Integrand size = 126, antiderivative size = 36 \[ \int \frac {216-36 x+\left (-351+e^3 (3-x)+117 x\right ) \log \left (-\frac {x^2}{-3+x}\right )+\left (108-18 x+(-162+54 x) \log \left (-\frac {x^2}{-3+x}\right )\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+(-27+9 x) \log \left (-\frac {x^2}{-3+x}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-9+3 x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx=\frac {1}{3} x \left (-e^3+9 \left (5+\left (2+\log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )\right )^2\right )\right ) \]
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.42 \[ \int \frac {216-36 x+\left (-351+e^3 (3-x)+117 x\right ) \log \left (-\frac {x^2}{-3+x}\right )+\left (108-18 x+(-162+54 x) \log \left (-\frac {x^2}{-3+x}\right )\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+(-27+9 x) \log \left (-\frac {x^2}{-3+x}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-9+3 x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx=-\frac {1}{3} \left (-81+e^3\right ) x+12 x \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+3 x \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right ) \]
Integrate[(216 - 36*x + (-351 + E^3*(3 - x) + 117*x)*Log[-(x^2/(-3 + x))] + (108 - 18*x + (-162 + 54*x)*Log[-(x^2/(-3 + x))])*Log[x/Log[-(x^2/(-3 + x))]] + (-27 + 9*x)*Log[-(x^2/(-3 + x))]*Log[x/Log[-(x^2/(-3 + x))]]^2)/(( -9 + 3*x)*Log[-(x^2/(-3 + x))]),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.
| \(\displaystyle \int \frac {(9 x-27) \log \left (-\frac {x^2}{x-3}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{x-3}\right )}\right )+\left ((54 x-162) \log \left (-\frac {x^2}{x-3}\right )-18 x+108\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{x-3}\right )}\right )+\left (e^3 (3-x)+117 x-351\right ) \log \left (-\frac {x^2}{x-3}\right )-36 x+216}{(3 x-9) \log \left (-\frac {x^2}{x-3}\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (3 \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{x-3}\right )}\right )+\frac {6 \left (3 x \log \left (-\frac {x^2}{x-3}\right )-9 \log \left (-\frac {x^2}{x-3}\right )-x+6\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{x-3}\right )}\right )}{(x-3) \log \left (-\frac {x^2}{x-3}\right )}+\frac {-117 \left (1-\frac {e^3}{117}\right ) x \log \left (-\frac {x^2}{x-3}\right )+351 \left (1-\frac {e^3}{117}\right ) \log \left (-\frac {x^2}{x-3}\right )+36 x-216}{3 (3-x) \log \left (-\frac {x^2}{x-3}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{x-3}\right )}\right )dx-18 \int \frac {6-x}{(x-3) \log \left (-\frac {x^2}{x-3}\right )}dx-12 \int \frac {x-6}{(x-3) \log \left (-\frac {x^2}{x-3}\right )}dx-6 \int \frac {\log \left (\frac {x}{\log \left (-\frac {x^2}{x-3}\right )}\right )}{\log \left (-\frac {x^2}{x-3}\right )}dx+18 \int \frac {\log \left (\frac {x}{\log \left (-\frac {x^2}{x-3}\right )}\right )}{(x-3) \log \left (-\frac {x^2}{x-3}\right )}dx+18 x \log \left (\frac {x}{\log \left (\frac {x^2}{3-x}\right )}\right )+\frac {1}{3} \left (117-e^3\right ) x-18 x\) |
