Integrand size = 149, antiderivative size = 25 \[ \int \frac {2 x^3-4 x^4+\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{\left (-x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-9+8 x+x^2\right ) \log ^3\left (-x+x^2\right )} \, dx=x+\log \left (4 \left (x+\left (3+\frac {x^2}{\log \left (-x+x^2\right )}\right )^2\right )\right ) \] Output:
ln(4*x+4*(x^2/ln(x^2-x)+3)^2)+x
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {2 x^3-4 x^4+\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{\left (-x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-9+8 x+x^2\right ) \log ^3\left (-x+x^2\right )} \, dx=x-2 \log (\log ((-1+x) x))+\log \left (x^4+6 x^2 \log ((-1+x) x)+9 \log ^2((-1+x) x)+x \log ^2((-1+x) x)\right ) \] Input:
Integrate[(2*x^3 - 4*x^4 + (6*x - 12*x^2 - 4*x^3 + 3*x^4 + x^5)*Log[-x + x ^2] + (-12*x + 6*x^2 + 6*x^3)*Log[-x + x^2]^2 + (-10 + 9*x + x^2)*Log[-x + x^2]^3)/((-x^4 + x^5)*Log[-x + x^2] + (-6*x^2 + 6*x^3)*Log[-x + x^2]^2 + (-9 + 8*x + x^2)*Log[-x + x^2]^3),x]
Output:
x - 2*Log[Log[(-1 + x)*x]] + Log[x^4 + 6*x^2*Log[(-1 + x)*x] + 9*Log[(-1 + x)*x]^2 + x*Log[(-1 + x)*x]^2]
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.
| \(\displaystyle \int \frac {-4 x^4+2 x^3+\left (x^2+9 x-10\right ) \log ^3\left (x^2-x\right )+\left (6 x^3+6 x^2-12 x\right ) \log ^2\left (x^2-x\right )+\left (x^5+3 x^4-4 x^3-12 x^2+6 x\right ) \log \left (x^2-x\right )}{\left (x^2+8 x-9\right ) \log ^3\left (x^2-x\right )+\left (6 x^3-6 x^2\right ) \log ^2\left (x^2-x\right )+\left (x^5-x^4\right ) \log \left (x^2-x\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {4 x^4-2 x^3-\left (x^2+9 x-10\right ) \log ^3\left (x^2-x\right )-\left (6 x^3+6 x^2-12 x\right ) \log ^2\left (x^2-x\right )-\left (x^5+3 x^4-4 x^3-12 x^2+6 x\right ) \log \left (x^2-x\right )}{(1-x) \log ((x-1) x) \left (x^4+6 x^2 \log ((x-1) x)+x \log ^2((x-1) x)+9 \log ^2((x-1) x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 (x+2) x \log ((x-1) x)}{x^4+6 x^2 \log ((x-1) x)+x \log ^2((x-1) x)+9 \log ^2((x-1) x)}+\frac {(x+10) \log ^2((x-1) x)}{x^4+6 x^2 \log ((x-1) x)+x \log ^2((x-1) x)+9 \log ^2((x-1) x)}-\frac {2 (2 x-1) x^3}{(x-1) \log ((x-1) x) \left (x^4+6 x^2 \log ((x-1) x)+x \log ^2((x-1) x)+9 \log ^2((x-1) x)\right )}+\frac {\left (x^4+3 x^3-4 x^2-12 x+6\right ) x}{(x-1) \left (x^4+6 x^2 \log ((x-1) x)+x \log ^2((x-1) x)+9 \log ^2((x-1) x)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 735 \int \frac {1}{x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)}dx+6 \int \frac {1}{(x-1) \left (x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)\right )}dx-69 \int \frac {x}{x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)}dx+9 \int \frac {x^2}{x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)}dx-6561 \int \frac {1}{(x+9) \left (x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)\right )}dx+58 \int \frac {\log ((x-1) x)}{x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)}dx+20 \int \frac {\log ((x-1) x)}{(x-1) \left (x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)\right )}dx+18 \int \frac {\log ((x-1) x)}{x \left (x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)\right )}dx+6 \int \frac {x \log ((x-1) x)}{x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)}dx-486 \int \frac {\log ((x-1) x)}{(x+9) \left (x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)\right )}dx+3 \int \frac {x^3}{x^4+6 \log ((x-1) x) x^2+\log ^2((x-1) x) x+9 \log ^2((x-1) x)}dx-2 \int \frac {2 x-1}{(x-1) x \log ((x-1) x)}dx+x+\log (x+9)\) |
