Integrand size = 115, antiderivative size = 25 \[ \int \frac {6-9 x^2+12 x^3-15 x^4+15 x^5+\left (-6+6 x-9 x^2+9 x^3\right ) \log (-1+x)+\left (-12 x+6 x^2+24 x^3-24 x^4+\left (12 x-12 x^2\right ) \log (-1+x)\right ) \log ^2(\log (5))+\left (3-9 x^2+9 x^3+(-3+3 x) \log (-1+x)\right ) \log ^4(\log (5))}{-1+x} \, dx=3 x \left (-1+x^2+\log (-1+x)\right ) \left (2+\left (-x+\log ^2(\log (5))\right )^2\right ) \] Output:
x*(3*x^2+3*ln(-1+x)-3)*(2+(ln(ln(5))^2-x)^2)
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {6-9 x^2+12 x^3-15 x^4+15 x^5+\left (-6+6 x-9 x^2+9 x^3\right ) \log (-1+x)+\left (-12 x+6 x^2+24 x^3-24 x^4+\left (12 x-12 x^2\right ) \log (-1+x)\right ) \log ^2(\log (5))+\left (3-9 x^2+9 x^3+(-3+3 x) \log (-1+x)\right ) \log ^4(\log (5))}{-1+x} \, dx=3 x \left (-1+x^2+\log (-1+x)\right ) \left (2+x^2-2 x \log ^2(\log (5))+\log ^4(\log (5))\right ) \] Input:
Integrate[(6 - 9*x^2 + 12*x^3 - 15*x^4 + 15*x^5 + (-6 + 6*x - 9*x^2 + 9*x^ 3)*Log[-1 + x] + (-12*x + 6*x^2 + 24*x^3 - 24*x^4 + (12*x - 12*x^2)*Log[-1 + x])*Log[Log[5]]^2 + (3 - 9*x^2 + 9*x^3 + (-3 + 3*x)*Log[-1 + x])*Log[Lo g[5]]^4)/(-1 + x),x]
Output:
3*x*(-1 + x^2 + Log[-1 + x])*(2 + x^2 - 2*x*Log[Log[5]]^2 + Log[Log[5]]^4)
Leaf count is larger than twice the leaf count of optimal. \(358\) vs. \(2(25)=50\).
Time = 0.86 (sec) , antiderivative size = 358, normalized size of antiderivative = 14.32, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {15 x^5-15 x^4+12 x^3-9 x^2+\log ^4(\log (5)) \left (9 x^3-9 x^2+(3 x-3) \log (x-1)+3\right )+\left (9 x^3-9 x^2+6 x-6\right ) \log (x-1)+\log ^2(\log (5)) \left (-24 x^4+24 x^3+6 x^2+\left (12 x-12 x^2\right ) \log (x-1)-12 x\right )+6}{x-1} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {15 x^5}{x-1}-\frac {15 x^4 \left (1+\frac {8}{5} \log ^2(\log (5))\right )}{x-1}+\frac {12 x^3 \left (1+\frac {3}{4} \log ^4(\log (5))+2 \log ^2(\log (5))\right )}{x-1}-\frac {9 x^2 \left (1+\log ^4(\log (5))-\frac {2}{3} \log ^2(\log (5))\right )}{x-1}+3 \log (x-1) \left (3 x^2-4 x \log ^2(\log (5))+2+\log ^4(\log (5))\right )+\frac {6 \left (1+\frac {1}{2} \log ^4(\log (5))\right )}{x-1}-\frac {12 x \log ^2(\log (5))}{x-1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 