Integrand size = 89, antiderivative size = 26 \[ \int \frac {-3 x+3 x^2-3 x^2 \log \left (-\frac {3}{2 x}\right )+\left (3-6 x+18 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )}{x^2+\left (2 x-12 x^2\right ) \log \left (-\frac {3}{2 x}\right )+\left (1-12 x+36 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )} \, dx=\frac {3 (-1+x)}{6-\frac {1}{x}-\frac {1}{\log \left (-\frac {3}{2 x}\right )}} \]
Time = 0.33 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \frac {-3 x+3 x^2-3 x^2 \log \left (-\frac {3}{2 x}\right )+\left (3-6 x+18 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )}{x^2+\left (2 x-12 x^2\right ) \log \left (-\frac {3}{2 x}\right )+\left (1-12 x+36 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )} \, dx=\frac {-5 x+\left (-5-6 x+36 x^2\right ) \log \left (-\frac {3}{2 x}\right )}{12 \left (-x+(-1+6 x) \log \left (-\frac {3}{2 x}\right )\right )} \]
Integrate[(-3*x + 3*x^2 - 3*x^2*Log[-3/(2*x)] + (3 - 6*x + 18*x^2)*Log[-3/ (2*x)]^2)/(x^2 + (2*x - 12*x^2)*Log[-3/(2*x)] + (1 - 12*x + 36*x^2)*Log[-3 /(2*x)]^2),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.
\(\displaystyle \int \frac {3 x^2+\left (18 x^2-6 x+3\right ) \log ^2\left (-\frac {3}{2 x}\right )-3 x^2 \log \left (-\frac {3}{2 x}\right )-3 x}{x^2+\left (36 x^2-12 x+1\right ) \log ^2\left (-\frac {3}{2 x}\right )+\left (2 x-12 x^2\right ) \log \left (-\frac {3}{2 x}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {3 x^2+\left (18 x^2-6 x+3\right ) \log ^2\left (-\frac {3}{2 x}\right )-3 x^2 \log \left (-\frac {3}{2 x}\right )-3 x}{\left (x-6 x \log \left (-\frac {3}{2 x}\right )+\log \left (-\frac {3}{2 x}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 \left (6 x^2-2 x+1\right )}{(6 x-1)^2}+\frac {3 x \left (6 x^2-3 x+2\right )}{(6 x-1)^2 \left (-x+6 x \log \left (-\frac {3}{2 x}\right )-\log \left (-\frac {3}{2 x}\right )\right )}+\frac {3 x \left (36 x^3-49 x^2+14 x-1\right )}{(6 x-1)^2 \left (-x+6 x \log \left (-\frac {3}{2 x}\right )-\log \left (-\frac {3}{2 x}\right )\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \frac {x^2}{\left (6 \log \left (-\frac {3}{2 x}\right ) x-x-\log \left (-\frac {3}{2 x}\right )\right )^2}dx+\frac {1}{18} \int \frac {1}{\left (6 \log \left (-\frac {3}{2 x}\right ) x-x-\log \left (-\frac {3}{2 x}\right )\right )^2}dx-\frac {37}{12} \int \frac {x}{\left (6 \log \left (-\frac {3}{2 x}\right ) x-x-\log \left (-\frac {3}{2 x}\right )\right )^2}dx+\frac {5}{72} \int \frac {1}{(6 x-1)^2 \left (6 \log \left (-\frac {3}{2 x}\right ) x-x-\log \left (-\frac {3}{2 x}\right )\right )^2}dx+\frac {1}{8} \int \frac {1}{(6 x-1) \left (6 \log \left (-\frac {3}{2 x}\right ) x-x-\log \left (-\frac {3}{2 x}\right )\right )^2}dx-\frac {1}{12} \int \frac {1}{6 \log \left (-\frac {3}{2 x}\right ) x-x-\log \left (-\frac {3}{2 x}\right )}dx+\frac {1}{2} \int \frac {x}{6 \log \left (-\frac {3}{2 x}\right ) x-x-\log \left (-\frac {3}{2 x}\right )}dx+\frac {5}{6} \int \frac {1}{(6 x-1)^2 \left (6 \log \left (-\frac {3}{2 x}\right ) x-x-\log \left (-\frac {3}{2 x}\right )\right )}dx+\frac {3}{4} \int \frac {1}{(6 x-1) \left (6 \log \left (-\frac {3}{2 x}\right ) x-x-\log \left (-\frac {3}{2 x}\right )\right )}dx+\frac {x}{2}+\frac {5}{12 (1-6 x)}\) |
