Integrand size = 116, antiderivative size = 32 \[ \int \frac {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx=\frac {-3-\frac {3}{x \log \left (\frac {3}{2}\right )}}{x^2 \left (-3+\frac {6}{x}+\frac {x}{\log (x)}\right )} \] Output:
(3/x/ln(2/3)-3)/x^2/(x/ln(x)-3+6/x)
Time = 0.75 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx=-\frac {3 \left (1+x \log \left (\frac {3}{2}\right )\right ) \log (x)}{\log \left (\frac {3}{2}\right ) \left (x^4-3 (-2+x) x^2 \log (x)\right )} \] Input:
Integrate[(3*x^2 + 3*x^3*Log[3/2] + (-12*x^2 - 9*x^3*Log[3/2])*Log[x] + (- 36 + 27*x - (18*x - 18*x^2)*Log[3/2])*Log[x]^2)/(-(x^7*Log[3/2]) - (12*x^5 - 6*x^6)*Log[3/2]*Log[x] - (36*x^3 - 36*x^4 + 9*x^5)*Log[3/2]*Log[x]^2),x ]
Output:
(-3*(1 + x*Log[3/2])*Log[x])/(Log[3/2]*(x^4 - 3*(-2 + x)*x^2*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.
| \(\displaystyle \int \frac {3 x^3 \log \left (\frac {3}{2}\right )+3 x^2+\left (-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )+27 x-36\right ) \log ^2(x)+\left (-9 x^3 \log \left (\frac {3}{2}\right )-12 x^2\right ) \log (x)}{x^7 \left (-\log \left (\frac {3}{2}\right )\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (9 x^5-36 x^4+36 x^3\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-3 x^3 \log \left (\frac {3}{2}\right )-3 x^2-\left (-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )+27 x-36\right ) \log ^2(x)-\left (-9 x^3 \log \left (\frac {3}{2}\right )-12 x^2\right ) \log (x)}{x^3 \log \left (\frac {3}{2}\right ) \left (x^2-3 x \log (x)+6 \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {3 \left (\log \left (\frac {3}{2}\right ) x^3+x^2-3 \left (-3 x+2 \left (x-x^2\right ) \log \left (\frac {3}{2}\right )+4\right ) \log ^2(x)-\left (3 \log \left (\frac {3}{2}\right ) x^3+4 x^2\right ) \log (x)\right )}{x^3 \left (x^2-3 \log (x) x+6 \log (x)\right )^2}dx}{\log \left (\frac {3}{2}\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \int \frac {\log \left (\frac {3}{2}\right ) x^3+x^2-3 \left (-3 x+2 \left (x-x^2\right ) \log \left (\frac {3}{2}\right )+4\right ) \log ^2(x)-\left (3 \log \left (\frac {3}{2}\right ) x^3+4 x^2\right ) \log (x)}{x^3 \left (x^2-3 \log (x) x+6 \log (x)\right )^2}dx}{\log \left (\frac {3}{2}\right )}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 \int \left (-\frac {\left (x^3-7 x^2+12 x-12\right ) \left (\log \left (\frac {3}{2}\right ) x+1\right )}{3 (x-2)^2 x \left (x^2-3 \log (x) x+6 \log (x)\right )^2}+\frac {\log \left (\frac {9}{4}\right ) x^2+\left (3-\log \left (\frac {9}{4}\right )\right ) x-4}{3 (2-x)^2 x^3}+\frac {-\log \left (\frac {3}{2}\right ) x-\log \left (\frac {9}{4}\right )-2}{3 (x-2)^2 \left (x^2-3 \log (x) x+6 \log (x)\right )}\right )dx}{\log \left (\frac {3}{2}\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \left (-\frac {1}{3} \left (1-\log \left (\frac {27}{8}\right )\right ) \int \frac {1}{\left (x^2-3 \log (x) x+6 \log (x)\right )^2}dx+\frac {4}{3} \left (1+\log \left (\frac {9}{4}\right )\right ) \int \frac {1}{(2-x)^2 \left (x^2-3 \log (x) x+6 \log (x)\right )^2}dx+\frac {4}{3} \log \left (\frac {3}{2}\right ) \int \frac {1}{(x-2) \left (x^2-3 \log (x) x+6 \log (x)\right )^2}dx+\int \frac {1}{x \left (x^2-3 \log (x) x+6 \log (x)\right )^2}dx-\frac {1}{3} \log \left (\frac {3}{2}\right ) \int \frac {x}{\left (x^2-3 \log (x) x+6 \log (x)\right )^2}dx-\frac {2}{3} \left (1+\log \left (\frac {9}{4}\right )\right ) \int \frac {1}{(x-2)^2 \left (x^2-3 \log (x) x+6 \log (x)\right )}dx-\frac {1}{3} \log \left (\frac {3}{2}\right ) \int \frac {1}{(x-2) \left (x^2-3 \log (x) x+6 \log (x)\right )}dx+\frac {1}{6 x^2}+\frac {1+\log \left (\frac {9}{4}\right )}{12 (2-x)}+\frac {1+\log \left (\frac {9}{4}\right )}{12 x}\right )}{\log \left (\frac {3}{2}\right )}\) |
Input:
Int[(3*x^2 + 3*x^3*Log[3/2] + (-12*x^2 - 9*x^3*Log[3/2])*Log[x] + (-36 + 2 7*x - (18*x - 18*x^2)*Log[3/2])*Log[x]^2)/(-(x^7*Log[3/2]) - (12*x^5 - 6*x ^6)*Log[3/2]*Log[x] - (36*x^3 - 36*x^4 + 9*x^5)*Log[3/2]*Log[x]^2),x]
Output:
$Aborted
Time = 1.36 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16
| method | result | size |
| parallelrisch | \(\frac {-9 \ln \left (\frac {2}{3}\right ) x \ln \left (x \right )+9 \ln \left (x \right )}{3 x^{2} \ln \left (\frac {2}{3}\right ) \left (x^{2}-3 x \ln \left (x \right )+6 \ln \left (x \right )\right )}\) | \(37\) |
| norman | \(\frac {-3 x \ln \left (x \right )-\frac {3 \ln \left (x \right )}{-\ln \left (2\right )+\ln \left (3\right )}}{x^{2} \left (x^{2}-3 x \ln \left (x \right )+6 \ln \left (x \right )\right )}\) | \(39\) |
| risch | \(\frac {x \ln \left (3\right )-x \ln \left (2\right )+1}{x^{2} \left (x \ln \left (3\right )-x \ln \left (2\right )-2 \ln \left (3\right )+2 \ln \left (2\right )\right )}-\frac {x \ln \left (3\right )-x \ln \left (2\right )+1}{\left (-2+x \right ) \left (-\ln \left (2\right )+\ln \left (3\right )\right ) \left (x^{2}-3 x \ln \left (x \right )+6 \ln \left (x \right )\right )}\) | \(79\) |
| default | \(-\frac {3 \left (-\frac {\ln \left (x \right )}{x^{2} \left (3 x \ln \left (x \right )-x^{2}-6 \ln \left (x \right )\right )}-\frac {\ln \left (3\right ) \ln \left (x \right )}{x \left (3 x \ln \left (x \right )-x^{2}-6 \ln \left (x \right )\right )}+\frac {\ln \left (2\right ) \ln \left (x \right )}{x \left (3 x \ln \left (x \right )-x^{2}-6 \ln \left (x \right )\right )}\right )}{-\ln \left (2\right )+\ln \left (3\right )}\) | \(88\) |
Input:
int((((-18*x^2+18*x)*ln(2/3)+27*x-36)*ln(x)^2+(9*x^3*ln(2/3)-12*x^2)*ln(x) -3*x^3*ln(2/3)+3*x^2)/((9*x^5-36*x^4+36*x^3)*ln(2/3)*ln(x)^2+(-6*x^6+12*x^ 5)*ln(2/3)*ln(x)+x^7*ln(2/3)),x,method=_RETURNVERBOSE)
Output:
1/3/x^2*(-9*ln(2/3)*x*ln(x)+9*ln(x))/ln(2/3)/(x^2-3*x*ln(x)+6*ln(x))
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x \log \left (\frac {2}{3}\right ) - 1\right )} \log \left (x\right )}{x^{4} \log \left (\frac {2}{3}\right ) - 3 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (\frac {2}{3}\right ) \log \left (x\right )} \] Input:
integrate((((-18*x^2+18*x)*log(2/3)+27*x-36)*log(x)^2+(9*x^3*log(2/3)-12*x ^2)*log(x)-3*x^3*log(2/3)+3*x^2)/((9*x^5-36*x^4+36*x^3)*log(2/3)*log(x)^2+ (-6*x^6+12*x^5)*log(2/3)*log(x)+x^7*log(2/3)),x, algorithm="fricas")
Output:
-3*(x*log(2/3) - 1)*log(x)/(x^4*log(2/3) - 3*(x^3 - 2*x^2)*log(2/3)*log(x) )
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (22) = 44\).