Input:
Int[(2*x^3 - 4*x^4 + (6*x - 12*x^2 - 4*x^3 + 3*x^4 + x^5)*Log[-x + x^2] + (-12*x + 6*x^2 + 6*x^3)*Log[-x + x^2]^2 + (-10 + 9*x + x^2)*Log[-x + x^2]^ 3)/((-x^4 + x^5)*Log[-x + x^2] + (-6*x^2 + 6*x^3)*Log[-x + x^2]^2 + (-9 + 8*x + x^2)*Log[-x + x^2]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
Time = 4.80 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24
| method | result | size |
| norman | \(x -2 \ln \left (\ln \left (x^{2}-x \right )\right )+\ln \left (x^{4}+\ln \left (x^{2}-x \right )^{2} x +6 x^{2} \ln \left (x^{2}-x \right )+9 \ln \left (x^{2}-x \right )^{2}\right )\) | \(56\) |
| risch | \(x +\ln \left (x +9\right )-2 \ln \left (\ln \left (x^{2}-x \right )\right )+\ln \left (\ln \left (x^{2}-x \right )^{2}+\frac {6 x^{2} \ln \left (x^{2}-x \right )}{x +9}+\frac {x^{4}}{x +9}\right )\) | \(57\) |
| parallelrisch | \(2-2 \ln \left (\ln \left (x^{2}-x \right )\right )+\ln \left (x^{4}+\ln \left (x^{2}-x \right )^{2} x +6 x^{2} \ln \left (x^{2}-x \right )+9 \ln \left (x^{2}-x \right )^{2}\right )+x\) | \(57\) |
Input:
int(((x^2+9*x-10)*ln(x^2-x)^3+(6*x^3+6*x^2-12*x)*ln(x^2-x)^2+(x^5+3*x^4-4* x^3-12*x^2+6*x)*ln(x^2-x)-4*x^4+2*x^3)/((x^2+8*x-9)*ln(x^2-x)^3+(6*x^3-6*x ^2)*ln(x^2-x)^2+(x^5-x^4)*ln(x^2-x)),x,method=_RETURNVERBOSE)
Output:
x-2*ln(ln(x^2-x))+ln(x^4+ln(x^2-x)^2*x+6*x^2*ln(x^2-x)+9*ln(x^2-x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {2 x^3-4 x^4+\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{\left (-x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-9+8 x+x^2\right ) \log ^3\left (-x+x^2\right )} \, dx=x + \log \left (x + 9\right ) + \log \left (\frac {x^{4} + 6 \, x^{2} \log \left (x^{2} - x\right ) + {\left (x + 9\right )} \log \left (x^{2} - x\right )^{2}}{x + 9}\right ) - 2 \, \log \left (\log \left (x^{2} - x\right )\right ) \] Input:
integrate(((x^2+9*x-10)*log(x^2-x)^3+(6*x^3+6*x^2-12*x)*log(x^2-x)^2+(x^5+ 3*x^4-4*x^3-12*x^2+6*x)*log(x^2-x)-4*x^4+2*x^3)/((x^2+8*x-9)*log(x^2-x)^3+ (6*x^3-6*x^2)*log(x^2-x)^2+(x^5-x^4)*log(x^2-x)),x, algorithm="fricas")
Output:
x + log(x + 9) + log((x^4 + 6*x^2*log(x^2 - x) + (x + 9)*log(x^2 - x)^2)/( x + 9)) - 2*log(log(x^2 - x))
Exception generated. \[ \int \frac {2 x^3-4 x^4+\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{\left (-x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-9+8 x+x^2\right ) \log ^3\left (-x+x^2\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate(((x**2+9*x-10)*ln(x**2-x)**3+(6*x**3+6*x**2-12*x)*ln(x**2-x)**2+ (x**5+3*x**4-4*x**3-12*x**2+6*x)*ln(x**2-x)-4*x**4+2*x**3)/((x**2+8*x-9)*l n(x**2-x)**3+(6*x**3-6*x**2)*ln(x**2-x)**2+(x**5-x**4)*ln(x**2-x)),x)
Output:
Exception raised: PolynomialError >> 1/(x**6 + 34*x**5 + 415*x**4 + 1980*x **3 + 1215*x**2 - 10206*x + 6561) contains an element of the set of genera tors.
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (27) = 54\).
