Integrand size = 114, antiderivative size = 16 \[ \int \frac {512-100 x+(2304+50 x) \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)}{512+2304 \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)} \, dx=\frac {3}{2}+x+\frac {25 x^2}{(2+\log (x))^8} \] Output:
3/2+25/(ln(x)+2)^8*x^2+x
Time = 0.11 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {512-100 x+(2304+50 x) \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)}{512+2304 \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)} \, dx=x+\frac {25 x^2}{(2+\log (x))^8} \] Input:
Integrate[(512 - 100*x + (2304 + 50*x)*Log[x] + 4608*Log[x]^2 + 5376*Log[x ]^3 + 4032*Log[x]^4 + 2016*Log[x]^5 + 672*Log[x]^6 + 144*Log[x]^7 + 18*Log [x]^8 + Log[x]^9)/(512 + 2304*Log[x] + 4608*Log[x]^2 + 5376*Log[x]^3 + 403 2*Log[x]^4 + 2016*Log[x]^5 + 672*Log[x]^6 + 144*Log[x]^7 + 18*Log[x]^8 + L og[x]^9),x]
Output:
x + (25*x^2)/(2 + Log[x])^8
Time = 0.85 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {7239, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-100 x+\log ^9(x)+18 \log ^8(x)+144 \log ^7(x)+672 \log ^6(x)+2016 \log ^5(x)+4032 \log ^4(x)+5376 \log ^3(x)+4608 \log ^2(x)+(50 x+2304) \log (x)+512}{\log ^9(x)+18 \log ^8(x)+144 \log ^7(x)+672 \log ^6(x)+2016 \log ^5(x)+4032 \log ^4(x)+5376 \log ^3(x)+4608 \log ^2(x)+2304 \log (x)+512} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-100 x+\log ^9(x)+18 \log ^8(x)+144 \log ^7(x)+672 \log ^6(x)+2016 \log ^5(x)+4032 \log ^4(x)+5376 \log ^3(x)+4608 \log ^2(x)+(50 x+2304) \log (x)+512}{(\log (x)+2)^9}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {50 x}{(\log (x)+2)^8}-\frac {200 x}{(\log (x)+2)^9}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {25 x^2}{(\log (x)+2)^8}+x\) |
Input:
Int[(512 - 100*x + (2304 + 50*x)*Log[x] + 4608*Log[x]^2 + 5376*Log[x]^3 + 4032*Log[x]^4 + 2016*Log[x]^5 + 672*Log[x]^6 + 144*Log[x]^7 + 18*Log[x]^8 + Log[x]^9)/(512 + 2304*Log[x] + 4608*Log[x]^2 + 5376*Log[x]^3 + 4032*Log[ x]^4 + 2016*Log[x]^5 + 672*Log[x]^6 + 144*Log[x]^7 + 18*Log[x]^8 + Log[x]^ 9),x]
Output:
x + (25*x^2)/(2 + Log[x])^8
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 4.92 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {25 x^{2}}{\left (\ln \left (x \right )+2\right )^{8}}+x\) | \(14\) |
| risch | \(\frac {25 x^{2}}{\left (\ln \left (x \right )+2\right )^{8}}+x\) | \(14\) |
| norman | \(\frac {x \ln \left (x \right )^{8}+256 x +25 x^{2}+1024 x \ln \left (x \right )+1792 x \ln \left (x \right )^{2}+1792 x \ln \left (x \right )^{3}+1120 x \ln \left (x \right )^{4}+448 x \ln \left (x \right )^{5}+112 x \ln \left (x \right )^{6}+16 x \ln \left (x \right )^{7}}{\left (\ln \left (x \right )+2\right )^{8}}\) | \(70\) |
| parallelrisch | \(\frac {286720 x +501760 x \ln \left (x \right )^{5}+125440 x \ln \left (x \right )^{6}+17920 x \ln \left (x \right )^{7}+1120 x \ln \left (x \right )^{8}+1254400 x \ln \left (x \right )^{4}+2007040 x \ln \left (x \right )^{3}+1146880 x \ln \left (x \right )+2007040 x \ln \left (x \right )^{2}+28000 x^{2}}{1120 \ln \left (x \right )^{8}+17920 \ln \left (x \right )^{7}+125440 \ln \left (x \right )^{6}+501760 \ln \left (x \right )^{5}+1254400 \ln \left (x \right )^{4}+2007040 \ln \left (x \right )^{3}+2007040 \ln \left (x \right )^{2}+1146880 \ln \left (x \right )+286720}\) | \(114\) |
Input:
int((ln(x)^9+18*ln(x)^8+144*ln(x)^7+672*ln(x)^6+2016*ln(x)^5+4032*ln(x)^4+ 5376*ln(x)^3+4608*ln(x)^2+(50*x+2304)*ln(x)-100*x+512)/(ln(x)^9+18*ln(x)^8 +144*ln(x)^7+672*ln(x)^6+2016*ln(x)^5+4032*ln(x)^4+5376*ln(x)^3+4608*ln(x) ^2+2304*ln(x)+512),x,method=_RETURNVERBOSE)
Output:
25/(ln(x)+2)^8*x^2+x
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (14) = 28\).
Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 6.94 \[ \int \frac {512-100 x+(2304+50 x) \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)}{512+2304 \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)} \, dx=\frac {x \log \left (x\right )^{8} + 16 \, x \log \left (x\right )^{7} + 112 \, x \log \left (x\right )^{6} + 448 \, x \log \left (x\right )^{5} + 1120 \, x \log \left (x\right )^{4} + 1792 \, x \log \left (x\right )^{3} + 1792 \, x \log \left (x\right )^{2} + 25 \, x^{2} + 1024 \, x \log \left (x\right ) + 256 \, x}{\log \left (x\right )^{8} + 16 \, \log \left (x\right )^{7} + 112 \, \log \left (x\right )^{6} + 448 \, \log \left (x\right )^{5} + 1120 \, \log \left (x\right )^{4} + 1792 \, \log \left (x\right )^{3} + 1792 \, \log \left (x\right )^{2} + 1024 \, \log \left (x\right ) + 256} \] Input:
integrate((log(x)^9+18*log(x)^8+144*log(x)^7+672*log(x)^6+2016*log(x)^5+40 32*log(x)^4+5376*log(x)^3+4608*log(x)^2+(50*x+2304)*log(x)-100*x+512)/(log (x)^9+18*log(x)^8+144*log(x)^7+672*log(x)^6+2016*log(x)^5+4032*log(x)^4+53 76*log(x)^3+4608*log(x)^2+2304*log(x)+512),x, algorithm="fricas")
Output:
(x*log(x)^8 + 16*x*log(x)^7 + 112*x*log(x)^6 + 448*x*log(x)^5 + 1120*x*log (x)^4 + 1792*x*log(x)^3 + 1792*x*log(x)^2 + 25*x^2 + 1024*x*log(x) + 256*x )/(log(x)^8 + 16*log(x)^7 + 112*log(x)^6 + 448*log(x)^5 + 1120*log(x)^4 + 1792*log(x)^3 + 1792*log(x)^2 + 1024*log(x) + 256)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (15) = 30\).
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.62 \[ \int \frac {512-100 x+(2304+50 x) \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)}{512+2304 \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)} \, dx=\frac {25 x^{2}}{\log {\left (x \right )}^{8} + 16 \log {\left (x \right )}^{7} + 112 \log {\left (x \right )}^{6} + 448 \log {\left (x \right )}^{5} + 1120 \log {\left (x \right )}^{4} + 1792 \log {\left (x \right )}^{3} + 1792 \log {\left (x \right )}^{2} + 1024 \log {\left (x \right )} + 256} + x \] Input:
integrate((ln(x)**9+18*ln(x)**8+144*ln(x)**7+672*ln(x)**6+2016*ln(x)**5+40 32*ln(x)**4+5376*ln(x)**3+4608*ln(x)**2+(50*x+2304)*ln(x)-100*x+512)/(ln(x )**9+18*ln(x)**8+144*ln(x)**7+672*ln(x)**6+2016*ln(x)**5+4032*ln(x)**4+537 6*ln(x)**3+4608*ln(x)**2+2304*ln(x)+512),x)
Output:
25*x**2/(log(x)**8 + 16*log(x)**7 + 112*log(x)**6 + 448*log(x)**5 + 1120*l og(x)**4 + 1792*log(x)**3 + 1792*log(x)**2 + 1024*log(x) + 256) + x
