Integrand size = 110, antiderivative size = 30 \[ \int \frac {x^3 \log (2)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)+\left (12 x-2 x^2+(-12+2 x) \log (2)\right ) \log ^2(6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (6 x^2-x^3+\left (6-13 x+2 x^2\right ) \log (2)\right ) \log ^2(6-x)} \, dx=\log \left (x^2-x \log (2) \left (2-\frac {1}{x}-\frac {x^2}{\log (6-x)}\right )\right ) \] Output:
ln(x^2-(2-1/x-x^2/ln(6-x))*x*ln(2))
\[ \int \frac {x^3 \log (2)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)+\left (12 x-2 x^2+(-12+2 x) \log (2)\right ) \log ^2(6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (6 x^2-x^3+\left (6-13 x+2 x^2\right ) \log (2)\right ) \log ^2(6-x)} \, dx=\int \frac {x^3 \log (2)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)+\left (12 x-2 x^2+(-12+2 x) \log (2)\right ) \log ^2(6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (6 x^2-x^3+\left (6-13 x+2 x^2\right ) \log (2)\right ) \log ^2(6-x)} \, dx \] Input:
Integrate[(x^3*Log[2] + (18*x^2 - 3*x^3)*Log[2]*Log[6 - x] + (12*x - 2*x^2 + (-12 + 2*x)*Log[2])*Log[6 - x]^2)/((6*x^3 - x^4)*Log[2]*Log[6 - x] + (6 *x^2 - x^3 + (6 - 13*x + 2*x^2)*Log[2])*Log[6 - x]^2),x]
Output:
Integrate[(x^3*Log[2] + (18*x^2 - 3*x^3)*Log[2]*Log[6 - x] + (12*x - 2*x^2 + (-12 + 2*x)*Log[2])*Log[6 - x]^2)/((6*x^3 - x^4)*Log[2]*Log[6 - x] + (6 *x^2 - x^3 + (6 - 13*x + 2*x^2)*Log[2])*Log[6 - 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.
| \(\displaystyle \int \frac {x^3 \log (2)+\left (-2 x^2+12 x+(2 x-12) \log (2)\right ) \log ^2(6-x)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (-x^3+6 x^2+\left (2 x^2-13 x+6\right ) \log (2)\right ) \log ^2(6-x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {x^3 \log (2)+\left (-2 x^2+12 x+(2 x-12) \log (2)\right ) \log ^2(6-x)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)}{(6-x) \log (6-x) \left (x^3 \log (2)+x^2 \log (6-x)-x \log (4) \log (6-x)+\log (2) \log (6-x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 (x-\log (2))}{x^2-x \log (4)+\log (2)}+\frac {x^2 \log (2) \left (x^2-2 x \log (4)+\log (8)\right )}{\left (x^2-x \log (4)+\log (2)\right ) \left (x^3 \log (2)+x^2 \log (6-x)-x \log (4) \log (6-x)+\log (2) \log (6-x)\right )}+\frac {x^2-x \log (4)+\log (2)}{(x-6) \left (x^3 \log (2)+x^2 \log (6-x)-x \log (4) \log (6-x)+\log (2) \log (6-x)\right )}-\frac {1}{(x-6) \log (6-x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\log (2) \log (4) \left (\log ^2(4)-\log (8)\right ) \left (1-\frac {i \log (4)}{\sqrt {\log (16)-\log ^2(4)}}\right ) \int \frac {1}{\left (2 x-i \sqrt {4 \log (2)-\log ^2(4)}-\log (4)\right ) \left (\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)\right )}dx-\log (2) \log (4) \left (\log ^2(4)-\log (8)\right ) \left (1+\frac {i \log (4)}{\sqrt {\log (16)-\log ^2(4)}}\right ) \int \frac {1}{\left (2 x+i \sqrt {4 \log (2)-\log ^2(4)}-\log (4)\right ) \left (\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)\right )}dx-\frac {2 i \log ^2(2) (1-\log (4)) \log (4) \int \frac {1}{\left (2 x+i \sqrt {4 \log (2)-\log ^2(4)}-\log (4)\right ) \left (\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)\right )}dx}{\sqrt {\log (16)-\log ^2(4)}}-\frac {2 i \log ^2(2) (1-\log (4)) \log (4) \int \frac {1}{\left (-2 x+i \sqrt {4 \log (2)-\log ^2(4)}+\log (4)\right ) \left (\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)\right )}dx}{\sqrt {\log (16)-\log ^2(4)}}+\log (2) (1-\log (4)) \log (4) \int \frac {1}{\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)}dx+2 (3-\log (2)) \int \frac {1}{\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)}dx+(36+\log (2)-6 \log (4)) \int \frac {1}{(x-6) \left (\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)\right )}dx-\log (2) \log (4) \int \frac {x}{\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)}dx+\int \frac {x}{\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)}dx+\log (2) \int \frac {x^2}{\log (2) x^3+\log (6-x) x^2-\log (4) \log (6-x) x+\log (2) \log (6-x)}dx+\frac {2 \log (2) \log \left (x^2-x \log (4)+\log (2)\right )}{\log (4)}-\log (\log (6-x))\) |
Input:
Int[(x^3*Log[2] + (18*x^2 - 3*x^3)*Log[2]*Log[6 - x] + (12*x - 2*x^2 + (-1 2 + 2*x)*Log[2])*Log[6 - x]^2)/((6*x^3 - x^4)*Log[2]*Log[6 - x] + (6*x^2 - x^3 + (6 - 13*x + 2*x^2)*Log[2])*Log[6 - x]^2),x]
Output:
$Aborted
Time = 1.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63
| method | result | size |
| norman | \(-\ln \left (\ln \left (-x +6\right )\right )+\ln \left (x^{3} \ln \left (2\right )-2 \ln \left (-x +6\right ) \ln \left (2\right ) x +\ln \left (-x +6\right ) x^{2}+\ln \left (2\right ) \ln \left (-x +6\right )\right )\) | \(49\) |
| parallelrisch | \(-\ln \left (\ln \left (-x +6\right )\right )+\ln \left (\frac {x^{3} \ln \left (2\right )-2 \ln \left (-x +6\right ) \ln \left (2\right ) x +\ln \left (-x +6\right ) x^{2}+\ln \left (2\right ) \ln \left (-x +6\right )}{\ln \left (2\right )}\right )\) | \(54\) |
| risch | \(\ln \left (-2 x \ln \left (2\right )+x^{2}+\ln \left (2\right )\right )-\ln \left (\ln \left (-x +6\right )\right )+\ln \left (\ln \left (-x +6\right )-\frac {x^{3} \ln \left (2\right )}{2 x \ln \left (2\right )-x^{2}-\ln \left (2\right )}\right )\) | \(55\) |
| derivativedivides | \(-\ln \left (\ln \left (-x +6\right )\right )+\ln \left (-\ln \left (2\right ) \left (-x +6\right )^{3}+2 \ln \left (-x +6\right ) \ln \left (2\right ) \left (-x +6\right )+\ln \left (-x +6\right ) \left (-x +6\right )^{2}+18 \ln \left (2\right ) \left (-x +6\right )^{2}-11 \ln \left (2\right ) \ln \left (-x +6\right )-12 \left (-x +6\right ) \ln \left (-x +6\right )-108 \left (-x +6\right ) \ln \left (2\right )+36 \ln \left (-x +6\right )+216 \ln \left (2\right )\right )\) | \(108\) |
| default | \(-\ln \left (\ln \left (-x +6\right )\right )+\ln \left (-\ln \left (2\right ) \left (-x +6\right )^{3}+2 \ln \left (-x +6\right ) \ln \left (2\right ) \left (-x +6\right )+\ln \left (-x +6\right ) \left (-x +6\right )^{2}+18 \ln \left (2\right ) \left (-x +6\right )^{2}-11 \ln \left (2\right ) \ln \left (-x +6\right )-12 \left (-x +6\right ) \ln \left (-x +6\right )-108 \left (-x +6\right ) \ln \left (2\right )+36 \ln \left (-x +6\right )+216 \ln \left (2\right )\right )\) | \(108\) |
Input:
int((((2*x-12)*ln(2)-2*x^2+12*x)*ln(-x+6)^2+(-3*x^3+18*x^2)*ln(2)*ln(-x+6) +x^3*ln(2))/(((2*x^2-13*x+6)*ln(2)-x^3+6*x^2)*ln(-x+6)^2+(-x^4+6*x^3)*ln(2 )*ln(-x+6)),x,method=_RETURNVERBOSE)
Output:
-ln(ln(-x+6))+ln(x^3*ln(2)-2*ln(-x+6)*ln(2)*x+ln(-x+6)*x^2+ln(2)*ln(-x+6))
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (26) = 52\).
