Integrand size = 52, antiderivative size = 35 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=\log \left (\frac {4}{\frac {x-x \left (-x^2+\left (x+x^4\right )^2\right )}{\log (3)}+\frac {5}{\log (x)}}\right ) \] Output:
ln(4/(5/ln(x)+(x-((x^4+x)^2-x^2)*x)/ln(3)))
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.83 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=-\log (x)-\log \left (1-2 x^5-x^8\right )+\log \left (x \left (1-2 x^5-x^8\right )\right )+\log (\log (x))-\log \left (5 \log (3)+x \log (x)-2 x^6 \log (x)-x^9 \log (x)\right ) \] Input:
Integrate[(-5*Log[3] + (x - 12*x^6 - 9*x^9)*Log[x]^2)/(-5*x*Log[3]*Log[x] + (-x^2 + 2*x^7 + x^10)*Log[x]^2),x]
Output:
-Log[x] - Log[1 - 2*x^5 - x^8] + Log[x*(1 - 2*x^5 - x^8)] + Log[Log[x]] - Log[5*Log[3] + x*Log[x] - 2*x^6*Log[x] - x^9*Log[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.
\(\displaystyle \int \frac {\left (-9 x^9-12 x^6+x\right ) \log ^2(x)-5 \log (3)}{\left (x^{10}+2 x^7-x^2\right ) \log ^2(x)-5 x \log (3) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {5 \log (3)-\left (-9 x^9-12 x^6+x\right ) \log ^2(x)}{x \log (x) \left (x^9 (-\log (x))-2 x^6 \log (x)+x \log (x)+5 \log (3)\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-9 x^8-12 x^5+1}{x \left (x^8+2 x^5-1\right )}+\frac {-x^8-2 x^5+1}{x^9 \log (x)+2 x^6 \log (x)-x \log (x)-5 \log (3)}-\frac {5 \left (9 x^8+12 x^5-1\right ) \log (3)}{x \left (x^8+2 x^5-1\right ) \left (x^9 \log (x)+2 x^6 \log (x)-x \log (x)-5 \log (3)\right )}+\frac {1}{x \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {1}{-\log (x) x^9-2 \log (x) x^6+\log (x) x+5 \log (3)}dx-5 \log (3) \int \frac {1}{x \left (\log (x) x^9+2 \log (x) x^6-\log (x) x-5 \log (3)\right )}dx-\int \frac {x^8}{\log (x) x^9+2 \log (x) x^6-\log (x) x-5 \log (3)}dx-2 \int \frac {x^5}{\log (x) x^9+2 \log (x) x^6-\log (x) x-5 \log (3)}dx-40 \log (3) \int \frac {x^7}{\left (x^8+2 x^5-1\right ) \left (\log (x) x^9+2 \log (x) x^6-\log (x) x-5 \log (3)\right )}dx-50 \log (3) \int \frac {x^4}{\left (x^8+2 x^5-1\right ) \left (\log (x) x^9+2 \log (x) x^6-\log (x) x-5 \log (3)\right )}dx-\log \left (-x \left (-x^8-2 x^5+1\right )\right )+\log (\log (x))\) |
Input:
Int[(-5*Log[3] + (x - 12*x^6 - 9*x^9)*Log[x]^2)/(-5*x*Log[3]*Log[x] + (-x^ 2 + 2*x^7 + x^10)*Log[x]^2),x]
Output:
$Aborted
Time = 0.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69
method | result | size |
default | \(-\ln \left (x^{9}+2 x^{6}-x -\frac {5 \ln \left (3\right )}{\ln \left (x \right )}\right )\) | \(24\) |
parallelrisch | \(\ln \left (\ln \left (x \right )\right )-\ln \left (x^{9} \ln \left (x \right )+2 x^{6} \ln \left (x \right )-x \ln \left (x \right )-5 \ln \left (3\right )\right )\) | \(31\) |
risch | \(-\ln \left (x^{9}+2 x^{6}-x \right )+\ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \right )-\frac {5 \ln \left (3\right )}{x \left (x^{8}+2 x^{5}-1\right )}\right )\) | \(45\) |
Input:
int(((-9*x^9-12*x^6+x)*ln(x)^2-5*ln(3))/((x^10+2*x^7-x^2)*ln(x)^2-5*x*ln(3 )*ln(x)),x,method=_RETURNVERBOSE)
Output:
-ln(x^9+2*x^6-x-5*ln(3)/ln(x))
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.63 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=-\log \left (x^{9} + 2 \, x^{6} - x\right ) - \log \left (\frac {{\left (x^{9} + 2 \, x^{6} - x\right )} \log \left (x\right ) - 5 \, \log \left (3\right )}{x^{9} + 2 \, x^{6} - x}\right ) + \log \left (\log \left (x\right )\right ) \] Input:
integrate(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2 -5*x*log(3)*log(x)),x, algorithm="fricas")
Output:
-log(x^9 + 2*x^6 - x) - log(((x^9 + 2*x^6 - x)*log(x) - 5*log(3))/(x^9 + 2 *x^6 - x)) + log(log(x))
Exception generated. \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate(((-9*x**9-12*x**6+x)*ln(x)**2-5*ln(3))/((x**10+2*x**7-x**2)*ln(x )**2-5*x*ln(3)*ln(x)),x)
Output:
Exception raised: PolynomialError >> 1/(x**18 + 4*x**15 + 4*x**12 - 2*x**1 0 - 4*x**7 + x**2) contains an element of the set of generators.