Time = 0.98 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.72 \[ \int \frac {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx=- \frac {x \left (- \log {\left (2 \right )} + \log {\left (3 \right )}\right ) + 1}{x^{3} \left (- \log {\left (3 \right )} + \log {\left (2 \right )}\right ) + x^{2} \left (- 2 \log {\left (2 \right )} + 2 \log {\left (3 \right )}\right )} + \frac {- x \log {\left (3 \right )} + x \log {\left (2 \right )} - 1}{- x^{3} \log {\left (2 \right )} + x^{3} \log {\left (3 \right )} - 2 x^{2} \log {\left (3 \right )} + 2 x^{2} \log {\left (2 \right )} + \left (- 3 x^{2} \log {\left (3 \right )} + 3 x^{2} \log {\left (2 \right )} - 12 x \log {\left (2 \right )} + 12 x \log {\left (3 \right )} - 12 \log {\left (3 \right )} + 12 \log {\left (2 \right )}\right ) \log {\left (x \right )}} \] Input:
integrate((((-18*x**2+18*x)*ln(2/3)+27*x-36)*ln(x)**2+(9*x**3*ln(2/3)-12*x **2)*ln(x)-3*x**3*ln(2/3)+3*x**2)/((9*x**5-36*x**4+36*x**3)*ln(2/3)*ln(x)* *2+(-6*x**6+12*x**5)*ln(2/3)*ln(x)+x**7*ln(2/3)),x)
Output:
-(x*(-log(2) + log(3)) + 1)/(x**3*(-log(3) + log(2)) + x**2*(-2*log(2) + 2 *log(3))) + (-x*log(3) + x*log(2) - 1)/(-x**3*log(2) + x**3*log(3) - 2*x** 2*log(3) + 2*x**2*log(2) + (-3*x**2*log(3) + 3*x**2*log(2) - 12*x*log(2) + 12*x*log(3) - 12*log(3) + 12*log(2))*log(x))
Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx=-\frac {3 \, {\left (x {\left (\log \left (3\right ) - \log \left (2\right )\right )} + 1\right )} \log \left (x\right )}{x^{4} {\left (\log \left (3\right ) - \log \left (2\right )\right )} - 3 \, {\left (x^{3} {\left (\log \left (3\right ) - \log \left (2\right )\right )} - 2 \, x^{2} {\left (\log \left (3\right ) - \log \left (2\right )\right )}\right )} \log \left (x\right )} \] Input:
integrate((((-18*x^2+18*x)*log(2/3)+27*x-36)*log(x)^2+(9*x^3*log(2/3)-12*x ^2)*log(x)-3*x^3*log(2/3)+3*x^2)/((9*x^5-36*x^4+36*x^3)*log(2/3)*log(x)^2+ (-6*x^6+12*x^5)*log(2/3)*log(x)+x^7*log(2/3)),x, algorithm="maxima")
Output:
-3*(x*(log(3) - log(2)) + 1)*log(x)/(x^4*(log(3) - log(2)) - 3*(x^3*(log(3 ) - log(2)) - 2*x^2*(log(3) - log(2)))*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (30) = 60\).