x^5+\frac {15 x^4}{4}-\frac {3}{4} x^4 \left (5+8 \log ^2(\log (5))\right )+4 x^3-x^3 \left (5+8 \log ^2(\log (5))\right )+x^3 \left (4+3 \log ^4(\log (5))+8 \log ^2(\log (5))\right )+3 x^3 \log (x-1)+6 x^2-\frac {3}{2} x^2 \left (5+8 \log ^2(\log (5))\right )-6 x^2 \log ^2(\log (5)) \log (x-1)+3 x^2 \log ^2(\log (5))+\frac {3}{2} x^2 \left (4+3 \log ^4(\log (5))+8 \log ^2(\log (5))\right )-\frac {3}{2} x^2 \left (3+3 \log ^4(\log (5))-2 \log ^2(\log (5))\right )+12 x-3 x \left (2+\log ^4(\log (5))\right )+3 \left (2+\log ^4(\log (5))\right ) \log (1-x)-3 (1-x) \left (2+\log ^4(\log (5))\right ) \log (x-1)-3 x \left (5+8 \log ^2(\log (5))\right )-6 x \log ^2(\log (5))-3 \left (5+8 \log ^2(\log (5))\right ) \log (1-x)-6 \log ^2(\log (5)) \log (1-x)+3 x \left (4+3 \log ^4(\log (5))+8 \log ^2(\log (5))\right )-3 x \left (3+3 \log ^4(\log (5))-2 \log ^2(\log (5))\right )+3 \left (4+3 \log ^4(\log (5))+8 \log ^2(\log (5))\right ) \log (1-x)-3 \left (3+3 \log ^4(\log (5))-2 \log ^2(\log (5))\right ) \log (1-x)+12 \log (1-x)\) |
Input:
Int[(6 - 9*x^2 + 12*x^3 - 15*x^4 + 15*x^5 + (-6 + 6*x - 9*x^2 + 9*x^3)*Log [-1 + x] + (-12*x + 6*x^2 + 24*x^3 - 24*x^4 + (12*x - 12*x^2)*Log[-1 + x]) *Log[Log[5]]^2 + (3 - 9*x^2 + 9*x^3 + (-3 + 3*x)*Log[-1 + x])*Log[Log[5]]^ 4)/(-1 + x),x]
Output:
12*x + 6*x^2 + 4*x^3 + (15*x^4)/4 + 3*x^5 + 12*Log[1 - x] + 3*x^3*Log[-1 + x] - 6*x*Log[Log[5]]^2 + 3*x^2*Log[Log[5]]^2 - 6*Log[1 - x]*Log[Log[5]]^2 - 6*x^2*Log[-1 + x]*Log[Log[5]]^2 - 3*x*(5 + 8*Log[Log[5]]^2) - (3*x^2*(5 + 8*Log[Log[5]]^2))/2 - x^3*(5 + 8*Log[Log[5]]^2) - (3*x^4*(5 + 8*Log[Log [5]]^2))/4 - 3*Log[1 - x]*(5 + 8*Log[Log[5]]^2) - 3*x*(2 + Log[Log[5]]^4) + 3*Log[1 - x]*(2 + Log[Log[5]]^4) - 3*(1 - x)*Log[-1 + x]*(2 + Log[Log[5] ]^4) - 3*x*(3 - 2*Log[Log[5]]^2 + 3*Log[Log[5]]^4) - (3*x^2*(3 - 2*Log[Log [5]]^2 + 3*Log[Log[5]]^4))/2 - 3*Log[1 - x]*(3 - 2*Log[Log[5]]^2 + 3*Log[L og[5]]^4) + 3*x*(4 + 8*Log[Log[5]]^2 + 3*Log[Log[5]]^4) + (3*x^2*(4 + 8*Lo g[Log[5]]^2 + 3*Log[Log[5]]^4))/2 + x^3*(4 + 8*Log[Log[5]]^2 + 3*Log[Log[5 ]]^4) + 3*Log[1 - x]*(4 + 8*Log[Log[5]]^2 + 3*Log[Log[5]]^4)
Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs. \(2(28)=56\).