\[ \int \frac {512-100 x+(2304+50 x) \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)}{512+2304 \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)} \, dx=\int { \frac {\log \left (x\right )^{9} + 18 \, \log \left (x\right )^{8} + 144 \, \log \left (x\right )^{7} + 672 \, \log \left (x\right )^{6} + 2016 \, \log \left (x\right )^{5} + 4032 \, \log \left (x\right )^{4} + 5376 \, \log \left (x\right )^{3} + 2 \, {\left (25 \, x + 1152\right )} \log \left (x\right ) + 4608 \, \log \left (x\right )^{2} - 100 \, x + 512}{\log \left (x\right )^{9} + 18 \, \log \left (x\right )^{8} + 144 \, \log \left (x\right )^{7} + 672 \, \log \left (x\right )^{6} + 2016 \, \log \left (x\right )^{5} + 4032 \, \log \left (x\right )^{4} + 5376 \, \log \left (x\right )^{3} + 4608 \, \log \left (x\right )^{2} + 2304 \, \log \left (x\right ) + 512} \,d x } \] Input:
integrate((log(x)^9+18*log(x)^8+144*log(x)^7+672*log(x)^6+2016*log(x)^5+40 32*log(x)^4+5376*log(x)^3+4608*log(x)^2+(50*x+2304)*log(x)-100*x+512)/(log (x)^9+18*log(x)^8+144*log(x)^7+672*log(x)^6+2016*log(x)^5+4032*log(x)^4+53 76*log(x)^3+4608*log(x)^2+2304*log(x)+512),x, algorithm="maxima")
Output:
1/630*(630*x*log(x)^8 - 8*(25*x^2 - 1261*x)*log(x)^7 - 20*(145*x^2 - 3534* x)*log(x)^6 - 4*(4525*x^2 - 70756*x)*log(x)^5 - 6*(10525*x^2 - 118088*x)*l og(x)^4 - 4*(33375*x^2 - 283984*x)*log(x)^3 - 50*(3451*x^2 - 22816*x)*log( x)^2 - 39625*x^2 - 10*(13045*x^2 - 66496*x)*log(x) + 221312*x)/(log(x)^8 + 16*log(x)^7 + 112*log(x)^6 + 448*log(x)^5 + 1120*log(x)^4 + 1792*log(x)^3 + 1792*log(x)^2 + 1024*log(x) + 256) - 512*e^(-2)*exp_integral_e(9, -log( x) - 2)/(log(x) + 2)^8 + 100*e^(-4)*exp_integral_e(9, -2*log(x) - 4)/(log( x) + 2)^8 + integrate(4/315*(50*x - 1)/(log(x) + 2), x)
Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (14) = 28\).
Time = 0.14 (sec) , antiderivative size = 542, normalized size of antiderivative = 33.88 \[ \int \frac {512-100 x+(2304+50 x) \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)}{512+2304 \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)} \, dx =\text {Too large to display} \] Input:
integrate((log(x)^9+18*log(x)^8+144*log(x)^7+672*log(x)^6+2016*log(x)^5+40 32*log(x)^4+5376*log(x)^3+4608*log(x)^2+(50*x+2304)*log(x)-100*x+512)/(log (x)^9+18*log(x)^8+144*log(x)^7+672*log(x)^6+2016*log(x)^5+4032*log(x)^4+53 76*log(x)^3+4608*log(x)^2+2304*log(x)+512),x, algorithm="giac")
Output:
x*log(x)^8/(log(x)^8 + 16*log(x)^7 + 112*log(x)^6 + 448*log(x)^5 + 1120*lo g(x)^4 + 1792*log(x)^3 + 1792*log(x)^2 + 1024*log(x) + 256) + 16*x*log(x)^ 7/(log(x)^8 + 16*log(x)^7 + 112*log(x)^6 + 448*log(x)^5 + 1120*log(x)^4 + 1792*log(x)^3 + 1792*log(x)^2 + 1024*log(x) + 256) + 112*x*log(x)^6/(log(x )^8 + 16*log(x)^7 + 112*log(x)^6 + 448*log(x)^5 + 1120*log(x)^4 + 1792*log (x)^3 + 1792*log(x)^2 + 1024*log(x) + 256) + 448*x*log(x)^5/(log(x)^8 + 16 *log(x)^7 + 112*log(x)^6 + 448*log(x)^5 + 1120*log(x)^4 + 1792*log(x)^3 + 1792*log(x)^2 + 1024*log(x) + 256) + 1120*x*log(x)^4/(log(x)^8 + 16*log(x) ^7 + 112*log(x)^6 + 448*log(x)^5 + 1120*log(x)^4 + 1792*log(x)^3 + 1792*lo g(x)^2 + 1024*log(x) + 256) + 1792*x*log(x)^3/(log(x)^8 + 16*log(x)^7 + 11 2*log(x)^6 + 448*log(x)^5 + 1120*log(x)^4 + 1792*log(x)^3 + 1792*log(x)^2 + 1024*log(x) + 256) + 1792*x*log(x)^2/(log(x)^8 + 16*log(x)^7 + 112*log(x )^6 + 448*log(x)^5 + 1120*log(x)^4 + 1792*log(x)^3 + 1792*log(x)^2 + 1024* log(x) + 256) + 25*x^2/(log(x)^8 + 16*log(x)^7 + 112*log(x)^6 + 448*log(x) ^5 + 1120*log(x)^4 + 1792*log(x)^3 + 1792*log(x)^2 + 1024*log(x) + 256) + 1024*x*log(x)/(log(x)^8 + 16*log(x)^7 + 112*log(x)^6 + 448*log(x)^5 + 1120 *log(x)^4 + 1792*log(x)^3 + 1792*log(x)^2 + 1024*log(x) + 256) + 256*x/(lo g(x)^8 + 16*log(x)^7 + 112*log(x)^6 + 448*log(x)^5 + 1120*log(x)^4 + 1792* log(x)^3 + 1792*log(x)^2 + 1024*log(x) + 256)
Time = 3.15 (sec) , antiderivative size = 337, normalized size of antiderivative = 21.06 \[ \int \frac {512-100 x+(2304+50 x) \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)}{512+2304 \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)} \, dx=x-\frac {\frac {25\,x^2\,\ln \left (x\right )}{4}-\frac {25\,x^2}{2}}{{\ln \left (x\right )}^8+16\,{\ln \left (x\right )}^7+112\,{\ln \left (x\right )}^6+448\,{\ln \left (x\right )}^5+1120\,{\ln \left (x\right )}^4+1792\,{\ln \left (x\right )}^3+1792\,{\ln \left (x\right )}^2+1024\,\ln \left (x\right )+256}-\frac {\frac {5\,x^2\,\ln \left (x\right )}{21}-\frac {5\,x^2}{42}}{{\ln \left (x\right )}^5+10\,{\ln \left (x\right )}^4+40\,{\ln \left (x\right )}^3+80\,{\ln \left (x\right )}^2+80\,\ln \left (x\right )+32}-\frac {\frac {5\,x^2\,\ln \left (x\right )}{63}+\frac {5\,x^2}{63}}{{\ln \left (x\right )}^2+4\,\ln \left (x\right )+4}-\frac {\frac {25\,x^2\,\ln \left (x\right )}{14}-\frac {75\,x^2}{28}}{{\ln \left (x\right )}^7+14\,{\ln \left (x\right )}^6+84\,{\ln \left (x\right )}^5+280\,{\ln \left (x\right )}^4+560\,{\ln \left (x\right )}^3+672\,{\ln \left (x\right )}^2+448\,\ln \left (x\right )+128}-\frac {\frac {10\,x^2\,\ln \left (x\right )}{63}+\frac {5\,x^2}{21}}{\ln \left (x\right )+2}+\frac {10\,x^2}{63}-\frac {\frac {25\,x^2\,\ln \left (x\right )}{42}-\frac {25\,x^2}{42}}{{\ln \left (x\right )}^6+12\,{\ln \left (x\right )}^5+60\,{\ln \left (x\right )}^4+160\,{\ln \left (x\right )}^3+240\,{\ln \left (x\right )}^2+192\,\ln \left (x\right )+64}-\frac {\frac {5\,x^2\,\ln \left (x\right )}{63}+\frac {5\,x^2}{126}}{{\ln \left (x\right )}^3+6\,{\ln \left (x\right )}^2+12\,\ln \left (x\right )+8}-\frac {5\,x^2\,\ln \left (x\right )}{42\,\left ({\ln \left (x\right )}^4+8\,{\ln \left (x\right )}^3+24\,{\ln \left (x\right )}^2+32\,\ln \left (x\right )+16\right )} \] Input:
