Integrand size = 102, antiderivative size = 31 \[ \int \frac {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx=x \left (\frac {x+\frac {(-1+\log (x))^2}{x}}{3+2 x \log (x)}-\log (2 x)\right ) \] Output:
x*(((ln(x)-1)^2/x+x)/(3+2*x*ln(x))-ln(2*x))
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx=\frac {1+x^2+\log ^2(x)-3 x \log (2 x)-2 \log (x) \left (1+x^2 \log (2 x)\right )}{3+2 x \log (x)} \] Input:
Integrate[(-6 - 11*x + 6*x^2 - 2*x^3 + (6 - 2*x - 12*x^2 + 2*x^3)*Log[x] + (6*x - 4*x^3)*Log[x]^2 - 2*x*Log[x]^3 + (-9*x - 12*x^2*Log[x] - 4*x^3*Log [x]^2)*Log[2*x])/(9*x + 12*x^2*Log[x] + 4*x^3*Log[x]^2),x]
Output:
(1 + x^2 + Log[x]^2 - 3*x*Log[2*x] - 2*Log[x]*(1 + x^2*Log[2*x]))/(3 + 2*x *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.
\(\displaystyle \int \frac {-2 x^3+\left (6 x-4 x^3\right ) \log ^2(x)+6 x^2+\left (-4 x^3 \log ^2(x)-12 x^2 \log (x)-9 x\right ) \log (2 x)+\left (2 x^3-12 x^2-2 x+6\right ) \log (x)-11 x-2 x \log ^3(x)-6}{4 x^3 \log ^2(x)+12 x^2 \log (x)+9 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-2 x^3+\left (6 x-4 x^3\right ) \log ^2(x)+6 x^2+\left (-4 x^3 \log ^2(x)-12 x^2 \log (x)-9 x\right ) \log (2 x)+\left (2 x^3-12 x^2-2 x+6\right ) \log (x)-11 x-2 x \log ^3(x)-6}{x (2 x \log (x)+3)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2 \left (2 x^2-3\right ) \log ^2(x)}{(2 x \log (x)+3)^2}-\frac {2 x^2}{(2 x \log (x)+3)^2}+\frac {2 \left (x^3-6 x^2-x+3\right ) \log (x)}{x (2 x \log (x)+3)^2}-\frac {2 \log ^3(x)}{(2 x \log (x)+3)^2}-\log (2 x)+\frac {6 x}{(2 x \log (x)+3)^2}-\frac {6}{x (2 x \log (x)+3)^2}-\frac {11}{(2 x \log (x)+3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {27}{4} \int \frac {1}{x^3 (2 x \log (x)+3)^2}dx-\frac {27}{4} \int \frac {1}{x^3 (2 x \log (x)+3)}dx+\frac {9}{2} \int \frac {1}{x^2 (2 x \log (x)+3)^2}dx-2 \int \frac {x^2}{(2 x \log (x)+3)^2}dx-6 \int \frac {1}{x^2 (2 x \log (x)+3)}dx-2 \int \frac {1}{(2 x \log (x)+3)^2}dx-3 \int \frac {1}{x (2 x \log (x)+3)^2}dx+3 \int \frac {x}{(2 x \log (x)+3)^2}dx-\int \frac {1}{x (2 x \log (x)+3)}dx+\int \frac {x}{2 x \log (x)+3}dx-\frac {3}{4 x^2}-\frac {1}{x}+\frac {\log (x)}{2 x}-x \log (2 x)\) |
Input:
Int[(-6 - 11*x + 6*x^2 - 2*x^3 + (6 - 2*x - 12*x^2 + 2*x^3)*Log[x] + (6*x - 4*x^3)*Log[x]^2 - 2*x*Log[x]^3 + (-9*x - 12*x^2*Log[x] - 4*x^3*Log[x]^2) *Log[2*x])/(9*x + 12*x^2*Log[x] + 4*x^3*Log[x]^2),x]
Output:
$Aborted
Time = 3.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42
