Integrand size = 126, antiderivative size = 21 \[ \int \frac {44+16 x^2+2 x^4+\left (32 x+8 x^3\right ) \log (3)+\left (16+12 x^2\right ) \log ^2(3)+8 x \log ^3(3)+2 \log ^4(3)+\left (-16 x+16 x^2-4 x^3+4 x^4+\left (-16+16 x-12 x^2+12 x^3\right ) \log (3)+\left (-12 x+12 x^2\right ) \log ^2(3)+(-4+4 x) \log ^3(3)\right ) \log \left (1-2 x+x^2\right )}{-1+x} \, dx=\left (6+\left (4+(x+\log (3))^2\right )^2\right ) \log \left ((1-x)^2\right ) \] Output:
((4+(ln(3)+x)^2)^2+6)*ln((1-x)^2)
Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(21)=42\).
Time = 0.05 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.48 \[ \int \frac {44+16 x^2+2 x^4+\left (32 x+8 x^3\right ) \log (3)+\left (16+12 x^2\right ) \log ^2(3)+8 x \log ^3(3)+2 \log ^4(3)+\left (-16 x+16 x^2-4 x^3+4 x^4+\left (-16+16 x-12 x^2+12 x^3\right ) \log (3)+\left (-12 x+12 x^2\right ) \log ^2(3)+(-4+4 x) \log ^3(3)\right ) \log \left (1-2 x+x^2\right )}{-1+x} \, dx=2 \left (\left (31+14 \log ^2(3)+4 \log ^3(3)+\log ^4(3)+5 \log (81)\right ) \log (-1+x)+\frac {1}{6} (-1+x) \left (27+3 x^3+18 \log ^2(3)+12 \log ^3(3)+4 \log (27)+x^2 (3+4 \log (27))+x \left (27+18 \log ^2(3)+4 \log (27)\right )+12 \log (81)\right ) \log \left ((-1+x)^2\right )\right ) \] Input:
Integrate[(44 + 16*x^2 + 2*x^4 + (32*x + 8*x^3)*Log[3] + (16 + 12*x^2)*Log [3]^2 + 8*x*Log[3]^3 + 2*Log[3]^4 + (-16*x + 16*x^2 - 4*x^3 + 4*x^4 + (-16 + 16*x - 12*x^2 + 12*x^3)*Log[3] + (-12*x + 12*x^2)*Log[3]^2 + (-4 + 4*x) *Log[3]^3)*Log[1 - 2*x + x^2])/(-1 + x),x]
Output:
2*((31 + 14*Log[3]^2 + 4*Log[3]^3 + Log[3]^4 + 5*Log[81])*Log[-1 + x] + (( -1 + x)*(27 + 3*x^3 + 18*Log[3]^2 + 12*Log[3]^3 + 4*Log[27] + x^2*(3 + 4*L og[27]) + x*(27 + 18*Log[3]^2 + 4*Log[27]) + 12*Log[81])*Log[(-1 + x)^2])/ 6)
Leaf count is larger than twice the leaf count of optimal. \(263\) vs. \(2(21)=42\).
Time = 1.11 (sec) , antiderivative size = 263, normalized size of antiderivative = 12.52, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {7292, 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.