Time = 0.14 (sec) , antiderivative size = 151, normalized size of antiderivative = 4.72 \[ \int \frac {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx=-\frac {2 \, x \log \left (3\right ) - 2 \, x \log \left (2\right ) + x + 2}{4 \, {\left (x^{2} \log \left (3\right ) - x^{2} \log \left (2\right )\right )}} - \frac {x \log \left (3\right ) - x \log \left (2\right ) + 1}{x^{3} \log \left (3\right ) - x^{3} \log \left (2\right ) - 3 \, x^{2} \log \left (3\right ) \log \left (x\right ) + 3 \, x^{2} \log \left (2\right ) \log \left (x\right ) - 2 \, x^{2} \log \left (3\right ) + 2 \, x^{2} \log \left (2\right ) + 12 \, x \log \left (3\right ) \log \left (x\right ) - 12 \, x \log \left (2\right ) \log \left (x\right ) - 12 \, \log \left (3\right ) \log \left (x\right ) + 12 \, \log \left (2\right ) \log \left (x\right )} + \frac {2 \, \log \left (3\right ) - 2 \, \log \left (2\right ) + 1}{4 \, {\left (x \log \left (3\right ) - x \log \left (2\right ) - 2 \, \log \left (3\right ) + 2 \, \log \left (2\right )\right )}} \] Input:
integrate((((-18*x^2+18*x)*log(2/3)+27*x-36)*log(x)^2+(9*x^3*log(2/3)-12*x ^2)*log(x)-3*x^3*log(2/3)+3*x^2)/((9*x^5-36*x^4+36*x^3)*log(2/3)*log(x)^2+ (-6*x^6+12*x^5)*log(2/3)*log(x)+x^7*log(2/3)),x, algorithm="giac")
Output:
-1/4*(2*x*log(3) - 2*x*log(2) + x + 2)/(x^2*log(3) - x^2*log(2)) - (x*log( 3) - x*log(2) + 1)/(x^3*log(3) - x^3*log(2) - 3*x^2*log(3)*log(x) + 3*x^2* log(2)*log(x) - 2*x^2*log(3) + 2*x^2*log(2) + 12*x*log(3)*log(x) - 12*x*lo g(2)*log(x) - 12*log(3)*log(x) + 12*log(2)*log(x)) + 1/4*(2*log(3) - 2*log (2) + 1)/(x*log(3) - x*log(2) - 2*log(3) + 2*log(2))
Timed out. \[ \int \frac {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx=\int \frac {{\ln \left (x\right )}^2\,\left (27\,x+\ln \left (\frac {2}{3}\right )\,\left (18\,x-18\,x^2\right )-36\right )+\ln \left (x\right )\,\left (9\,x^3\,\ln \left (\frac {2}{3}\right )-12\,x^2\right )-3\,x^3\,\ln \left (\frac {2}{3}\right )+3\,x^2}{x^7\,\ln \left (\frac {2}{3}\right )+\ln \left (\frac {2}{3}\right )\,{\ln \left (x\right )}^2\,\left (9\,x^5-36\,x^4+36\,x^3\right )+\ln \left (\frac {2}{3}\right )\,\ln \left (x\right )\,\left (12\,x^5-6\,x^6\right )} \,d x \] Input:
int((log(x)^2*(27*x + log(2/3)*(18*x - 18*x^2) - 36) + log(x)*(9*x^3*log(2 /3) - 12*x^2) - 3*x^3*log(2/3) + 3*x^2)/(x^7*log(2/3) + log(2/3)*log(x)^2* (36*x^3 - 36*x^4 + 9*x^5) + log(2/3)*log(x)*(12*x^5 - 6*x^6)),x)
Output:
int((log(x)^2*(27*x + log(2/3)*(18*x - 18*x^2) - 36) + log(x)*(9*x^3*log(2 /3) - 12*x^2) - 3*x^3*log(2/3) + 3*x^2)/(x^7*log(2/3) + log(2/3)*log(x)^2* (36*x^3 - 36*x^4 + 9*x^5) + log(2/3)*log(x)*(12*x^5 - 6*x^6)), x)
Time = 0.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {3 x^2+3 x^3 \log \left (\frac {3}{2}\right )+\left (-12 x^2-9 x^3 \log \left (\frac {3}{2}\right )\right ) \log (x)+\left (-36+27 x-\left (18 x-18 x^2\right ) \log \left (\frac {3}{2}\right )\right ) \log ^2(x)}{-x^7 \log \left (\frac {3}{2}\right )-\left (12 x^5-6 x^6\right ) \log \left (\frac {3}{2}\right ) \log (x)-\left (36 x^3-36 x^4+9 x^5\right ) \log \left (\frac {3}{2}\right ) \log ^2(x)} \, dx=\frac {3 \,\mathrm {log}\left (x \right ) \left (\mathrm {log}\left (\frac {2}{3}\right ) x -1\right )}{\mathrm {log}\left (\frac {2}{3}\right ) x^{2} \left (3 \,\mathrm {log}\left (x \right ) x -6 \,\mathrm {log}\left (x \right )-x^{2}\right )} \] Input:
int((((-18*x^2+18*x)*log(2/3)+27*x-36)*log(x)^2+(9*x^3*log(2/3)-12*x^2)*lo g(x)-3*x^3*log(2/3)+3*x^2)/((9*x^5-36*x^4+36*x^3)*log(2/3)*log(x)^2+(-6*x^ 6+12*x^5)*log(2/3)*log(x)+x^7*log(2/3)),x)
Output:
(3*log(x)*(log(2/3)*x - 1))/(log(2/3)*x**2*(3*log(x)*x - 6*log(x) - x**2))