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.80 \[ \int \frac {2 x^3-4 x^4+\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{\left (-x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-9+8 x+x^2\right ) \log ^3\left (-x+x^2\right )} \, dx=x + \log \left (x + 9\right ) + \log \left (\frac {x^{4} + {\left (x + 9\right )} \log \left (x - 1\right )^{2} + 6 \, x^{2} \log \left (x\right ) + {\left (x + 9\right )} \log \left (x\right )^{2} + 2 \, {\left (3 \, x^{2} + {\left (x + 9\right )} \log \left (x\right )\right )} \log \left (x - 1\right )}{x + 9}\right ) - 2 \, \log \left (\log \left (x - 1\right ) + \log \left (x\right )\right ) \] Input:
integrate(((x^2+9*x-10)*log(x^2-x)^3+(6*x^3+6*x^2-12*x)*log(x^2-x)^2+(x^5+ 3*x^4-4*x^3-12*x^2+6*x)*log(x^2-x)-4*x^4+2*x^3)/((x^2+8*x-9)*log(x^2-x)^3+ (6*x^3-6*x^2)*log(x^2-x)^2+(x^5-x^4)*log(x^2-x)),x, algorithm="maxima")
Output:
x + log(x + 9) + log((x^4 + (x + 9)*log(x - 1)^2 + 6*x^2*log(x) + (x + 9)* log(x)^2 + 2*(3*x^2 + (x + 9)*log(x))*log(x - 1))/(x + 9)) - 2*log(log(x - 1) + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {2 x^3-4 x^4+\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{\left (-x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-9+8 x+x^2\right ) \log ^3\left (-x+x^2\right )} \, dx=x + \log \left (x^{4} + 6 \, x^{2} \log \left (x^{2} - x\right ) + x \log \left (x^{2} - x\right )^{2} + 9 \, \log \left (x^{2} - x\right )^{2}\right ) - 2 \, \log \left (\log \left (x^{2} - x\right )\right ) \] Input:
integrate(((x^2+9*x-10)*log(x^2-x)^3+(6*x^3+6*x^2-12*x)*log(x^2-x)^2+(x^5+ 3*x^4-4*x^3-12*x^2+6*x)*log(x^2-x)-4*x^4+2*x^3)/((x^2+8*x-9)*log(x^2-x)^3+ (6*x^3-6*x^2)*log(x^2-x)^2+(x^5-x^4)*log(x^2-x)),x, algorithm="giac")
Output:
x + log(x^4 + 6*x^2*log(x^2 - x) + x*log(x^2 - x)^2 + 9*log(x^2 - x)^2) - 2*log(log(x^2 - x))
Time = 3.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 7.36 \[ \int \frac {2 x^3-4 x^4+\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{\left (-x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-9+8 x+x^2\right ) \log ^3\left (-x+x^2\right )} \, dx=x-3\,\ln \left (x-1\right )-\ln \left (x-\frac {1}{2}\right )-2\,\ln \left (\frac {x^3\,\ln \left (x\,\left (x-1\right )\right )\,\left (9\,x^7+126\,x^6-231\,x^5+952\,x^4-1000\,x^3+1444\,x^2-1224\,x+324\right )}{{\left (x-1\right )}^2\,{\left (x+9\right )}^4}\right )+2\,\ln \left (9\,x^7+126\,x^6-231\,x^5+952\,x^4-1000\,x^3+1444\,x^2-1224\,x+324\right )+\ln \left (\frac {\left (2\,x-1\right )\,\left (x^4+6\,x^2\,\ln \left (x\,\left (x-1\right )\right )+x\,{\ln \left (x\,\left (x-1\right )\right )}^2+9\,{\ln \left (x\,\left (x-1\right )\right )}^2\right )}{x\,\left (x^2+8\,x-9\right )}\right )-\mathrm {atan}\left (\frac {x\,735002345703105317{}\mathrm {i}-177147{}\mathrm {i}}{735002345703144683\,x+177147}\right )\,14{}\mathrm {i} \] Input:
int(-(log(x^2 - x)^3*(9*x + x^2 - 10) + log(x^2 - x)*(6*x - 12*x^2 - 4*x^3 + 3*x^4 + x^5) + 2*x^3 - 4*x^4 + log(x^2 - x)^2*(6*x^2 - 12*x + 6*x^3))/( log(x^2 - x)^2*(6*x^2 - 6*x^3) + log(x^2 - x)*(x^4 - x^5) - log(x^2 - x)^3 *(8*x + x^2 - 9)),x)
Output:
x - atan((x*735002345703105317i - 177147i)/(735002345703144683*x + 177147) )*14i - 3*log(x - 1) - log(x - 1/2) - 2*log((x^3*log(x*(x - 1))*(1444*x^2 - 1224*x - 1000*x^3 + 952*x^4 - 231*x^5 + 126*x^6 + 9*x^7 + 324))/((x - 1) ^2*(x + 9)^4)) + 2*log(1444*x^2 - 1224*x - 1000*x^3 + 952*x^4 - 231*x^5 + 126*x^6 + 9*x^7 + 324) + log(((2*x - 1)*(x*log(x*(x - 1))^2 + 6*x^2*log(x* (x - 1)) + 9*log(x*(x - 1))^2 + x^4))/(x*(8*x + x^2 - 9)))
\[ \int \frac {2 x^3-4 x^4+\left (6 x-12 x^2-4 x^3+3 x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-12 x+6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-10+9 x+x^2\right ) \log ^3\left (-x+x^2\right )}{\left (-x^4+x^5\right ) \log \left (-x+x^2\right )+\left (-6 x^2+6 x^3\right ) \log ^2\left (-x+x^2\right )+\left (-9+8 x+x^2\right ) \log ^3\left (-x+x^2\right )} \, dx =\text {Too large to display} \] Input:
int(((x^2+9*x-10)*log(x^2-x)^3+(6*x^3+6*x^2-12*x)*log(x^2-x)^2+(x^5+3*x^4- 4*x^3-12*x^2+6*x)*log(x^2-x)-4*x^4+2*x^3)/((x^2+8*x-9)*log(x^2-x)^3+(6*x^3 -6*x^2)*log(x^2-x)^2+(x^5-x^4)*log(x^2-x)),x)
Output:
- 10*int(log(x**2 - x)**2/(log(x**2 - x)**2*x**2 + 8*log(x**2 - x)**2*x - 9*log(x**2 - x)**2 + 6*log(x**2 - x)*x**3 - 6*log(x**2 - x)*x**2 + x**5 - x**4),x) + int(x**5/(log(x**2 - x)**2*x**2 + 8*log(x**2 - x)**2*x - 9*log (x**2 - x)**2 + 6*log(x**2 - x)*x**3 - 6*log(x**2 - x)*x**2 + x**5 - x**4) ,x) - 4*int(x**4/(log(x**2 - x)**3*x**2 + 8*log(x**2 - x)**3*x - 9*log(x** 2 - x)**3 + 6*log(x**2 - x)**2*x**3 - 6*log(x**2 - x)**2*x**2 + log(x**2 - x)*x**5 - log(x**2 - x)*x**4),x) + 3*int(x**4/(log(x**2 - x)**2*x**2 + 8* log(x**2 - x)**2*x - 9*log(x**2 - x)**2 + 6*log(x**2 - x)*x**3 - 6*log(x** 2 - x)*x**2 + x**5 - x**4),x) + 2*int(x**3/(log(x**2 - x)**3*x**2 + 8*log( x**2 - x)**3*x - 9*log(x**2 - x)**3 + 6*log(x**2 - x)**2*x**3 - 6*log(x**2 - x)**2*x**2 + log(x**2 - x)*x**5 - log(x**2 - x)*x**4),x) - 4*int(x**3/( log(x**2 - x)**2*x**2 + 8*log(x**2 - x)**2*x - 9*log(x**2 - x)**2 + 6*log( x**2 - x)*x**3 - 6*log(x**2 - x)*x**2 + x**5 - x**4),x) - 12*int(x**2/(log (x**2 - x)**2*x**2 + 8*log(x**2 - x)**2*x - 9*log(x**2 - x)**2 + 6*log(x** 2 - x)*x**3 - 6*log(x**2 - x)*x**2 + x**5 - x**4),x) + int((log(x**2 - x)* *2*x**2)/(log(x**2 - x)**2*x**2 + 8*log(x**2 - x)**2*x - 9*log(x**2 - x)** 2 + 6*log(x**2 - x)*x**3 - 6*log(x**2 - x)*x**2 + x**5 - x**4),x) + 9*int( (log(x**2 - x)**2*x)/(log(x**2 - x)**2*x**2 + 8*log(x**2 - x)**2*x - 9*log (x**2 - x)**2 + 6*log(x**2 - x)*x**3 - 6*log(x**2 - x)*x**2 + x**5 - x**4) ,x) + 6*int((log(x**2 - x)*x**3)/(log(x**2 - x)**2*x**2 + 8*log(x**2 - ...