| \(\displaystyle \int \frac {2 x^4+\left (8 x^3+32 x\right ) \log (3)+16 x^2+\left (12 x^2+16\right ) \log ^2(3)+\left (4 x^4-4 x^3+16 x^2+\left (12 x^2-12 x\right ) \log ^2(3)+\left (12 x^3-12 x^2+16 x-16\right ) \log (3)-16 x+(4 x-4) \log ^3(3)\right ) \log \left (x^2-2 x+1\right )+8 x \log ^3(3)+44+2 \log ^4(3)}{x-1} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 x^4-\left (8 x^3+32 x\right ) \log (3)-16 x^2-\left (12 x^2+16\right ) \log ^2(3)-\left (4 x^4-4 x^3+16 x^2+\left (12 x^2-12 x\right ) \log ^2(3)+\left (12 x^3-12 x^2+16 x-16\right ) \log (3)-16 x+(4 x-4) \log ^3(3)\right ) \log \left (x^2-2 x+1\right )-8 x \log ^3(3)-44 \left (1+\frac {\log ^4(3)}{22}\right )}{1-x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (4 (x+\log (3)) \left (x^2+x \log (9)+4+\log ^2(3)\right ) \log \left ((x-1)^2\right )+\frac {2 \left (-x^4-x^3 \log (81)-2 x^2 \left (4+3 \log ^2(3)\right )-4 x \left (\log ^3(3)+\log (81)\right )-22-\log ^4(3)-8 \log ^2(3)\right )}{1-x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^4 \log \left ((x-1)^2\right )-\frac {2 x^3}{3}+\frac {4}{3} x^3 \log (27) \log \left ((x-1)^2\right )+\frac {2}{3} x^3 (1+\log (81))-\frac {8}{9} x^3 \log (27)-x^2+2 x^2 \left (4+\log ^2(3)+\log (3) \log (9)\right ) \log \left ((x-1)^2\right )+x^2 \left (9+6 \log ^2(3)+\log (81)\right )-2 x^2 \left (4+\log ^2(3)+\log (3) \log (9)\right )-\frac {4}{3} x^2 \log (27)-2 x-8 x \left (\log ^3(3)+\log (81)\right )-4 (1-x) \left (\log ^3(3)+\log (81)\right ) \log \left ((x-1)^2\right )-4 x \left (4+\log ^2(3)+\log (3) \log (9)\right )-4 \left (4+\log ^2(3)+\log (3) \log (9)\right ) \log (1-x)+2 x \left (9+4 \log ^3(3)+6 \log ^2(3)+5 \log (81)\right )+2 \left (31+\log ^4(3)+4 \log ^3(3)+14 \log ^2(3)+5 \log (81)\right ) \log (1-x)-\frac {8}{3} x \log (27)-\frac {8}{3} \log (27) \log (1-x)-2 \log (1-x)\) |
Input:
Int[(44 + 16*x^2 + 2*x^4 + (32*x + 8*x^3)*Log[3] + (16 + 12*x^2)*Log[3]^2 + 8*x*Log[3]^3 + 2*Log[3]^4 + (-16*x + 16*x^2 - 4*x^3 + 4*x^4 + (-16 + 16* x - 12*x^2 + 12*x^3)*Log[3] + (-12*x + 12*x^2)*Log[3]^2 + (-4 + 4*x)*Log[3 ]^3)*Log[1 - 2*x + x^2])/(-1 + x),x]
Output:
-2*x - x^2 - (2*x^3)/3 - 4*x*(4 + Log[3]^2 + Log[3]*Log[9]) - 2*x^2*(4 + L og[3]^2 + Log[3]*Log[9]) - (8*x*Log[27])/3 - (4*x^2*Log[27])/3 - (8*x^3*Lo g[27])/9 + (2*x^3*(1 + Log[81]))/3 + x^2*(9 + 6*Log[3]^2 + Log[81]) - 8*x* (Log[3]^3 + Log[81]) + 2*x*(9 + 6*Log[3]^2 + 4*Log[3]^3 + 5*Log[81]) - 2*L og[1 - x] - 4*(4 + Log[3]^2 + Log[3]*Log[9])*Log[1 - x] - (8*Log[27]*Log[1 - x])/3 + 2*(31 + 14*Log[3]^2 + 4*Log[3]^3 + Log[3]^4 + 5*Log[81])*Log[1 - x] + x^4*Log[(-1 + x)^2] + 2*x^2*(4 + Log[3]^2 + Log[3]*Log[9])*Log[(-1 + x)^2] + (4*x^3*Log[27]*Log[(-1 + x)^2])/3 - 4*(1 - x)*(Log[3]^3 + Log[81 ])*Log[(-1 + x)^2]
Time = 2.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62
| method | result | size |
| risch | \(\left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}+4\right )^{2} \ln \left (x^{2}-2 x +1\right )+12 \ln \left (-1+x \right )\) | \(34\) |
| norman | \(\left (\ln \left (3\right )^{4}+8 \ln \left (3\right )^{2}+22\right ) \ln \left (x^{2}-2 x +1\right )+\ln \left (x^{2}-2 x +1\right ) x^{4}+\left (8+6 \ln \left (3\right )^{2}\right ) x^{2} \ln \left (x^{2}-2 x +1\right )+\left (4 \ln \left (3\right )^{3}+16 \ln \left (3\right )\right ) x \ln \left (x^{2}-2 x +1\right )+4 \ln \left (3\right ) \ln \left (x^{2}-2 x +1\right ) x^{3}\) | \(96\) |