Int[(-3*x + 3*x^2 - 3*x^2*Log[-3/(2*x)] + (3 - 6*x + 18*x^2)*Log[-3/(2*x)] ^2)/(x^2 + (2*x - 12*x^2)*Log[-3/(2*x)] + (1 - 12*x + 36*x^2)*Log[-3/(2*x) ]^2),x]
3.22.24.3.1 Defintions of rubi rules used
Time = 0.43 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81
method | result | size |
derivativedivides | \(-\frac {3 \left (\frac {3 \ln \left (-\frac {3}{2 x}\right )}{x}-3 \ln \left (-\frac {3}{2 x}\right )\right ) x}{-\frac {3 \ln \left (-\frac {3}{2 x}\right )}{x}+18 \ln \left (-\frac {3}{2 x}\right )-3}\) | \(47\) |
default | \(-\frac {3 \left (\frac {3 \ln \left (-\frac {3}{2 x}\right )}{x}-3 \ln \left (-\frac {3}{2 x}\right )\right ) x}{-\frac {3 \ln \left (-\frac {3}{2 x}\right )}{x}+18 \ln \left (-\frac {3}{2 x}\right )-3}\) | \(47\) |
norman | \(\frac {-\frac {x}{2}-\frac {\ln \left (-\frac {3}{2 x}\right )}{2}+3 x^{2} \ln \left (-\frac {3}{2 x}\right )}{6 x \ln \left (-\frac {3}{2 x}\right )-\ln \left (-\frac {3}{2 x}\right )-x}\) | \(48\) |
parallelrisch | \(\frac {18 x^{2} \ln \left (-\frac {3}{2 x}\right )-3 x -3 \ln \left (-\frac {3}{2 x}\right )}{36 x \ln \left (-\frac {3}{2 x}\right )-6 \ln \left (-\frac {3}{2 x}\right )-6 x}\) | \(49\) |
risch | \(\frac {36 x^{2}-6 x -5}{72 x -12}+\frac {3 \left (-1+x \right ) x^{2}}{\left (-1+6 x \right ) \left (6 x \ln \left (-\frac {3}{2 x}\right )-\ln \left (-\frac {3}{2 x}\right )-x \right )}\) | \(59\) |
int(((18*x^2-6*x+3)*ln(-3/2/x)^2-3*x^2*ln(-3/2/x)+3*x^2-3*x)/((36*x^2-12*x +1)*ln(-3/2/x)^2+(-12*x^2+2*x)*ln(-3/2/x)+x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {-3 x+3 x^2-3 x^2 \log \left (-\frac {3}{2 x}\right )+\left (3-6 x+18 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )}{x^2+\left (2 x-12 x^2\right ) \log \left (-\frac {3}{2 x}\right )+\left (1-12 x+36 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )} \, dx=\frac {{\left (36 \, x^{2} - 6 \, x - 5\right )} \log \left (-\frac {3}{2 \, x}\right ) - 5 \, x}{12 \, {\left ({\left (6 \, x - 1\right )} \log \left (-\frac {3}{2 \, x}\right ) - x\right )}} \]
integrate(((18*x^2-6*x+3)*log(-3/2/x)^2-3*x^2*log(-3/2/x)+3*x^2-3*x)/((36* x^2-12*x+1)*log(-3/2/x)^2+(-12*x^2+2*x)*log(-3/2/x)+x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {-3 x+3 x^2-3 x^2 \log \left (-\frac {3}{2 x}\right )+\left (3-6 x+18 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )}{x^2+\left (2 x-12 x^2\right ) \log \left (-\frac {3}{2 x}\right )+\left (1-12 x+36 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )} \, dx=\frac {x}{2} + \frac {3 x^{3} - 3 x^{2}}{- 6 x^{2} + x + \left (36 x^{2} - 12 x + 1\right ) \log {\left (- \frac {3}{2 x} \right )}} - \frac {5}{72 x - 12} \]
integrate(((18*x**2-6*x+3)*ln(-3/2/x)**2-3*x**2*ln(-3/2/x)+3*x**2-3*x)/((3 6*x**2-12*x+1)*ln(-3/2/x)**2+(-12*x**2+2*x)*ln(-3/2/x)+x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (20) = 40\).