int((4608*log(x)^2 - 100*x + 5376*log(x)^3 + 4032*log(x)^4 + 2016*log(x)^5 + 672*log(x)^6 + 144*log(x)^7 + 18*log(x)^8 + log(x)^9 + log(x)*(50*x + 2 304) + 512)/(2304*log(x) + 4608*log(x)^2 + 5376*log(x)^3 + 4032*log(x)^4 + 2016*log(x)^5 + 672*log(x)^6 + 144*log(x)^7 + 18*log(x)^8 + log(x)^9 + 51 2),x)
Output:
x - ((25*x^2*log(x))/4 - (25*x^2)/2)/(1024*log(x) + 1792*log(x)^2 + 1792*l og(x)^3 + 1120*log(x)^4 + 448*log(x)^5 + 112*log(x)^6 + 16*log(x)^7 + log( x)^8 + 256) - ((5*x^2*log(x))/21 - (5*x^2)/42)/(80*log(x) + 80*log(x)^2 + 40*log(x)^3 + 10*log(x)^4 + log(x)^5 + 32) - ((5*x^2*log(x))/63 + (5*x^2)/ 63)/(4*log(x) + log(x)^2 + 4) - ((25*x^2*log(x))/14 - (75*x^2)/28)/(448*lo g(x) + 672*log(x)^2 + 560*log(x)^3 + 280*log(x)^4 + 84*log(x)^5 + 14*log(x )^6 + log(x)^7 + 128) - ((10*x^2*log(x))/63 + (5*x^2)/21)/(log(x) + 2) + ( 10*x^2)/63 - ((25*x^2*log(x))/42 - (25*x^2)/42)/(192*log(x) + 240*log(x)^2 + 160*log(x)^3 + 60*log(x)^4 + 12*log(x)^5 + log(x)^6 + 64) - ((5*x^2*log (x))/63 + (5*x^2)/126)/(12*log(x) + 6*log(x)^2 + log(x)^3 + 8) - (5*x^2*lo g(x))/(42*(32*log(x) + 24*log(x)^2 + 8*log(x)^3 + log(x)^4 + 16))
Time = 0.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 6.19 \[ \int \frac {512-100 x+(2304+50 x) \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)}{512+2304 \log (x)+4608 \log ^2(x)+5376 \log ^3(x)+4032 \log ^4(x)+2016 \log ^5(x)+672 \log ^6(x)+144 \log ^7(x)+18 \log ^8(x)+\log ^9(x)} \, dx=\frac {x \left (\mathrm {log}\left (x \right )^{8}+16 \mathrm {log}\left (x \right )^{7}+112 \mathrm {log}\left (x \right )^{6}+448 \mathrm {log}\left (x \right )^{5}+1120 \mathrm {log}\left (x \right )^{4}+1792 \mathrm {log}\left (x \right )^{3}+1792 \mathrm {log}\left (x \right )^{2}+1024 \,\mathrm {log}\left (x \right )+25 x +256\right )}{\mathrm {log}\left (x \right )^{8}+16 \mathrm {log}\left (x \right )^{7}+112 \mathrm {log}\left (x \right )^{6}+448 \mathrm {log}\left (x \right )^{5}+1120 \mathrm {log}\left (x \right )^{4}+1792 \mathrm {log}\left (x \right )^{3}+1792 \mathrm {log}\left (x \right )^{2}+1024 \,\mathrm {log}\left (x \right )+256} \] Input:
int((log(x)^9+18*log(x)^8+144*log(x)^7+672*log(x)^6+2016*log(x)^5+4032*log (x)^4+5376*log(x)^3+4608*log(x)^2+(50*x+2304)*log(x)-100*x+512)/(log(x)^9+ 18*log(x)^8+144*log(x)^7+672*log(x)^6+2016*log(x)^5+4032*log(x)^4+5376*log (x)^3+4608*log(x)^2+2304*log(x)+512),x)
Output:
(x*(log(x)**8 + 16*log(x)**7 + 112*log(x)**6 + 448*log(x)**5 + 1120*log(x) **4 + 1792*log(x)**3 + 1792*log(x)**2 + 1024*log(x) + 25*x + 256))/(log(x) **8 + 16*log(x)**7 + 112*log(x)**6 + 448*log(x)**5 + 1120*log(x)**4 + 1792 *log(x)**3 + 1792*log(x)**2 + 1024*log(x) + 256)