Integrand size = 108, antiderivative size = 30 \[ \int \frac {\left (54-99 x-15 x^2+23 x^3+3 x^4\right ) \log (5)+\left (-54+18 x^2+2 x^3\right ) \log (5) \log \left (\frac {3}{x}\right )}{81 x^4-18 x^6+x^8+\left (162 x^3-36 x^5+2 x^7\right ) \log \left (\frac {3}{x}\right )+\left (81 x^2-18 x^4+x^6\right ) \log ^2\left (\frac {3}{x}\right )} \, dx=-\frac {(6+x) \log (5)}{(-3+x) x (3+x) \left (x+\log \left (\frac {3}{x}\right )\right )} \] Output:
-(6+x)/(-3+x)/(ln(3/x)+x)/(3+x)/x*ln(5)
Time = 0.42 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {\left (54-99 x-15 x^2+23 x^3+3 x^4\right ) \log (5)+\left (-54+18 x^2+2 x^3\right ) \log (5) \log \left (\frac {3}{x}\right )}{81 x^4-18 x^6+x^8+\left (162 x^3-36 x^5+2 x^7\right ) \log \left (\frac {3}{x}\right )+\left (81 x^2-18 x^4+x^6\right ) \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {(-6-x) \log (5)}{x \left (-9+x^2\right ) \left (x+\log \left (\frac {3}{x}\right )\right )} \] Input:
Integrate[((54 - 99*x - 15*x^2 + 23*x^3 + 3*x^4)*Log[5] + (-54 + 18*x^2 + 2*x^3)*Log[5]*Log[3/x])/(81*x^4 - 18*x^6 + x^8 + (162*x^3 - 36*x^5 + 2*x^7 )*Log[3/x] + (81*x^2 - 18*x^4 + x^6)*Log[3/x]^2),x]
Output:
((-6 - x)*Log[5])/(x*(-9 + x^2)*(x + Log[3/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.
\(\displaystyle \int \frac {\left (2 x^3+18 x^2-54\right ) \log (5) \log \left (\frac {3}{x}\right )+\left (3 x^4+23 x^3-15 x^2-99 x+54\right ) \log (5)}{x^8-18 x^6+81 x^4+\left (2 x^7-36 x^5+162 x^3\right ) \log \left (\frac {3}{x}\right )+\left (x^6-18 x^4+81 x^2\right ) \log ^2\left (\frac {3}{x}\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\log (5) \left (3 x^4+23 x^3-15 x^2+2 \left (x^3+9 x^2-27\right ) \log \left (\frac {3}{x}\right )-99 x+54\right )}{x^2 \left (9-x^2\right )^2 \left (x+\log \left (\frac {3}{x}\right )\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \log (5) \int \frac {3 x^4+23 x^3-15 x^2-99 x-2 \left (-x^3-9 x^2+27\right ) \log \left (\frac {3}{x}\right )+54}{x^2 \left (9-x^2\right )^2 \left (x+\log \left (\frac {3}{x}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \log (5) \int \left (\frac {x^2+5 x-6}{x^2 \left (x^2-9\right ) \left (x+\log \left (\frac {3}{x}\right )\right )^2}+\frac {2 \left (x^3+9 x^2-27\right )}{x^2 \left (x^2-9\right )^2 \left (x+\log \left (\frac {3}{x}\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log (5) \left (\frac {2}{3} \int \frac {1}{x^2 \left (x+\log \left (\frac {3}{x}\right )\right )^2}dx-\frac {2}{3} \int \frac {1}{x^2 \left (x+\log \left (\frac {3}{x}\right )\right )}dx+\frac {1}{3} \int \frac {1}{(x-3) \left (x+\log \left (\frac {3}{x}\right )\right )^2}dx-\frac {5}{9} \int \frac {1}{x \left (x+\log \left (\frac {3}{x}\right )\right )^2}dx+\frac {2}{9} \int \frac {1}{(x+3) \left (x+\log \left (\frac {3}{x}\right )\right )^2}dx+\frac {1}{2} \int \frac {1}{(x-3)^2 \left (x+\log \left (\frac {3}{x}\right )\right )}dx+\frac {1}{6} \int \frac {1}{(x+3)^2 \left (x+\log \left (\frac {3}{x}\right )\right )}dx\right )\) |