Int[(216 - 36*x + (-351 + E^3*(3 - x) + 117*x)*Log[-(x^2/(-3 + x))] + (108 - 18*x + (-162 + 54*x)*Log[-(x^2/(-3 + x))])*Log[x/Log[-(x^2/(-3 + x))]] + (-27 + 9*x)*Log[-(x^2/(-3 + x))]*Log[x/Log[-(x^2/(-3 + x))]]^2)/((-9 + 3 *x)*Log[-(x^2/(-3 + x))]),x]
3.29.30.3.1 Defintions of rubi rules used
Time = 2.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39
| method | result | size |
| norman | \(\left (-\frac {{\mathrm e}^{3}}{3}+27\right ) x +3 x {\ln \left (\frac {x}{\ln \left (-\frac {x^{2}}{-3+x}\right )}\right )}^{2}+12 \ln \left (\frac {x}{\ln \left (-\frac {x^{2}}{-3+x}\right )}\right ) x\) | \(50\) |
| parallelrisch | \(3 x {\ln \left (\frac {x}{\ln \left (-\frac {x^{2}}{-3+x}\right )}\right )}^{2}-\frac {x \,{\mathrm e}^{3}}{3}+162+12 \ln \left (\frac {x}{\ln \left (-\frac {x^{2}}{-3+x}\right )}\right ) x -2 \,{\mathrm e}^{3}+27 x\) | \(55\) |
int(((9*x-27)*ln(-x^2/(-3+x))*ln(x/ln(-x^2/(-3+x)))^2+((54*x-162)*ln(-x^2/ (-3+x))-18*x+108)*ln(x/ln(-x^2/(-3+x)))+((-x+3)*exp(3)+117*x-351)*ln(-x^2/ (-3+x))-36*x+216)/(3*x-9)/ln(-x^2/(-3+x)),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36 \[ \int \frac {216-36 x+\left (-351+e^3 (3-x)+117 x\right ) \log \left (-\frac {x^2}{-3+x}\right )+\left (108-18 x+(-162+54 x) \log \left (-\frac {x^2}{-3+x}\right )\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+(-27+9 x) \log \left (-\frac {x^2}{-3+x}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-9+3 x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx=3 \, x \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right )^{2} - \frac {1}{3} \, x e^{3} + 12 \, x \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right ) + 27 \, x \]
integrate(((9*x-27)*log(-x^2/(-3+x))*log(x/log(-x^2/(-3+x)))^2+((54*x-162) *log(-x^2/(-3+x))-18*x+108)*log(x/log(-x^2/(-3+x)))+((-x+3)*exp(3)+117*x-3 51)*log(-x^2/(-3+x))-36*x+216)/(3*x-9)/log(-x^2/(-3+x)),x, algorithm=\
Time = 0.45 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {216-36 x+\left (-351+e^3 (3-x)+117 x\right ) \log \left (-\frac {x^2}{-3+x}\right )+\left (108-18 x+(-162+54 x) \log \left (-\frac {x^2}{-3+x}\right )\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+(-27+9 x) \log \left (-\frac {x^2}{-3+x}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-9+3 x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx=3 x \log {\left (\frac {x}{\log {\left (- \frac {x^{2}}{x - 3} \right )}} \right )}^{2} + 12 x \log {\left (\frac {x}{\log {\left (- \frac {x^{2}}{x - 3} \right )}} \right )} + x \left (27 - \frac {e^{3}}{3}\right ) \]
integrate(((9*x-27)*ln(-x**2/(-3+x))*ln(x/ln(-x**2/(-3+x)))**2+((54*x-162) *ln(-x**2/(-3+x))-18*x+108)*ln(x/ln(-x**2/(-3+x)))+((-x+3)*exp(3)+117*x-35 1)*ln(-x**2/(-3+x))-36*x+216)/(3*x-9)/ln(-x**2/(-3+x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (31) = 62\).