method | result | size |
default | \(-\frac {2 \ln \left (x \right )-\ln \left (x \right )^{2}-x^{2}+2 x^{2} \ln \left (x \right )^{2}-\frac {11}{2}}{3+2 x \ln \left (x \right )}-x \ln \left (2\right )\) | \(44\) |
parallelrisch | \(-\frac {-36+108 x \ln \left (2 x \right )+72 \ln \left (x \right )-36 \ln \left (x \right )^{2}-36 x^{2}+72 \ln \left (x \right ) \ln \left (2 x \right ) x^{2}}{36 \left (3+2 x \ln \left (x \right )\right )}\) | \(47\) |
risch | \(-\frac {\left (2 x^{2}-1\right ) \ln \left (x \right )}{2 x}-\frac {3+4 x^{3} \ln \left (2\right )+4 x}{4 x^{2}}+\frac {4 x^{4}+4 x^{2}+12 x +9}{4 x^{2} \left (3+2 x \ln \left (x \right )\right )}\) | \(62\) |
Input:
int(((-4*x^3*ln(x)^2-12*x^2*ln(x)-9*x)*ln(2*x)-2*x*ln(x)^3+(-4*x^3+6*x)*ln (x)^2+(2*x^3-12*x^2-2*x+6)*ln(x)-2*x^3+6*x^2-11*x-6)/(4*x^3*ln(x)^2+12*x^2 *ln(x)+9*x),x,method=_RETURNVERBOSE)
Output:
-(2*ln(x)-ln(x)^2-x^2+2*x^2*ln(x)^2-11/2)/(3+2*x*ln(x))-x*ln(2)
Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx=-\frac {{\left (2 \, x^{2} - 1\right )} \log \left (x\right )^{2} - x^{2} + 3 \, x \log \left (2\right ) + {\left (2 \, x^{2} \log \left (2\right ) + 3 \, x + 2\right )} \log \left (x\right ) - 1}{2 \, x \log \left (x\right ) + 3} \] Input:
integrate(((-4*x^3*log(x)^2-12*x^2*log(x)-9*x)*log(2*x)-2*x*log(x)^3+(-4*x ^3+6*x)*log(x)^2+(2*x^3-12*x^2-2*x+6)*log(x)-2*x^3+6*x^2-11*x-6)/(4*x^3*lo g(x)^2+12*x^2*log(x)+9*x),x, algorithm="fricas")
Output:
-((2*x^2 - 1)*log(x)^2 - x^2 + 3*x*log(2) + (2*x^2*log(2) + 3*x + 2)*log(x ) - 1)/(2*x*log(x) + 3)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx=- x \log {\left (2 \right )} + \frac {4 x^{4} + 4 x^{2} + 12 x + 9}{8 x^{3} \log {\left (x \right )} + 12 x^{2}} + \frac {\left (1 - 2 x^{2}\right ) \log {\left (x \right )}}{2 x} - \frac {4 x + 3}{4 x^{2}} \] Input:
integrate(((-4*x**3*ln(x)**2-12*x**2*ln(x)-9*x)*ln(2*x)-2*x*ln(x)**3+(-4*x **3+6*x)*ln(x)**2+(2*x**3-12*x**2-2*x+6)*ln(x)-2*x**3+6*x**2-11*x-6)/(4*x* *3*ln(x)**2+12*x**2*ln(x)+9*x),x)
Output:
-x*log(2) + (4*x**4 + 4*x**2 + 12*x + 9)/(8*x**3*log(x) + 12*x**2) + (1 - 2*x**2)*log(x)/(2*x) - (4*x + 3)/(4*x**2)
Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx=-\frac {{\left (2 \, x^{2} - 1\right )} \log \left (x\right )^{2} - x^{2} + 3 \, x \log \left (2\right ) + {\left (2 \, x^{2} \log \left (2\right ) + 3 \, x + 2\right )} \log \left (x\right ) - 1}{2 \, x \log \left (x\right ) + 3} \] Input:
integrate(((-4*x^3*log(x)^2-12*x^2*log(x)-9*x)*log(2*x)-2*x*log(x)^3+(-4*x ^3+6*x)*log(x)^2+(2*x^3-12*x^2-2*x+6)*log(x)-2*x^3+6*x^2-11*x-6)/(4*x^3*lo g(x)^2+12*x^2*log(x)+9*x),x, algorithm="maxima")
Output:
-((2*x^2 - 1)*log(x)^2 - x^2 + 3*x*log(2) + (2*x^2*log(2) + 3*x + 2)*log(x ) - 1)/(2*x*log(x) + 3)
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx=-x \log \left (2\right ) - \frac {1}{2} \, {\left (2 \, x - \frac {1}{x}\right )} \log \left (x\right ) + \frac {4 \, x^{4} + 4 \, x^{2} + 12 \, x + 9}{4 \, {\left (2 \, x^{3} \log \left (x\right ) + 3 \, x^{2}\right )}} - \frac {4 \, x + 3}{4 \, x^{2}} \] Input:
integrate(((-4*x^3*log(x)^2-12*x^2*log(x)-9*x)*log(2*x)-2*x*log(x)^3+(-4*x ^3+6*x)*log(x)^2+(2*x^3-12*x^2-2*x+6)*log(x)-2*x^3+6*x^2-11*x-6)/(4*x^3*lo g(x)^2+12*x^2*log(x)+9*x),x, algorithm="giac")
Output:
-x*log(2) - 1/2*(2*x - 1/x)*log(x) + 1/4*(4*x^4 + 4*x^2 + 12*x + 9)/(2*x^3 *log(x) + 3*x^2) - 1/4*(4*x + 3)/x^2
Time = 3.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.74 \[ \int \frac {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx=-\frac {2\,x^2\,\ln \left (x\right )+\frac {11\,x^3\,\ln \left (x\right )}{3}+3\,x^3\,\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )-x^2\,{\ln \left (x\right )}^2+2\,x^4\,{\ln \left (x\right )}^2-x^4+2\,x^4\,\ln \left (x\right )\,\left (\ln \left (2\,x\right )-\ln \left (x\right )\right )}{2\,x^3\,\ln \left (x\right )+3\,x^2} \] Input:
int(-(11*x - log(x)^2*(6*x - 4*x^3) + 2*x*log(x)^3 + log(2*x)*(9*x + 12*x^ 2*log(x) + 4*x^3*log(x)^2) - 6*x^2 + 2*x^3 + log(x)*(2*x + 12*x^2 - 2*x^3 - 6) + 6)/(9*x + 12*x^2*log(x) + 4*x^3*log(x)^2),x)
Output:
-(2*x^2*log(x) + (11*x^3*log(x))/3 + 3*x^3*(log(2*x) - log(x)) - x^2*log(x )^2 + 2*x^4*log(x)^2 - x^4 + 2*x^4*log(x)*(log(2*x) - log(x)))/(2*x^3*log( x) + 3*x^2)
Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(x)} \, dx=\frac {-6 \,\mathrm {log}\left (2 x \right ) \mathrm {log}\left (x \right ) x^{2}-9 \,\mathrm {log}\left (2 x \right ) x +3 \mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) x -6 \,\mathrm {log}\left (x \right )+3 x^{2}}{6 \,\mathrm {log}\left (x \right ) x +9} \] Input:
int(((-4*x^3*log(x)^2-12*x^2*log(x)-9*x)*log(2*x)-2*x*log(x)^3+(-4*x^3+6*x )*log(x)^2+(2*x^3-12*x^2-2*x+6)*log(x)-2*x^3+6*x^2-11*x-6)/(4*x^3*log(x)^2 +12*x^2*log(x)+9*x),x)
Output:
( - 6*log(2*x)*log(x)*x**2 - 9*log(2*x)*x + 3*log(x)**2 - 2*log(x)*x - 6*l og(x) + 3*x**2)/(3*(2*log(x)*x + 3))