x\right )^{2} + 128000 \, x + 160000 \, \log \left (x\right ) + 160000\right ) - \log \left (x\right )\right ) \]