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=-\log \left (x^{8} + 2 \, x^{5} - 1\right ) - \log \left (x\right ) - \log \left (\frac {{\left (x^{9} + 2 \, x^{6} - x\right )} \log \left (x\right ) - 5 \, \log \left (3\right )}{x^{9} + 2 \, x^{6} - x}\right ) + \log \left (\log \left (x\right )\right ) \] Input:
integrate(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2 -5*x*log(3)*log(x)),x, algorithm="maxima")
Output:
-log(x^8 + 2*x^5 - 1) - log(x) - log(((x^9 + 2*x^6 - x)*log(x) - 5*log(3)) /(x^9 + 2*x^6 - x)) + log(log(x))
Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=-\log \left (-x^{9} \log \left (x\right ) - 2 \, x^{6} \log \left (x\right ) + x \log \left (x\right ) + 5 \, \log \left (3\right )\right ) + \log \left (\log \left (x\right )\right ) \] Input:
integrate(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2 -5*x*log(3)*log(x)),x, algorithm="giac")
Output:
-log(-x^9*log(x) - 2*x^6*log(x) + x*log(x) + 5*log(3)) + log(log(x))
Time = 26.03 (sec) , antiderivative size = 186, normalized size of antiderivative = 5.31 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=\ln \left (8\,x^6\,\ln \left (x\right )+4\,x^9\,\ln \left (x\right )-8\,x^{11}\,\ln \left (x\right )-8\,x^{14}\,\ln \left (x\right )-2\,x^{17}\,\ln \left (x\right )+10\,\ln \left (3\right )\,\ln \left (x\right )-2\,x\,\ln \left (x\right )-120\,x^5\,\ln \left (3\right )\,\ln \left (x\right )-90\,x^8\,\ln \left (3\right )\,\ln \left (x\right )\right )-\ln \left (10\,\ln \left (3\right )-8\,x^6\,\ln \left (x\right )-4\,x^9\,\ln \left (x\right )+8\,x^{11}\,\ln \left (x\right )+8\,x^{14}\,\ln \left (x\right )+2\,x^{17}\,\ln \left (x\right )-20\,x^5\,\ln \left (3\right )-10\,x^8\,\ln \left (3\right )+2\,x\,\ln \left (x\right )\right )-\ln \left (x^{17}+4\,x^{14}+4\,x^{11}-2\,x^9+45\,\ln \left (3\right )\,x^8-4\,x^6+60\,\ln \left (3\right )\,x^5+x-5\,\ln \left (3\right )\right )+\ln \left (x^8+2\,x^5-1\right ) \] Input:
int(-(5*log(3) + log(x)^2*(12*x^6 - x + 9*x^9))/(log(x)^2*(2*x^7 - x^2 + x ^10) - 5*x*log(3)*log(x)),x)
Output:
log(8*x^6*log(x) + 4*x^9*log(x) - 8*x^11*log(x) - 8*x^14*log(x) - 2*x^17*l og(x) + 10*log(3)*log(x) - 2*x*log(x) - 120*x^5*log(3)*log(x) - 90*x^8*log (3)*log(x)) - log(10*log(3) - 8*x^6*log(x) - 4*x^9*log(x) + 8*x^11*log(x) + 8*x^14*log(x) + 2*x^17*log(x) - 20*x^5*log(3) - 10*x^8*log(3) + 2*x*log( x)) - log(x - 5*log(3) + 60*x^5*log(3) + 45*x^8*log(3) - 4*x^6 - 2*x^9 + 4 *x^11 + 4*x^14 + x^17) + log(2*x^5 + x^8 - 1)
Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {-5 \log (3)+\left (x-12 x^6-9 x^9\right ) \log ^2(x)}{-5 x \log (3) \log (x)+\left (-x^2+2 x^7+x^{10}\right ) \log ^2(x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (x \right )\right )-\mathrm {log}\left (\mathrm {log}\left (x \right ) x^{9}+2 \,\mathrm {log}\left (x \right ) x^{6}-\mathrm {log}\left (x \right ) x -5 \,\mathrm {log}\left (3\right )\right ) \] Input:
int(((-9*x^9-12*x^6+x)*log(x)^2-5*log(3))/((x^10+2*x^7-x^2)*log(x)^2-5*x*l og(3)*log(x)),x)
Output:
log(log(x)) - log(log(x)*x**9 + 2*log(x)*x**6 - log(x)*x - 5*log(3))