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.75 \[ \int \frac {216-36 x+\left (-351+e^3 (3-x)+117 x\right ) \log \left (-\frac {x^2}{-3+x}\right )+\left (108-18 x+(-162+54 x) \log \left (-\frac {x^2}{-3+x}\right )\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+(-27+9 x) \log \left (-\frac {x^2}{-3+x}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-9+3 x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx=3 \, x \log \left (x\right )^{2} + 3 \, x \log \left (2 \, \log \left (x\right ) - \log \left (-x + 3\right )\right )^{2} - \frac {1}{3} \, x {\left (e^{3} - 81\right )} + 12 \, x \log \left (x\right ) - 6 \, {\left (x \log \left (x\right ) + 2 \, x\right )} \log \left (2 \, \log \left (x\right ) - \log \left (-x + 3\right )\right ) \]
integrate(((9*x-27)*log(-x^2/(-3+x))*log(x/log(-x^2/(-3+x)))^2+((54*x-162) *log(-x^2/(-3+x))-18*x+108)*log(x/log(-x^2/(-3+x)))+((-x+3)*exp(3)+117*x-3 51)*log(-x^2/(-3+x))-36*x+216)/(3*x-9)/log(-x^2/(-3+x)),x, algorithm=\
3*x*log(x)^2 + 3*x*log(2*log(x) - log(-x + 3))^2 - 1/3*x*(e^3 - 81) + 12*x *log(x) - 6*(x*log(x) + 2*x)*log(2*log(x) - log(-x + 3))
\[ \int \frac {216-36 x+\left (-351+e^3 (3-x)+117 x\right ) \log \left (-\frac {x^2}{-3+x}\right )+\left (108-18 x+(-162+54 x) \log \left (-\frac {x^2}{-3+x}\right )\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+(-27+9 x) \log \left (-\frac {x^2}{-3+x}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-9+3 x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx=\int { \frac {9 \, {\left (x - 3\right )} \log \left (-\frac {x^{2}}{x - 3}\right ) \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right )^{2} - {\left ({\left (x - 3\right )} e^{3} - 117 \, x + 351\right )} \log \left (-\frac {x^{2}}{x - 3}\right ) + 18 \, {\left (3 \, {\left (x - 3\right )} \log \left (-\frac {x^{2}}{x - 3}\right ) - x + 6\right )} \log \left (\frac {x}{\log \left (-\frac {x^{2}}{x - 3}\right )}\right ) - 36 \, x + 216}{3 \, {\left (x - 3\right )} \log \left (-\frac {x^{2}}{x - 3}\right )} \,d x } \]
integrate(((9*x-27)*log(-x^2/(-3+x))*log(x/log(-x^2/(-3+x)))^2+((54*x-162) *log(-x^2/(-3+x))-18*x+108)*log(x/log(-x^2/(-3+x)))+((-x+3)*exp(3)+117*x-3 51)*log(-x^2/(-3+x))-36*x+216)/(3*x-9)/log(-x^2/(-3+x)),x, algorithm=\
integrate(1/3*(9*(x - 3)*log(-x^2/(x - 3))*log(x/log(-x^2/(x - 3)))^2 - (( x - 3)*e^3 - 117*x + 351)*log(-x^2/(x - 3)) + 18*(3*(x - 3)*log(-x^2/(x - 3)) - x + 6)*log(x/log(-x^2/(x - 3))) - 36*x + 216)/((x - 3)*log(-x^2/(x - 3))), x)
Time = 10.74 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {216-36 x+\left (-351+e^3 (3-x)+117 x\right ) \log \left (-\frac {x^2}{-3+x}\right )+\left (108-18 x+(-162+54 x) \log \left (-\frac {x^2}{-3+x}\right )\right ) \log \left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )+(-27+9 x) \log \left (-\frac {x^2}{-3+x}\right ) \log ^2\left (\frac {x}{\log \left (-\frac {x^2}{-3+x}\right )}\right )}{(-9+3 x) \log \left (-\frac {x^2}{-3+x}\right )} \, dx=3\,x\,{\ln \left (\frac {x}{\ln \left (-\frac {x^2}{x-3}\right )}\right )}^2+12\,x\,\ln \left (\frac {x}{\ln \left (-\frac {x^2}{x-3}\right )}\right )-x\,\left (\frac {{\mathrm {e}}^3}{3}-27\right ) \]