Time = 1.61 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.40
| method | result | size |
| risch | \(\left (3 \ln \left (\ln \left (5\right )\right )^{4} x -6 \ln \left (\ln \left (5\right )\right )^{2} x^{2}+3 x^{3}+6 x \right ) \ln \left (-1+x \right )+3 \ln \left (\ln \left (5\right )\right )^{4} x^{3}-6 \ln \left (\ln \left (5\right )\right )^{2} x^{4}-3 \ln \left (\ln \left (5\right )\right )^{4} x +3 x^{5}+6 \ln \left (\ln \left (5\right )\right )^{2} x^{2}+3 x^{3}-6 x\) | \(85\) |
| norman | \(\left (-6-3 \ln \left (\ln \left (5\right )\right )^{4}\right ) x +\left (3+3 \ln \left (\ln \left (5\right )\right )^{4}\right ) x^{3}+\left (3 \ln \left (\ln \left (5\right )\right )^{4}+6\right ) x \ln \left (-1+x \right )+3 x^{5}+3 \ln \left (-1+x \right ) x^{3}+6 \ln \left (\ln \left (5\right )\right )^{2} x^{2}-6 \ln \left (\ln \left (5\right )\right )^{2} x^{4}-6 \ln \left (-1+x \right ) \ln \left (\ln \left (5\right )\right )^{2} x^{2}\) | \(89\) |
| parallelrisch | \(3 \ln \left (\ln \left (5\right )\right )^{4} x^{3}+3 \ln \left (-1+x \right ) \ln \left (\ln \left (5\right )\right )^{4} x -6 \ln \left (\ln \left (5\right )\right )^{2} x^{4}-3 \ln \left (\ln \left (5\right )\right )^{4} x -6 \ln \left (-1+x \right ) \ln \left (\ln \left (5\right )\right )^{2} x^{2}+3 x^{5}-6 \ln \left (\ln \left (5\right )\right )^{4}+6 \ln \left (\ln \left (5\right )\right )^{2} x^{2}+3 \ln \left (-1+x \right ) x^{3}+3 x^{3}-6 \ln \left (\ln \left (5\right )\right )^{2}+6 \ln \left (-1+x \right ) x -6 x -12\) | \(110\) |
| parts | \(3 \ln \left (\ln \left (5\right )\right )^{4} \left (\left (-1+x \right ) \ln \left (-1+x \right )+1-x \right )-12 \ln \left (\ln \left (5\right )\right )^{2} \left (\frac {\ln \left (-1+x \right ) \left (-1+x \right )^{2}}{2}-\frac {\left (-1+x \right )^{2}}{4}\right )-12 \ln \left (\ln \left (5\right )\right )^{2} \left (\left (-1+x \right ) \ln \left (-1+x \right )+1-x \right )+3 \ln \left (-1+x \right ) \left (-1+x \right )^{3}-\left (-1+x \right )^{3}+9 \ln \left (-1+x \right ) \left (-1+x \right )^{2}-\frac {9 \left (-1+x \right )^{2}}{2}+15 \left (-1+x \right ) \ln \left (-1+x \right )-12 x +15+3 \ln \left (\ln \left (5\right )\right )^{4} x^{3}-6 \ln \left (\ln \left (5\right )\right )^{2} x^{4}+3 x^{5}+3 \ln \left (\ln \left (5\right )\right )^{2} x^{2}-6 \ln \left (\ln \left (5\right )\right )^{2} x +4 x^{3}+\frac {3 x^{2}}{2}+3 \left (3-2 \ln \left (\ln \left (5\right )\right )^{2}+\ln \left (\ln \left (5\right )\right )^{4}\right ) \ln \left (-1+x \right )\) | \(190\) |
| derivativedivides | \(3 \ln \left (\ln \left (5\right )\right )^{4} \left (-1+x \right )^{3}+3 \ln \left (\ln \left (5\right )\right )^{4} \left (\left (-1+x \right ) \ln \left (-1+x \right )+1-x \right )+9 \ln \left (\ln \left (5\right )\right )^{4} \left (-1+x \right )^{2}-6 \ln \left (\ln \left (5\right )\right )^{2} \left (-1+x \right )^{4}-12 \ln \left (\ln \left (5\right )\right )^{2} \left (\frac {\ln \left (-1+x \right ) \left (-1+x \right )^{2}}{2}-\frac {\left (-1+x \right )^{2}}{4}\right )+9 \ln \left (\ln \left (5\right )\right )^{4} \left (-1+x \right )-24 \ln \left (\ln \left (5\right )\right )^{2} \left (-1+x \right )^{3}+3 \left (-1+x \right )^{5}-12 \ln \left (\ln \left (5\right )\right )^{2} \left (\left (-1+x \right ) \ln \left (-1+x \right )+1-x \right )+3 \ln \left (-1+x \right ) \left (-1+x \right )^{3}+33 \left (-1+x \right )^{3}+3 \ln \left (-1+x \right ) \ln \left (\ln \left (5\right )\right )^{4}-33 \ln \left (\ln \left (5\right )\right )^{2} \left (-1+x \right )^{2}+15 \left (-1+x \right )^{4}+9 \ln \left (-1+x \right ) \left (-1+x \right )^{2}+39 \left (-1+x \right )^{2}-24 \ln \left (\ln \left (5\right )\right )^{2} \left (-1+x \right )+15 \left (-1+x \right ) \ln \left (-1+x \right )-18+18 x -6 \ln \left (-1+x \right ) \ln \left (\ln \left (5\right )\right )^{2}+9 \ln \left (-1+x \right )\) | \(239\) |
| default | \(3 \ln \left (\ln \left (5\right )\right )^{4} \left (-1+x \right )^{3}+3 \ln \left (\ln \left (5\right )\right )^{4} \left (\left (-1+x \right ) \ln \left (-1+x \right )+1-x \right )+9 \ln \left (\ln \left (5\right )\right )^{4} \left (-1+x \right )^{2}-6 \ln \left (\ln \left (5\right )\right )^{2} \left (-1+x \right )^{4}-12 \ln \left (\ln \left (5\right )\right )^{2} \left (\frac {\ln \left (-1+x \right ) \left (-1+x \right )^{2}}{2}-\frac {\left (-1+x \right )^{2}}{4}\right )+9 \ln \left (\ln \left (5\right )\right )^{4} \left (-1+x \right )-24 \ln \left (\ln \left (5\right )\right )^{2} \left (-1+x \right )^{3}+3 \left (-1+x \right )^{5}-12 \ln \left (\ln \left (5\right )\right )^{2} \left (\left (-1+x \right ) \ln \left (-1+x \right )+1-x \right )+3 \ln \left (-1+x \right ) \left (-1+x \right )^{3}+33 \left (-1+x \right )^{3}+3 \ln \left (-1+x \right ) \ln \left (\ln \left (5\right )\right )^{4}-33 \ln \left (\ln \left (5\right )\right )^{2} \left (-1+x \right )^{2}+15 \left (-1+x \right )^{4}+9 \ln \left (-1+x \right ) \left (-1+x \right )^{2}+39 \left (-1+x \right )^{2}-24 \ln \left (\ln \left (5\right )\right )^{2} \left (-1+x \right )+15 \left (-1+x \right ) \ln \left (-1+x \right )-18+18 x -6 \ln \left (-1+x \right ) \ln \left (\ln \left (5\right )\right )^{2}+9 \ln \left (-1+x \right )\) | \(239\) |
Input:
int((((-3+3*x)*ln(-1+x)+9*x^3-9*x^2+3)*ln(ln(5))^4+((-12*x^2+12*x)*ln(-1+x )-24*x^4+24*x^3+6*x^2-12*x)*ln(ln(5))^2+(9*x^3-9*x^2+6*x-6)*ln(-1+x)+15*x^ 5-15*x^4+12*x^3-9*x^2+6)/(-1+x),x,method=_RETURNVERBOSE)
Output:
(3*ln(ln(5))^4*x-6*ln(ln(5))^2*x^2+3*x^3+6*x)*ln(-1+x)+3*ln(ln(5))^4*x^3-6 *ln(ln(5))^2*x^4-3*ln(ln(5))^4*x+3*x^5+6*ln(ln(5))^2*x^2+3*x^3-6*x
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \frac {6-9 x^2+12 x^3-15 x^4+15 x^5+\left (-6+6 x-9 x^2+9 x^3\right ) \log (-1+x)+\left (-12 x+6 x^2+24 x^3-24 x^4+\left (12 x-12 x^2\right ) \log (-1+x)\right ) \log ^2(\log (5))+\left (3-9 x^2+9 x^3+(-3+3 x) \log (-1+x)\right ) \log ^4(\log (5))}{-1+x} \, dx=3 \, x^{5} + 3 \, {\left (x^{3} + x \log \left (x - 1\right ) - x\right )} \log \left (\log \left (5\right )\right )^{4} + 3 \, x^{3} - 6 \, {\left (x^{4} + x^{2} \log \left (x - 1\right ) - x^{2}\right )} \log \left (\log \left (5\right )\right )^{2} + 3 \, {\left (x^{3} + 2 \, x\right )} \log \left (x - 1\right ) - 6 \, x \] Input:
integrate((((-3+3*x)*log(-1+x)+9*x^3-9*x^2+3)*log(log(5))^4+((-12*x^2+12*x )*log(-1+x)-24*x^4+24*x^3+6*x^2-12*x)*log(log(5))^2+(9*x^3-9*x^2+6*x-6)*lo g(-1+x)+15*x^5-15*x^4+12*x^3-9*x^2+6)/(-1+x),x, algorithm="fricas")
Output:
3*x^5 + 3*(x^3 + x*log(x - 1) - x)*log(log(5))^4 + 3*x^3 - 6*(x^4 + x^2*lo g(x - 1) - x^2)*log(log(5))^2 + 3*(x^3 + 2*x)*log(x - 1) - 6*x
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (26) = 52\).