Input:
Int[((54 - 99*x - 15*x^2 + 23*x^3 + 3*x^4)*Log[5] + (-54 + 18*x^2 + 2*x^3) *Log[5]*Log[3/x])/(81*x^4 - 18*x^6 + x^8 + (162*x^3 - 36*x^5 + 2*x^7)*Log[ 3/x] + (81*x^2 - 18*x^4 + x^6)*Log[3/x]^2),x]
Output:
$Aborted
Time = 0.61 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {\ln \left (5\right ) \left (6+x \right )}{\left (x^{2}-9\right ) x \left (\ln \left (\frac {3}{x}\right )+x \right )}\) | \(28\) |
derivativedivides | \(\frac {\ln \left (5\right ) \left (\frac {162}{x^{4}}+\frac {27}{x^{3}}\right )}{9 \left (\frac {9}{x^{2}}-1\right ) \left (\frac {3 \ln \left (\frac {3}{x}\right )}{x}+3\right )}\) | \(40\) |
default | \(\frac {\ln \left (5\right ) \left (\frac {162}{x^{4}}+\frac {27}{x^{3}}\right )}{9 \left (\frac {9}{x^{2}}-1\right ) \left (\frac {3 \ln \left (\frac {3}{x}\right )}{x}+3\right )}\) | \(40\) |
parallelrisch | \(-\frac {x \ln \left (5\right )+6 \ln \left (5\right )}{x \left (x^{2} \ln \left (\frac {3}{x}\right )+x^{3}-9 \ln \left (\frac {3}{x}\right )-9 x \right )}\) | \(42\) |
Input:
int(((2*x^3+18*x^2-54)*ln(5)*ln(3/x)+(3*x^4+23*x^3-15*x^2-99*x+54)*ln(5))/ ((x^6-18*x^4+81*x^2)*ln(3/x)^2+(2*x^7-36*x^5+162*x^3)*ln(3/x)+x^8-18*x^6+8 1*x^4),x,method=_RETURNVERBOSE)
Output:
-ln(5)*(6+x)/(x^2-9)/x/(ln(3/x)+x)
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {\left (54-99 x-15 x^2+23 x^3+3 x^4\right ) \log (5)+\left (-54+18 x^2+2 x^3\right ) \log (5) \log \left (\frac {3}{x}\right )}{81 x^4-18 x^6+x^8+\left (162 x^3-36 x^5+2 x^7\right ) \log \left (\frac {3}{x}\right )+\left (81 x^2-18 x^4+x^6\right ) \log ^2\left (\frac {3}{x}\right )} \, dx=-\frac {{\left (x + 6\right )} \log \left (5\right )}{x^{4} - 9 \, x^{2} + {\left (x^{3} - 9 \, x\right )} \log \left (\frac {3}{x}\right )} \] Input:
integrate(((2*x^3+18*x^2-54)*log(5)*log(3/x)+(3*x^4+23*x^3-15*x^2-99*x+54) *log(5))/((x^6-18*x^4+81*x^2)*log(3/x)^2+(2*x^7-36*x^5+162*x^3)*log(3/x)+x ^8-18*x^6+81*x^4),x, algorithm="fricas")
Output:
-(x + 6)*log(5)/(x^4 - 9*x^2 + (x^3 - 9*x)*log(3/x))
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\left (54-99 x-15 x^2+23 x^3+3 x^4\right ) \log (5)+\left (-54+18 x^2+2 x^3\right ) \log (5) \log \left (\frac {3}{x}\right )}{81 x^4-18 x^6+x^8+\left (162 x^3-36 x^5+2 x^7\right ) \log \left (\frac {3}{x}\right )+\left (81 x^2-18 x^4+x^6\right ) \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {- x \log {\left (5 \right )} - 6 \log {\left (5 \right )}}{x^{4} - 9 x^{2} + \left (x^{3} - 9 x\right ) \log {\left (\frac {3}{x} \right )}} \] Input:
integrate(((2*x**3+18*x**2-54)*ln(5)*ln(3/x)+(3*x**4+23*x**3-15*x**2-99*x+ 54)*ln(5))/((x**6-18*x**4+81*x**2)*ln(3/x)**2+(2*x**7-36*x**5+162*x**3)*ln (3/x)+x**8-18*x**6+81*x**4),x)
Output:
(-x*log(5) - 6*log(5))/(x**4 - 9*x**2 + (x**3 - 9*x)*log(3/x))
Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {\left (54-99 x-15 x^2+23 x^3+3 x^4\right ) \log (5)+\left (-54+18 x^2+2 x^3\right ) \log (5) \log \left (\frac {3}{x}\right )}{81 x^4-18 x^6+x^8+\left (162 x^3-36 x^5+2 x^7\right ) \log \left (\frac {3}{x}\right )+\left (81 x^2-18 x^4+x^6\right ) \log ^2\left (\frac {3}{x}\right )} \, dx=-\frac {x \log \left (5\right ) + 6 \, \log \left (5\right )}{x^{4} + x^{3} \log \left (3\right ) - 9 \, x^{2} - 9 \, x \log \left (3\right ) - {\left (x^{3} - 9 \, x\right )} \log \left (x\right )} \] Input:
integrate(((2*x^3+18*x^2-54)*log(5)*log(3/x)+(3*x^4+23*x^3-15*x^2-99*x+54) *log(5))/((x^6-18*x^4+81*x^2)*log(3/x)^2+(2*x^7-36*x^5+162*x^3)*log(3/x)+x ^8-18*x^6+81*x^4),x, algorithm="maxima")
Output:
-(x*log(5) + 6*log(5))/(x^4 + x^3*log(3) - 9*x^2 - 9*x*log(3) - (x^3 - 9*x )*log(x))
Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {\left (54-99 x-15 x^2+23 x^3+3 x^4\right ) \log (5)+\left (-54+18 x^2+2 x^3\right ) \log (5) \log \left (\frac {3}{x}\right )}{81 x^4-18 x^6+x^8+\left (162 x^3-36 x^5+2 x^7\right ) \log \left (\frac {3}{x}\right )+\left (81 x^2-18 x^4+x^6\right ) \log ^2\left (\frac {3}{x}\right )} \, dx=-\frac {\frac {\log \left (5\right )}{x^{3}} + \frac {6 \, \log \left (5\right )}{x^{4}}}{\frac {\log \left (\frac {3}{x}\right )}{x} - \frac {9}{x^{2}} - \frac {9 \, \log \left (\frac {3}{x}\right )}{x^{3}} + 1} \] Input:
integrate(((2*x^3+18*x^2-54)*log(5)*log(3/x)+(3*x^4+23*x^3-15*x^2-99*x+54) *log(5))/((x^6-18*x^4+81*x^2)*log(3/x)^2+(2*x^7-36*x^5+162*x^3)*log(3/x)+x ^8-18*x^6+81*x^4),x, algorithm="giac")
Output:
-(log(5)/x^3 + 6*log(5)/x^4)/(log(3/x)/x - 9/x^2 - 9*log(3/x)/x^3 + 1)
Time = 3.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {\left (54-99 x-15 x^2+23 x^3+3 x^4\right ) \log (5)+\left (-54+18 x^2+2 x^3\right ) \log (5) \log \left (\frac {3}{x}\right )}{81 x^4-18 x^6+x^8+\left (162 x^3-36 x^5+2 x^7\right ) \log \left (\frac {3}{x}\right )+\left (81 x^2-18 x^4+x^6\right ) \log ^2\left (\frac {3}{x}\right )} \, dx=-\frac {\ln \left (5\right )\,\left (x+6\right )}{x\,\left (x^2-9\right )\,\left (x+\ln \left (\frac {3}{x}\right )\right )} \] Input:
int((log(5)*(23*x^3 - 15*x^2 - 99*x + 3*x^4 + 54) + log(5)*log(3/x)*(18*x^ 2 + 2*x^3 - 54))/(log(3/x)*(162*x^3 - 36*x^5 + 2*x^7) + log(3/x)^2*(81*x^2 - 18*x^4 + x^6) + 81*x^4 - 18*x^6 + x^8),x)
Output:
-(log(5)*(x + 6))/(x*(x^2 - 9)*(x + log(3/x)))
Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {\left (54-99 x-15 x^2+23 x^3+3 x^4\right ) \log (5)+\left (-54+18 x^2+2 x^3\right ) \log (5) \log \left (\frac {3}{x}\right )}{81 x^4-18 x^6+x^8+\left (162 x^3-36 x^5+2 x^7\right ) \log \left (\frac {3}{x}\right )+\left (81 x^2-18 x^4+x^6\right ) \log ^2\left (\frac {3}{x}\right )} \, dx=\frac {\mathrm {log}\left (5\right ) \left (-x -6\right )}{x \left (\mathrm {log}\left (\frac {3}{x}\right ) x^{2}-9 \,\mathrm {log}\left (\frac {3}{x}\right )+x^{3}-9 x \right )} \] Input:
int(((2*x^3+18*x^2-54)*log(5)*log(3/x)+(3*x^4+23*x^3-15*x^2-99*x+54)*log(5 ))/((x^6-18*x^4+81*x^2)*log(3/x)^2+(2*x^7-36*x^5+162*x^3)*log(3/x)+x^8-18* x^6+81*x^4),x)
Output:
(log(5)*( - x - 6))/(x*(log(3/x)*x**2 - 9*log(3/x) + x**3 - 9*x))