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.23 \[ \int \frac {-3 x+3 x^2-3 x^2 \log \left (-\frac {3}{2 x}\right )+\left (3-6 x+18 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )}{x^2+\left (2 x-12 x^2\right ) \log \left (-\frac {3}{2 x}\right )+\left (1-12 x+36 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )} \, dx=\frac {36 \, x^{2} {\left (\log \left (3\right ) - \log \left (2\right )\right )} - x {\left (6 \, \log \left (3\right ) - 6 \, \log \left (2\right ) + 5\right )} - {\left (36 \, x^{2} - 6 \, x - 5\right )} \log \left (-x\right ) - 5 \, \log \left (3\right ) + 5 \, \log \left (2\right )}{12 \, {\left (x {\left (6 \, \log \left (3\right ) - 6 \, \log \left (2\right ) - 1\right )} - {\left (6 \, x - 1\right )} \log \left (-x\right ) - \log \left (3\right ) + \log \left (2\right )\right )}} \]
integrate(((18*x^2-6*x+3)*log(-3/2/x)^2-3*x^2*log(-3/2/x)+3*x^2-3*x)/((36* x^2-12*x+1)*log(-3/2/x)^2+(-12*x^2+2*x)*log(-3/2/x)+x^2),x, algorithm=\
1/12*(36*x^2*(log(3) - log(2)) - x*(6*log(3) - 6*log(2) + 5) - (36*x^2 - 6 *x - 5)*log(-x) - 5*log(3) + 5*log(2))/(x*(6*log(3) - 6*log(2) - 1) - (6*x - 1)*log(-x) - log(3) + log(2))
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {-3 x+3 x^2-3 x^2 \log \left (-\frac {3}{2 x}\right )+\left (3-6 x+18 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )}{x^2+\left (2 x-12 x^2\right ) \log \left (-\frac {3}{2 x}\right )+\left (1-12 x+36 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )} \, dx=\frac {1}{2} \, x - \frac {3 \, {\left (\frac {1}{x} - 1\right )}}{\frac {36 \, \log \left (-\frac {3}{2 \, x}\right )}{x} - \frac {6}{x} - \frac {12 \, \log \left (-\frac {3}{2 \, x}\right )}{x^{2}} + \frac {1}{x^{2}} + \frac {\log \left (-\frac {3}{2 \, x}\right )}{x^{3}}} + \frac {5}{2 \, {\left (\frac {1}{x} - 6\right )}} \]
integrate(((18*x^2-6*x+3)*log(-3/2/x)^2-3*x^2*log(-3/2/x)+3*x^2-3*x)/((36* x^2-12*x+1)*log(-3/2/x)^2+(-12*x^2+2*x)*log(-3/2/x)+x^2),x, algorithm=\
1/2*x - 3*(1/x - 1)/(36*log(-3/2/x)/x - 6/x - 12*log(-3/2/x)/x^2 + 1/x^2 + log(-3/2/x)/x^3) + 5/2/(1/x - 6)
Time = 10.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-3 x+3 x^2-3 x^2 \log \left (-\frac {3}{2 x}\right )+\left (3-6 x+18 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )}{x^2+\left (2 x-12 x^2\right ) \log \left (-\frac {3}{2 x}\right )+\left (1-12 x+36 x^2\right ) \log ^2\left (-\frac {3}{2 x}\right )} \, dx=-\frac {3\,x\,\ln \left (-\frac {3}{2\,x}\right )\,\left (x-1\right )}{x+\ln \left (-\frac {3}{2\,x}\right )-6\,x\,\ln \left (-\frac {3}{2\,x}\right )} \]