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.60 \[ \int \frac {6-9 x^2+12 x^3-15 x^4+15 x^5+\left (-6+6 x-9 x^2+9 x^3\right ) \log (-1+x)+\left (-12 x+6 x^2+24 x^3-24 x^4+\left (12 x-12 x^2\right ) \log (-1+x)\right ) \log ^2(\log (5))+\left (3-9 x^2+9 x^3+(-3+3 x) \log (-1+x)\right ) \log ^4(\log (5))}{-1+x} \, dx=3 x^{5} - 6 x^{4} \log {\left (\log {\left (5 \right )} \right )}^{2} + x^{3} \cdot \left (3 \log {\left (\log {\left (5 \right )} \right )}^{4} + 3\right ) + 6 x^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + x \left (-6 - 3 \log {\left (\log {\left (5 \right )} \right )}^{4}\right ) + \left (3 x^{3} - 6 x^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + 3 x \log {\left (\log {\left (5 \right )} \right )}^{4} + 6 x\right ) \log {\left (x - 1 \right )} \] Input:
integrate((((-3+3*x)*ln(-1+x)+9*x**3-9*x**2+3)*ln(ln(5))**4+((-12*x**2+12* x)*ln(-1+x)-24*x**4+24*x**3+6*x**2-12*x)*ln(ln(5))**2+(9*x**3-9*x**2+6*x-6 )*ln(-1+x)+15*x**5-15*x**4+12*x**3-9*x**2+6)/(-1+x),x)
Output:
3*x**5 - 6*x**4*log(log(5))**2 + x**3*(3*log(log(5))**4 + 3) + 6*x**2*log( log(5))**2 + x*(-6 - 3*log(log(5))**4) + (3*x**3 - 6*x**2*log(log(5))**2 + 3*x*log(log(5))**4 + 6*x)*log(x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (25) = 50\).
Time = 0.04 (sec) , antiderivative size = 374, normalized size of antiderivative = 14.96 \[ \int \frac {6-9 x^2+12 x^3-15 x^4+15 x^5+\left (-6+6 x-9 x^2+9 x^3\right ) \log (-1+x)+\left (-12 x+6 x^2+24 x^3-24 x^4+\left (12 x-12 x^2\right ) \log (-1+x)\right ) \log ^2(\log (5))+\left (3-9 x^2+9 x^3+(-3+3 x) \log (-1+x)\right ) \log ^4(\log (5))}{-1+x} \, dx=3 \, {\left (x + \log \left (x - 1\right )\right )} \log \left (x - 1\right ) \log \left (\log \left (5\right )\right )^{4} - \frac {3}{2} \, \log \left (x - 1\right )^{2} \log \left (\log \left (5\right )\right )^{4} + 3 \, x^{5} + \frac {3}{2} \, {\left (2 \, x^{3} + 3 \, x^{2} + 6 \, x + 6 \, \log \left (x - 1\right )\right )} \log \left (\log \left (5\right )\right )^{4} - \frac {9}{2} \, {\left (x^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \left (\log \left (5\right )\right )^{4} - \frac {3}{2} \, {\left (\log \left (x - 1\right )^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \left (\log \left (5\right )\right )^{4} + 3 \, \log \left (x - 1\right ) \log \left (\log \left (5\right )\right )^{4} - 6 \, {\left (x^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \left (x - 1\right ) \log \left (\log \left (5\right )\right )^{2} + 12 \, {\left (x + \log \left (x - 1\right )\right )} \log \left (x - 1\right ) \log \left (\log \left (5\right )\right )^{2} + 3 \, x^{3} - 2 \, {\left (3 \, x^{4} + 4 \, x^{3} + 6 \, x^{2} + 12 \, x + 12 \, \log \left (x - 1\right )\right )} \log \left (\log \left (5\right )\right )^{2} + 4 \, {\left (2 \, x^{3} + 3 \, x^{2} + 6 \, x + 6 \, \log \left (x - 1\right )\right )} \log \left (\log \left (5\right )\right )^{2} + 3 \, {\left (x^{2} + 2 \, \log \left (x - 1\right )^{2} + 6 \, x + 6 \, \log \left (x - 1\right )\right )} \log \left (\log \left (5\right )\right )^{2} + 3 \, {\left (x^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \left (\log \left (5\right )\right )^{2} - 6 \, {\left (\log \left (x - 1\right )^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \left (\log \left (5\right )\right )^{2} - 12 \, {\left (x + \log \left (x - 1\right )\right )} \log \left (\log \left (5\right )\right )^{2} + \frac {3}{2} \, {\left (2 \, x^{3} + 3 \, x^{2} + 6 \, x + 6 \, \log \left (x - 1\right )\right )} \log \left (x - 1\right ) - \frac {9}{2} \, {\left (x^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \left (x - 1\right ) + 6 \, {\left (x + \log \left (x - 1\right )\right )} \log \left (x - 1\right ) - 6 \, \log \left (x - 1\right )^{2} - 6 \, x \] Input:
integrate((((-3+3*x)*log(-1+x)+9*x^3-9*x^2+3)*log(log(5))^4+((-12*x^2+12*x )*log(-1+x)-24*x^4+24*x^3+6*x^2-12*x)*log(log(5))^2+(9*x^3-9*x^2+6*x-6)*lo g(-1+x)+15*x^5-15*x^4+12*x^3-9*x^2+6)/(-1+x),x, algorithm="maxima")
Output:
3*(x + log(x - 1))*log(x - 1)*log(log(5))^4 - 3/2*log(x - 1)^2*log(log(5)) ^4 + 3*x^5 + 3/2*(2*x^3 + 3*x^2 + 6*x + 6*log(x - 1))*log(log(5))^4 - 9/2* (x^2 + 2*x + 2*log(x - 1))*log(log(5))^4 - 3/2*(log(x - 1)^2 + 2*x + 2*log (x - 1))*log(log(5))^4 + 3*log(x - 1)*log(log(5))^4 - 6*(x^2 + 2*x + 2*log (x - 1))*log(x - 1)*log(log(5))^2 + 12*(x + log(x - 1))*log(x - 1)*log(log (5))^2 + 3*x^3 - 2*(3*x^4 + 4*x^3 + 6*x^2 + 12*x + 12*log(x - 1))*log(log( 5))^2 + 4*(2*x^3 + 3*x^2 + 6*x + 6*log(x - 1))*log(log(5))^2 + 3*(x^2 + 2* log(x - 1)^2 + 6*x + 6*log(x - 1))*log(log(5))^2 + 3*(x^2 + 2*x + 2*log(x - 1))*log(log(5))^2 - 6*(log(x - 1)^2 + 2*x + 2*log(x - 1))*log(log(5))^2 - 12*(x + log(x - 1))*log(log(5))^2 + 3/2*(2*x^3 + 3*x^2 + 6*x + 6*log(x - 1))*log(x - 1) - 9/2*(x^2 + 2*x + 2*log(x - 1))*log(x - 1) + 6*(x + log(x - 1))*log(x - 1) - 6*log(x - 1)^2 - 6*x
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.20 \[ \int \frac {6-9 x^2+12 x^3-15 x^4+15 x^5+\left (-6+6 x-9 x^2+9 x^3\right ) \log (-1+x)+\left (-12 x+6 x^2+24 x^3-24 x^4+\left (12 x-12 x^2\right ) \log (-1+x)\right ) \log ^2(\log (5))+\left (3-9 x^2+9 x^3+(-3+3 x) \log (-1+x)\right ) \log ^4(\log (5))}{-1+x} \, dx=-6 \, x^{4} \log \left (\log \left (5\right )\right )^{2} + 3 \, x^{5} + 3 \, {\left (\log \left (\log \left (5\right )\right )^{4} + 1\right )} x^{3} + 6 \, x^{2} \log \left (\log \left (5\right )\right )^{2} - 3 \, {\left (\log \left (\log \left (5\right )\right )^{4} + 2\right )} x - 3 \, {\left (2 \, x^{2} \log \left (\log \left (5\right )\right )^{2} - x^{3} - {\left (\log \left (\log \left (5\right )\right )^{4} + 2\right )} x\right )} \log \left (x - 1\right ) \] Input:
integrate((((-3+3*x)*log(-1+x)+9*x^3-9*x^2+3)*log(log(5))^4+((-12*x^2+12*x )*log(-1+x)-24*x^4+24*x^3+6*x^2-12*x)*log(log(5))^2+(9*x^3-9*x^2+6*x-6)*lo g(-1+x)+15*x^5-15*x^4+12*x^3-9*x^2+6)/(-1+x),x, algorithm="giac")
Output:
-6*x^4*log(log(5))^2 + 3*x^5 + 3*(log(log(5))^4 + 1)*x^3 + 6*x^2*log(log(5 ))^2 - 3*(log(log(5))^4 + 2)*x - 3*(2*x^2*log(log(5))^2 - x^3 - (log(log(5 ))^4 + 2)*x)*log(x - 1)
Time = 3.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {6-9 x^2+12 x^3-15 x^4+15 x^5+\left (-6+6 x-9 x^2+9 x^3\right ) \log (-1+x)+\left (-12 x+6 x^2+24 x^3-24 x^4+\left (12 x-12 x^2\right ) \log (-1+x)\right ) \log ^2(\log (5))+\left (3-9 x^2+9 x^3+(-3+3 x) \log (-1+x)\right ) \log ^4(\log (5))}{-1+x} \, dx=3\,x\,\left (\ln \left (x-1\right )+x^2-1\right )\,\left (x^2-2\,{\ln \left (\ln \left (5\right )\right )}^2\,x+{\ln \left (\ln \left (5\right )\right )}^4+2\right ) \] Input:
int((log(log(5))^4*(9*x^3 - 9*x^2 + log(x - 1)*(3*x - 3) + 3) + log(x - 1) *(6*x - 9*x^2 + 9*x^3 - 6) + log(log(5))^2*(log(x - 1)*(12*x - 12*x^2) - 1 2*x + 6*x^2 + 24*x^3 - 24*x^4) - 9*x^2 + 12*x^3 - 15*x^4 + 15*x^5 + 6)/(x - 1),x)
Output:
3*x*(log(x - 1) + x^2 - 1)*(log(log(5))^4 - 2*x*log(log(5))^2 + x^2 + 2)
Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.24 \[ \int \frac {6-9 x^2+12 x^3-15 x^4+15 x^5+\left (-6+6 x-9 x^2+9 x^3\right ) \log (-1+x)+\left (-12 x+6 x^2+24 x^3-24 x^4+\left (12 x-12 x^2\right ) \log (-1+x)\right ) \log ^2(\log (5))+\left (3-9 x^2+9 x^3+(-3+3 x) \log (-1+x)\right ) \log ^4(\log (5))}{-1+x} \, dx=3 x \left (\mathrm {log}\left (\mathrm {log}\left (5\right )\right )^{4} \mathrm {log}\left (x -1\right )+\mathrm {log}\left (\mathrm {log}\left (5\right )\right )^{4} x^{2}-\mathrm {log}\left (\mathrm {log}\left (5\right )\right )^{4}-2 \mathrm {log}\left (\mathrm {log}\left (5\right )\right )^{2} \mathrm {log}\left (x -1\right ) x -2 \mathrm {log}\left (\mathrm {log}\left (5\right )\right )^{2} x^{3}+2 \mathrm {log}\left (\mathrm {log}\left (5\right )\right )^{2} x +\mathrm {log}\left (x -1\right ) x^{2}+2 \,\mathrm {log}\left (x -1\right )+x^{4}+x^{2}-2\right ) \] Input:
int((((-3+3*x)*log(-1+x)+9*x^3-9*x^2+3)*log(log(5))^4+((-12*x^2+12*x)*log( -1+x)-24*x^4+24*x^3+6*x^2-12*x)*log(log(5))^2+(9*x^3-9*x^2+6*x-6)*log(-1+x )+15*x^5-15*x^4+12*x^3-9*x^2+6)/(-1+x),x)
Output:
3*x*(log(log(5))**4*log(x - 1) + log(log(5))**4*x**2 - log(log(5))**4 - 2* log(log(5))**2*log(x - 1)*x - 2*log(log(5))**2*x**3 + 2*log(log(5))**2*x + log(x - 1)*x**2 + 2*log(x - 1) + x**4 + x**2 - 2)