Integrand size = 103, antiderivative size = 25 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {\left (1-\frac {1}{x}\right ) x^2}{x+(2 (5+x)+\log (x))^2} \]
Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {(-1+x) x}{100+41 x+4 x^2+4 (5+x) \log (x)+\log ^2(x)} \]
Integrate[(-80 + 184*x + 41*x^2 + (-18 + 38*x + 4*x^2)*Log[x] + (-1 + 2*x) *Log[x]^2)/(10000 + 8200*x + 2481*x^2 + 328*x^3 + 16*x^4 + (4000 + 2440*x + 488*x^2 + 32*x^3)*Log[x] + (600 + 242*x + 24*x^2)*Log[x]^2 + (40 + 8*x)* Log[x]^3 + Log[x]^4),x]
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.
\(\displaystyle \int \frac {41 x^2+\left (4 x^2+38 x-18\right ) \log (x)+184 x+(2 x-1) \log ^2(x)-80}{16 x^4+328 x^3+2481 x^2+\left (24 x^2+242 x+600\right ) \log ^2(x)+\left (32 x^3+488 x^2+2440 x+4000\right ) \log (x)+8200 x+\log ^4(x)+(8 x+40) \log ^3(x)+10000} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {41 x^2+2 \left (2 x^2+19 x-9\right ) \log (x)+184 x+(2 x-1) \log ^2(x)-80}{\left (4 x^2+41 x+\log ^2(x)+4 (x+5) \log (x)+100\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x-1}{4 x^2+41 x+\log ^2(x)+4 x \log (x)+20 \log (x)+100}-\frac {(x-1) \left (8 x^2+45 x+4 x \log (x)+2 \log (x)+20\right )}{\left (4 x^2+41 x+\log ^2(x)+4 x \log (x)+20 \log (x)+100\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 20 \int \frac {1}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx+25 \int \frac {x}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx-37 \int \frac {x^2}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx+2 \int \frac {\log (x)}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx+2 \int \frac {x \log (x)}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx-4 \int \frac {x^2 \log (x)}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx-\int \frac {1}{4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100}dx+2 \int \frac {x}{4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100}dx-8 \int \frac {x^3}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx\) |
Int[(-80 + 184*x + 41*x^2 + (-18 + 38*x + 4*x^2)*Log[x] + (-1 + 2*x)*Log[x ]^2)/(10000 + 8200*x + 2481*x^2 + 328*x^3 + 16*x^4 + (4000 + 2440*x + 488* x^2 + 32*x^3)*Log[x] + (600 + 242*x + 24*x^2)*Log[x]^2 + (40 + 8*x)*Log[x] ^3 + Log[x]^4),x]
3.17.99.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {x \left (-1+x \right )}{\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}+20 \ln \left (x \right )+41 x +100}\) | \(31\) |
default | \(\frac {x^{2}-x}{\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}+20 \ln \left (x \right )+41 x +100}\) | \(34\) |
parallelrisch | \(\frac {x^{2}-x}{\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}+20 \ln \left (x \right )+41 x +100}\) | \(34\) |
norman | \(\frac {-5 \ln \left (x \right )-\frac {45 x}{4}-\frac {\ln \left (x \right )^{2}}{4}-x \ln \left (x \right )-25}{\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}+20 \ln \left (x \right )+41 x +100}\) | \(47\) |
int(((-1+2*x)*ln(x)^2+(4*x^2+38*x-18)*ln(x)+41*x^2+184*x-80)/(ln(x)^4+(8*x +40)*ln(x)^3+(24*x^2+242*x+600)*ln(x)^2+(32*x^3+488*x^2+2440*x+4000)*ln(x) +16*x^4+328*x^3+2481*x^2+8200*x+10000),x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {x^{2} - x}{4 \, x^{2} + 4 \, {\left (x + 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 41 \, x + 100} \]
integrate(((-1+2*x)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log( x)^4+(8*x+40)*log(x)^3+(24*x^2+242*x+600)*log(x)^2+(32*x^3+488*x^2+2440*x+ 4000)*log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {x^{2} - x}{4 x^{2} + 41 x + \left (4 x + 20\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 100} \]
integrate(((-1+2*x)*ln(x)**2+(4*x**2+38*x-18)*ln(x)+41*x**2+184*x-80)/(ln( x)**4+(8*x+40)*ln(x)**3+(24*x**2+242*x+600)*ln(x)**2+(32*x**3+488*x**2+244 0*x+4000)*ln(x)+16*x**4+328*x**3+2481*x**2+8200*x+10000),x)
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {x^{2} - x}{4 \, x^{2} + 4 \, {\left (x + 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 41 \, x + 100} \]
integrate(((-1+2*x)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log( x)^4+(8*x+40)*log(x)^3+(24*x^2+242*x+600)*log(x)^2+(32*x^3+488*x^2+2440*x+ 4000)*log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {x^{2} - x}{4 \, x^{2} + 4 \, x \log \left (x\right ) + \log \left (x\right )^{2} + 41 \, x + 20 \, \log \left (x\right ) + 100} \]
integrate(((-1+2*x)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log( x)^4+(8*x+40)*log(x)^3+(24*x^2+242*x+600)*log(x)^2+(32*x^3+488*x^2+2440*x+ 4000)*log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x, algorithm=\
Timed out. \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\int \frac {184\,x+\ln \left (x\right )\,\left (4\,x^2+38\,x-18\right )+41\,x^2+{\ln \left (x\right )}^2\,\left (2\,x-1\right )-80}{8200\,x+{\ln \left (x\right )}^2\,\left (24\,x^2+242\,x+600\right )+{\ln \left (x\right )}^4+2481\,x^2+328\,x^3+16\,x^4+{\ln \left (x\right )}^3\,\left (8\,x+40\right )+\ln \left (x\right )\,\left (32\,x^3+488\,x^2+2440\,x+4000\right )+10000} \,d x \]
int((184*x + log(x)*(38*x + 4*x^2 - 18) + 41*x^2 + log(x)^2*(2*x - 1) - 80 )/(8200*x + log(x)^2*(242*x + 24*x^2 + 600) + log(x)^4 + 2481*x^2 + 328*x^ 3 + 16*x^4 + log(x)^3*(8*x + 40) + log(x)*(2440*x + 488*x^2 + 32*x^3 + 400 0) + 10000),x)