Integrand size = 75, antiderivative size = 31 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=-\frac {49}{16}+x-\frac {-\frac {5}{x}+\frac {x}{1-\frac {9}{x^4 \log ^2(x)}}}{x} \] Output:
x-49/16-(x/(1-9/x^4/ln(x)^2)-5/x)/x
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=\frac {5}{x^2}+x+\frac {9}{9-x^4 \log ^2(x)} \] Input:
Integrate[(-810 + 81*x^3 + 18*x^6*Log[x] + (180*x^4 + 36*x^6 - 18*x^7)*Log [x]^2 + (-10*x^8 + x^11)*Log[x]^4)/(81*x^3 - 18*x^7*Log[x]^2 + x^11*Log[x] ^4),x]
Output:
5/x^2 + x + 9/(9 - x^4*Log[x]^2)
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.
\(\displaystyle \int \frac {18 x^6 \log (x)+81 x^3+\left (x^{11}-10 x^8\right ) \log ^4(x)+\left (-18 x^7+36 x^6+180 x^4\right ) \log ^2(x)-810}{x^{11} \log ^4(x)-18 x^7 \log ^2(x)+81 x^3} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {18 x^6 \log (x)+81 x^3+\left (x^{11}-10 x^8\right ) \log ^4(x)+\left (-18 x^7+36 x^6+180 x^4\right ) \log ^2(x)-810}{x^3 \left (9-x^4 \log ^2(x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^3-10}{x^3}-\frac {3 \left (x^2-6\right )}{2 x \left (x^2 \log (x)+3\right )^2}+\frac {3}{x \left (x^2 \log (x)-3\right )}-\frac {3}{x \left (x^2 \log (x)+3\right )}+\frac {3 \left (x^2+6\right )}{2 x \left (x^2 \log (x)-3\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 9 \int \frac {1}{x \left (x^2 \log (x)-3\right )^2}dx+\frac {3}{2} \int \frac {x}{\left (x^2 \log (x)-3\right )^2}dx+3 \int \frac {1}{x \left (x^2 \log (x)-3\right )}dx+9 \int \frac {1}{x \left (\log (x) x^2+3\right )^2}dx-\frac {3}{2} \int \frac {x}{\left (\log (x) x^2+3\right )^2}dx-3 \int \frac {1}{x \left (\log (x) x^2+3\right )}dx+\frac {5}{x^2}+x\) |
Input:
Int[(-810 + 81*x^3 + 18*x^6*Log[x] + (180*x^4 + 36*x^6 - 18*x^7)*Log[x]^2 + (-10*x^8 + x^11)*Log[x]^4)/(81*x^3 - 18*x^7*Log[x]^2 + x^11*Log[x]^4),x]
Output:
$Aborted
Time = 2.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71
method | result | size |
default | \(x +\frac {5}{x^{2}}-\frac {9}{x^{4} \ln \left (x \right )^{2}-9}\) | \(22\) |
risch | \(\frac {x^{3}+5}{x^{2}}-\frac {9}{x^{4} \ln \left (x \right )^{2}-9}\) | \(25\) |
parallelrisch | \(-\frac {-x^{7} \ln \left (x \right )^{2}+45-5 x^{4} \ln \left (x \right )^{2}+9 x^{3}+9 x^{2}}{x^{2} \left (x^{4} \ln \left (x \right )^{2}-9\right )}\) | \(48\) |
Input:
int(((x^11-10*x^8)*ln(x)^4+(-18*x^7+36*x^6+180*x^4)*ln(x)^2+18*x^6*ln(x)+8 1*x^3-810)/(x^11*ln(x)^4-18*x^7*ln(x)^2+81*x^3),x,method=_RETURNVERBOSE)
Output:
x+5/x^2-9/(x^4*ln(x)^2-9)
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=-\frac {9 \, x^{3} - {\left (x^{7} + 5 \, x^{4}\right )} \log \left (x\right )^{2} + 9 \, x^{2} + 45}{x^{6} \log \left (x\right )^{2} - 9 \, x^{2}} \] Input:
integrate(((x^11-10*x^8)*log(x)^4+(-18*x^7+36*x^6+180*x^4)*log(x)^2+18*x^6 *log(x)+81*x^3-810)/(x^11*log(x)^4-18*x^7*log(x)^2+81*x^3),x, algorithm="f ricas")
Output:
-(9*x^3 - (x^7 + 5*x^4)*log(x)^2 + 9*x^2 + 45)/(x^6*log(x)^2 - 9*x^2)
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=x - \frac {9}{x^{4} \log {\left (x \right )}^{2} - 9} + \frac {5}{x^{2}} \] Input:
integrate(((x**11-10*x**8)*ln(x)**4+(-18*x**7+36*x**6+180*x**4)*ln(x)**2+1 8*x**6*ln(x)+81*x**3-810)/(x**11*ln(x)**4-18*x**7*ln(x)**2+81*x**3),x)
Output:
x - 9/(x**4*log(x)**2 - 9) + 5/x**2
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=-\frac {9 \, x^{3} - {\left (x^{7} + 5 \, x^{4}\right )} \log \left (x\right )^{2} + 9 \, x^{2} + 45}{x^{6} \log \left (x\right )^{2} - 9 \, x^{2}} \] Input:
integrate(((x^11-10*x^8)*log(x)^4+(-18*x^7+36*x^6+180*x^4)*log(x)^2+18*x^6 *log(x)+81*x^3-810)/(x^11*log(x)^4-18*x^7*log(x)^2+81*x^3),x, algorithm="m axima")
Output:
-(9*x^3 - (x^7 + 5*x^4)*log(x)^2 + 9*x^2 + 45)/(x^6*log(x)^2 - 9*x^2)
Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=x - \frac {9}{x^{4} \log \left (x\right )^{2} - 9} + \frac {5}{x^{2}} \] Input:
integrate(((x^11-10*x^8)*log(x)^4+(-18*x^7+36*x^6+180*x^4)*log(x)^2+18*x^6 *log(x)+81*x^3-810)/(x^11*log(x)^4-18*x^7*log(x)^2+81*x^3),x, algorithm="g iac")
Output:
x - 9/(x^4*log(x)^2 - 9) + 5/x^2
Time = 3.93 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=x-\frac {9}{x^4\,{\ln \left (x\right )}^2-9}+\frac {5}{x^2} \] Input:
int((18*x^6*log(x) - log(x)^4*(10*x^8 - x^11) + log(x)^2*(180*x^4 + 36*x^6 - 18*x^7) + 81*x^3 - 810)/(x^11*log(x)^4 - 18*x^7*log(x)^2 + 81*x^3),x)
Output:
x - 9/(x^4*log(x)^2 - 9) + 5/x^2
\[ \int \frac {-810+81 x^3+18 x^6 \log (x)+\left (180 x^4+36 x^6-18 x^7\right ) \log ^2(x)+\left (-10 x^8+x^{11}\right ) \log ^4(x)}{81 x^3-18 x^7 \log ^2(x)+x^{11} \log ^4(x)} \, dx=\int \frac {\mathrm {log}\left (x \right )^{4} x^{8}}{x^{8} \mathrm {log}\left (x \right )^{4}-18 \mathrm {log}\left (x \right )^{2} x^{4}+81}d x -10 \left (\int \frac {\mathrm {log}\left (x \right )^{4} x^{5}}{x^{8} \mathrm {log}\left (x \right )^{4}-18 \mathrm {log}\left (x \right )^{2} x^{4}+81}d x \right )-18 \left (\int \frac {\mathrm {log}\left (x \right )^{2} x^{4}}{x^{8} \mathrm {log}\left (x \right )^{4}-18 \mathrm {log}\left (x \right )^{2} x^{4}+81}d x \right )+36 \left (\int \frac {\mathrm {log}\left (x \right )^{2} x^{3}}{x^{8} \mathrm {log}\left (x \right )^{4}-18 \mathrm {log}\left (x \right )^{2} x^{4}+81}d x \right )+180 \left (\int \frac {\mathrm {log}\left (x \right )^{2} x}{x^{8} \mathrm {log}\left (x \right )^{4}-18 \mathrm {log}\left (x \right )^{2} x^{4}+81}d x \right )+18 \left (\int \frac {\mathrm {log}\left (x \right ) x^{3}}{x^{8} \mathrm {log}\left (x \right )^{4}-18 \mathrm {log}\left (x \right )^{2} x^{4}+81}d x \right )-810 \left (\int \frac {1}{\mathrm {log}\left (x \right )^{4} x^{11}-18 \mathrm {log}\left (x \right )^{2} x^{7}+81 x^{3}}d x \right )+81 \left (\int \frac {1}{x^{8} \mathrm {log}\left (x \right )^{4}-18 \mathrm {log}\left (x \right )^{2} x^{4}+81}d x \right ) \] Input:
int(((x^11-10*x^8)*log(x)^4+(-18*x^7+36*x^6+180*x^4)*log(x)^2+18*x^6*log(x )+81*x^3-810)/(x^11*log(x)^4-18*x^7*log(x)^2+81*x^3),x)
Output:
int((log(x)**4*x**8)/(log(x)**4*x**8 - 18*log(x)**2*x**4 + 81),x) - 10*int ((log(x)**4*x**5)/(log(x)**4*x**8 - 18*log(x)**2*x**4 + 81),x) - 18*int((l og(x)**2*x**4)/(log(x)**4*x**8 - 18*log(x)**2*x**4 + 81),x) + 36*int((log( x)**2*x**3)/(log(x)**4*x**8 - 18*log(x)**2*x**4 + 81),x) + 180*int((log(x) **2*x)/(log(x)**4*x**8 - 18*log(x)**2*x**4 + 81),x) + 18*int((log(x)*x**3) /(log(x)**4*x**8 - 18*log(x)**2*x**4 + 81),x) - 810*int(1/(log(x)**4*x**11 - 18*log(x)**2*x**7 + 81*x**3),x) + 81*int(1/(log(x)**4*x**8 - 18*log(x)* *2*x**4 + 81),x)