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.30 \[ \int \frac {x^3 \log (2)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)+\left (12 x-2 x^2+(-12+2 x) \log (2)\right ) \log ^2(6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (6 x^2-x^3+\left (6-13 x+2 x^2\right ) \log (2)\right ) \log ^2(6-x)} \, dx=\log \left (x^{2} - {\left (2 \, x - 1\right )} \log \left (2\right )\right ) + \log \left (-\frac {x^{3} \log \left (2\right ) + {\left (x^{2} - {\left (2 \, x - 1\right )} \log \left (2\right )\right )} \log \left (-x + 6\right )}{x^{2} - {\left (2 \, x - 1\right )} \log \left (2\right )}\right ) - \log \left (\log \left (-x + 6\right )\right ) \] Input:
integrate((((2*x-12)*log(2)-2*x^2+12*x)*log(6-x)^2+(-3*x^3+18*x^2)*log(2)* log(6-x)+x^3*log(2))/(((2*x^2-13*x+6)*log(2)-x^3+6*x^2)*log(6-x)^2+(-x^4+6 *x^3)*log(2)*log(6-x)),x, algorithm="fricas")
Output:
log(x^2 - (2*x - 1)*log(2)) + log(-(x^3*log(2) + (x^2 - (2*x - 1)*log(2))* log(-x + 6))/(x^2 - (2*x - 1)*log(2))) - log(log(-x + 6))
Exception generated. \[ \int \frac {x^3 \log (2)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)+\left (12 x-2 x^2+(-12+2 x) \log (2)\right ) \log ^2(6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (6 x^2-x^3+\left (6-13 x+2 x^2\right ) \log (2)\right ) \log ^2(6-x)} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((((2*x-12)*ln(2)-2*x**2+12*x)*ln(6-x)**2+(-3*x**3+18*x**2)*ln(2) *ln(6-x)+x**3*ln(2))/(((2*x**2-13*x+6)*ln(2)-x**3+6*x**2)*ln(6-x)**2+(-x** 4+6*x**3)*ln(2)*ln(6-x)),x)
Output:
Exception raised: PolynomialError >> 1/(x**5 - 6*x**4 - 4*x**4*log(2) + 4* x**3*log(2)**2 + 26*x**3*log(2) - 28*x**2*log(2)**2 - 12*x**2*log(2) + 25* x*log(2)**2 - 6*log(2)**2) contains an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (26) = 52\).
Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {x^3 \log (2)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)+\left (12 x-2 x^2+(-12+2 x) \log (2)\right ) \log ^2(6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (6 x^2-x^3+\left (6-13 x+2 x^2\right ) \log (2)\right ) \log ^2(6-x)} \, dx=\log \left (x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )\right ) + \log \left (\frac {x^{3} \log \left (2\right ) + {\left (x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )\right )} \log \left (-x + 6\right )}{x^{2} - 2 \, x \log \left (2\right ) + \log \left (2\right )}\right ) - \log \left (\log \left (-x + 6\right )\right ) \] Input:
integrate((((2*x-12)*log(2)-2*x^2+12*x)*log(6-x)^2+(-3*x^3+18*x^2)*log(2)* log(6-x)+x^3*log(2))/(((2*x^2-13*x+6)*log(2)-x^3+6*x^2)*log(6-x)^2+(-x^4+6 *x^3)*log(2)*log(6-x)),x, algorithm="maxima")
Output:
log(x^2 - 2*x*log(2) + log(2)) + log((x^3*log(2) + (x^2 - 2*x*log(2) + log (2))*log(-x + 6))/(x^2 - 2*x*log(2) + log(2))) - log(log(-x + 6))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (26) = 52\).
Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.20 \[ \int \frac {x^3 \log (2)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)+\left (12 x-2 x^2+(-12+2 x) \log (2)\right ) \log ^2(6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (6 x^2-x^3+\left (6-13 x+2 x^2\right ) \log (2)\right ) \log ^2(6-x)} \, dx=\log \left (-{\left (x - 6\right )}^{3} \log \left (2\right ) - 18 \, {\left (x - 6\right )}^{2} \log \left (2\right ) - {\left (x - 6\right )}^{2} \log \left (-x + 6\right ) + 2 \, {\left (x - 6\right )} \log \left (2\right ) \log \left (-x + 6\right ) - 108 \, {\left (x - 6\right )} \log \left (2\right ) - 12 \, {\left (x - 6\right )} \log \left (-x + 6\right ) + 11 \, \log \left (2\right ) \log \left (-x + 6\right ) - 216 \, \log \left (2\right ) - 36 \, \log \left (-x + 6\right )\right ) - \log \left (\log \left (-x + 6\right )\right ) \] Input:
integrate((((2*x-12)*log(2)-2*x^2+12*x)*log(6-x)^2+(-3*x^3+18*x^2)*log(2)* log(6-x)+x^3*log(2))/(((2*x^2-13*x+6)*log(2)-x^3+6*x^2)*log(6-x)^2+(-x^4+6 *x^3)*log(2)*log(6-x)),x, algorithm="giac")
Output:
log(-(x - 6)^3*log(2) - 18*(x - 6)^2*log(2) - (x - 6)^2*log(-x + 6) + 2*(x - 6)*log(2)*log(-x + 6) - 108*(x - 6)*log(2) - 12*(x - 6)*log(-x + 6) + 1 1*log(2)*log(-x + 6) - 216*log(2) - 36*log(-x + 6)) - log(log(-x + 6))
Timed out. \[ \int \frac {x^3 \log (2)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)+\left (12 x-2 x^2+(-12+2 x) \log (2)\right ) \log ^2(6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (6 x^2-x^3+\left (6-13 x+2 x^2\right ) \log (2)\right ) \log ^2(6-x)} \, dx=\int \frac {x^3\,\ln \left (2\right )+{\ln \left (6-x\right )}^2\,\left (12\,x+\ln \left (2\right )\,\left (2\,x-12\right )-2\,x^2\right )+\ln \left (2\right )\,\ln \left (6-x\right )\,\left (18\,x^2-3\,x^3\right )}{\left (\ln \left (2\right )\,\left (2\,x^2-13\,x+6\right )+6\,x^2-x^3\right )\,{\ln \left (6-x\right )}^2+\ln \left (2\right )\,\left (6\,x^3-x^4\right )\,\ln \left (6-x\right )} \,d x \] Input:
int((x^3*log(2) + log(6 - x)^2*(12*x + log(2)*(2*x - 12) - 2*x^2) + log(2) *log(6 - x)*(18*x^2 - 3*x^3))/(log(6 - x)^2*(log(2)*(2*x^2 - 13*x + 6) + 6 *x^2 - x^3) + log(2)*log(6 - x)*(6*x^3 - x^4)),x)
Output:
int((x^3*log(2) + log(6 - x)^2*(12*x + log(2)*(2*x - 12) - 2*x^2) + log(2) *log(6 - x)*(18*x^2 - 3*x^3))/(log(6 - x)^2*(log(2)*(2*x^2 - 13*x + 6) + 6 *x^2 - x^3) + log(2)*log(6 - x)*(6*x^3 - x^4)), x)
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {x^3 \log (2)+\left (18 x^2-3 x^3\right ) \log (2) \log (6-x)+\left (12 x-2 x^2+(-12+2 x) \log (2)\right ) \log ^2(6-x)}{\left (6 x^3-x^4\right ) \log (2) \log (6-x)+\left (6 x^2-x^3+\left (6-13 x+2 x^2\right ) \log (2)\right ) \log ^2(6-x)} \, dx=-\mathrm {log}\left (\mathrm {log}\left (-x +6\right )\right )+\mathrm {log}\left (2 \,\mathrm {log}\left (-x +6\right ) \mathrm {log}\left (2\right ) x -\mathrm {log}\left (-x +6\right ) \mathrm {log}\left (2\right )-\mathrm {log}\left (-x +6\right ) x^{2}-\mathrm {log}\left (2\right ) x^{3}\right ) \] Input:
int((((2*x-12)*log(2)-2*x^2+12*x)*log(6-x)^2+(-3*x^3+18*x^2)*log(2)*log(6- x)+x^3*log(2))/(((2*x^2-13*x+6)*log(2)-x^3+6*x^2)*log(6-x)^2+(-x^4+6*x^3)* log(2)*log(6-x)),x)
Output:
- log(log( - x + 6)) + log(2*log( - x + 6)*log(2)*x - log( - x + 6)*log(2 ) - log( - x + 6)*x**2 - log(2)*x**3)