| default | \(\ln \left (3\right ) \left (4 \ln \left (x^{2}-2 x +1\right ) x^{3}+16 \ln \left (x^{2}-2 x +1\right ) x \right )+4 \ln \left (3\right )^{2} \left (4 \ln \left (-1+x \right )+\frac {3 \ln \left (x^{2}-2 x +1\right ) x^{2}}{2}\right )+4 \ln \left (3\right )^{3} \ln \left (x^{2}-2 x +1\right ) x +2 \ln \left (3\right )^{4} \ln \left (-1+x \right )+44 \ln \left (-1+x \right )+\ln \left (x^{2}-2 x +1\right ) x^{4}+8 \ln \left (x^{2}-2 x +1\right ) x^{2}\) | \(118\) |
| parallelrisch | \(\ln \left (x^{2}-2 x +1\right ) \ln \left (3\right )^{4}+4 \ln \left (3\right )^{3} \ln \left (x^{2}-2 x +1\right ) x +6 \ln \left (3\right )^{2} \ln \left (x^{2}-2 x +1\right ) x^{2}+4 \ln \left (3\right ) \ln \left (x^{2}-2 x +1\right ) x^{3}+\ln \left (x^{2}-2 x +1\right ) x^{4}+8 \ln \left (3\right )^{2} \ln \left (x^{2}-2 x +1\right )+16 \ln \left (3\right ) \ln \left (x^{2}-2 x +1\right ) x +8 \ln \left (x^{2}-2 x +1\right ) x^{2}+22 \ln \left (x^{2}-2 x +1\right )\) | \(133\) |
| parts | \(\ln \left (x^{2}-2 x +1\right ) x^{4}-18 \ln \left (-1+x \right )+4 \ln \left (3\right )^{3} \left (\ln \left (x^{2}-2 x +1\right ) x -2 x -2 \ln \left (-1+x \right )\right )+8 \ln \left (x^{2}-2 x +1\right ) x^{2}+16 \ln \left (3\right ) \left (\ln \left (x^{2}-2 x +1\right ) x -2 x -2 \ln \left (-1+x \right )\right )+12 \ln \left (3\right )^{2} \left (\frac {\ln \left (x^{2}-2 x +1\right ) x^{2}}{2}-x -\frac {x^{2}}{2}-\ln \left (-1+x \right )\right )+12 \ln \left (3\right ) \left (\frac {\ln \left (x^{2}-2 x +1\right ) x^{3}}{3}-\frac {2 x^{3}}{9}-\frac {x^{2}}{3}-\frac {2 x}{3}-\frac {2 \ln \left (-1+x \right )}{3}\right )+8 x \ln \left (3\right )^{3}+6 x^{2} \ln \left (3\right )^{2}+\frac {8 x^{3} \ln \left (3\right )}{3}+12 x \ln \left (3\right )^{2}+4 x^{2} \ln \left (3\right )+40 x \ln \left (3\right )+2 \left (31+20 \ln \left (3\right )+14 \ln \left (3\right )^{2}+4 \ln \left (3\right )^{3}+\ln \left (3\right )^{4}\right ) \ln \left (-1+x \right )\) | \(230\) |
Input:
int((((-4+4*x)*ln(3)^3+(12*x^2-12*x)*ln(3)^2+(12*x^3-12*x^2+16*x-16)*ln(3) +4*x^4-4*x^3+16*x^2-16*x)*ln(x^2-2*x+1)+2*ln(3)^4+8*x*ln(3)^3+(12*x^2+16)* ln(3)^2+(8*x^3+32*x)*ln(3)+2*x^4+16*x^2+44)/(-1+x),x,method=_RETURNVERBOSE )
Output:
(ln(3)^2+2*x*ln(3)+x^2+4)^2*ln(x^2-2*x+1)+12*ln(-1+x)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (19) = 38\).
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.62 \[ \int \frac {44+16 x^2+2 x^4+\left (32 x+8 x^3\right ) \log (3)+\left (16+12 x^2\right ) \log ^2(3)+8 x \log ^3(3)+2 \log ^4(3)+\left (-16 x+16 x^2-4 x^3+4 x^4+\left (-16+16 x-12 x^2+12 x^3\right ) \log (3)+\left (-12 x+12 x^2\right ) \log ^2(3)+(-4+4 x) \log ^3(3)\right ) \log \left (1-2 x+x^2\right )}{-1+x} \, dx={\left (x^{4} + 4 \, x \log \left (3\right )^{3} + \log \left (3\right )^{4} + 2 \, {\left (3 \, x^{2} + 4\right )} \log \left (3\right )^{2} + 8 \, x^{2} + 4 \, {\left (x^{3} + 4 \, x\right )} \log \left (3\right ) + 22\right )} \log \left (x^{2} - 2 \, x + 1\right ) \] Input:
integrate((((-4+4*x)*log(3)^3+(12*x^2-12*x)*log(3)^2+(12*x^3-12*x^2+16*x-1 6)*log(3)+4*x^4-4*x^3+16*x^2-16*x)*log(x^2-2*x+1)+2*log(3)^4+8*x*log(3)^3+ (12*x^2+16)*log(3)^2+(8*x^3+32*x)*log(3)+2*x^4+16*x^2+44)/(-1+x),x, algori thm="fricas")
Output:
(x^4 + 4*x*log(3)^3 + log(3)^4 + 2*(3*x^2 + 4)*log(3)^2 + 8*x^2 + 4*(x^3 + 4*x)*log(3) + 22)*log(x^2 - 2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (17) = 34\).
Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.38 \[ \int \frac {44+16 x^2+2 x^4+\left (32 x+8 x^3\right ) \log (3)+\left (16+12 x^2\right ) \log ^2(3)+8 x \log ^3(3)+2 \log ^4(3)+\left (-16 x+16 x^2-4 x^3+4 x^4+\left (-16+16 x-12 x^2+12 x^3\right ) \log (3)+\left (-12 x+12 x^2\right ) \log ^2(3)+(-4+4 x) \log ^3(3)\right ) \log \left (1-2 x+x^2\right )}{-1+x} \, dx=\left (x^{4} + 4 x^{3} \log {\left (3 \right )} + 6 x^{2} \log {\left (3 \right )}^{2} + 8 x^{2} + 4 x \log {\left (3 \right )}^{3} + 16 x \log {\left (3 \right )}\right ) \log {\left (x^{2} - 2 x + 1 \right )} + \left (2 \log {\left (3 \right )}^{4} + 16 \log {\left (3 \right )}^{2} + 44\right ) \log {\left (x - 1 \right )} \] Input:
integrate((((-4+4*x)*ln(3)**3+(12*x**2-12*x)*ln(3)**2+(12*x**3-12*x**2+16* x-16)*ln(3)+4*x**4-4*x**3+16*x**2-16*x)*ln(x**2-2*x+1)+2*ln(3)**4+8*x*ln(3 )**3+(12*x**2+16)*ln(3)**2+(8*x**3+32*x)*ln(3)+2*x**4+16*x**2+44)/(-1+x),x )
Output:
(x**4 + 4*x**3*log(3) + 6*x**2*log(3)**2 + 8*x**2 + 4*x*log(3)**3 + 16*x*l og(3))*log(x**2 - 2*x + 1) + (2*log(3)**4 + 16*log(3)**2 + 44)*log(x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (19) = 38\).
Time = 0.05 (sec) , antiderivative size = 588, normalized size of antiderivative = 28.00 \[ \int \frac {44+16 x^2+2 x^4+\left (32 x+8 x^3\right ) \log (3)+\left (16+12 x^2\right ) \log ^2(3)+8 x \log ^3(3)+2 \log ^4(3)+\left (-16 x+16 x^2-4 x^3+4 x^4+\left (-16+16 x-12 x^2+12 x^3\right ) \log (3)+\left (-12 x+12 x^2\right ) \log ^2(3)+(-4+4 x) \log ^3(3)\right ) \log \left (1-2 x+x^2\right )}{-1+x} \, dx =\text {Too large to display} \] Input:
integrate((((-4+4*x)*log(3)^3+(12*x^2-12*x)*log(3)^2+(12*x^3-12*x^2+16*x-1 6)*log(3)+4*x^4-4*x^3+16*x^2-16*x)*log(x^2-2*x+1)+2*log(3)^4+8*x*log(3)^3+ (12*x^2+16)*log(3)^2+(8*x^3+32*x)*log(3)+2*x^4+16*x^2+44)/(-1+x),x, algori thm="maxima")
Output:
4*(x + log(x - 1))*log(3)^3*log(x^2 - 2*x + 1) + 2*log(3)^4*log(x - 1) - 4 *log(3)^3*log(x^2 - 2*x + 1)*log(x - 1) + 4*(log(x^2 - 2*x + 1)*log(x - 1) - log(x - 1)^2)*log(3)^3 - 4*(log(x - 1)^2 + 2*x + 2*log(x - 1))*log(3)^3 + 8*(x + log(x - 1))*log(3)^3 + 6*(x^2 + 2*x + 2*log(x - 1))*log(3)^2*log (x^2 - 2*x + 1) - 12*(x + log(x - 1))*log(3)^2*log(x^2 - 2*x + 1) - 6*(x^2 + 2*log(x - 1)^2 + 6*x + 6*log(x - 1))*log(3)^2 + 6*(x^2 + 2*x + 2*log(x - 1))*log(3)^2 + 12*(log(x - 1)^2 + 2*x + 2*log(x - 1))*log(3)^2 + 2*(2*x^ 3 + 3*x^2 + 6*x + 6*log(x - 1))*log(3)*log(x^2 - 2*x + 1) - 6*(x^2 + 2*x + 2*log(x - 1))*log(3)*log(x^2 - 2*x + 1) + 16*(x + log(x - 1))*log(3)*log( x^2 - 2*x + 1) + 16*log(3)^2*log(x - 1) - 16*log(3)*log(x^2 - 2*x + 1)*log (x - 1) - 2/3*(4*x^3 + 15*x^2 + 18*log(x - 1)^2 + 66*x + 66*log(x - 1))*lo g(3) + 4/3*(2*x^3 + 3*x^2 + 6*x + 6*log(x - 1))*log(3) + 6*(x^2 + 2*log(x - 1)^2 + 6*x + 6*log(x - 1))*log(3) + 16*(log(x^2 - 2*x + 1)*log(x - 1) - log(x - 1)^2)*log(3) - 16*(log(x - 1)^2 + 2*x + 2*log(x - 1))*log(3) + 32* (x + log(x - 1))*log(3) + 1/3*(3*x^4 + 4*x^3 + 6*x^2 + 12*x + 12*log(x - 1 ))*log(x^2 - 2*x + 1) - 2/3*(2*x^3 + 3*x^2 + 6*x + 6*log(x - 1))*log(x^2 - 2*x + 1) + 8*(x^2 + 2*x + 2*log(x - 1))*log(x^2 - 2*x + 1) - 16*(x + log( x - 1))*log(x^2 - 2*x + 1) + 44*log(x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (19) = 38\).
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.10 \[ \int \frac {44+16 x^2+2 x^4+\left (32 x+8 x^3\right ) \log (3)+\left (16+12 x^2\right ) \log ^2(3)+8 x \log ^3(3)+2 \log ^4(3)+\left (-16 x+16 x^2-4 x^3+4 x^4+\left (-16+16 x-12 x^2+12 x^3\right ) \log (3)+\left (-12 x+12 x^2\right ) \log ^2(3)+(-4+4 x) \log ^3(3)\right ) \log \left (1-2 x+x^2\right )}{-1+x} \, dx={\left (x^{4} + 4 \, x^{3} \log \left (3\right ) + 2 \, {\left (3 \, \log \left (3\right )^{2} + 4\right )} x^{2} + 4 \, {\left (\log \left (3\right )^{3} + 4 \, \log \left (3\right )\right )} x\right )} \log \left (x^{2} - 2 \, x + 1\right ) + 2 \, {\left (\log \left (3\right )^{4} + 8 \, \log \left (3\right )^{2} + 22\right )} \log \left (x - 1\right ) \] Input:
integrate((((-4+4*x)*log(3)^3+(12*x^2-12*x)*log(3)^2+(12*x^3-12*x^2+16*x-1 6)*log(3)+4*x^4-4*x^3+16*x^2-16*x)*log(x^2-2*x+1)+2*log(3)^4+8*x*log(3)^3+ (12*x^2+16)*log(3)^2+(8*x^3+32*x)*log(3)+2*x^4+16*x^2+44)/(-1+x),x, algori thm="giac")
Output:
(x^4 + 4*x^3*log(3) + 2*(3*log(3)^2 + 4)*x^2 + 4*(log(3)^3 + 4*log(3))*x)* log(x^2 - 2*x + 1) + 2*(log(3)^4 + 8*log(3)^2 + 22)*log(x - 1)
Time = 3.41 (sec) , antiderivative size = 135, normalized size of antiderivative = 6.43 \[ \int \frac {44+16 x^2+2 x^4+\left (32 x+8 x^3\right ) \log (3)+\left (16+12 x^2\right ) \log ^2(3)+8 x \log ^3(3)+2 \log ^4(3)+\left (-16 x+16 x^2-4 x^3+4 x^4+\left (-16+16 x-12 x^2+12 x^3\right ) \log (3)+\left (-12 x+12 x^2\right ) \log ^2(3)+(-4+4 x) \log ^3(3)\right ) \log \left (1-2 x+x^2\right )}{-1+x} \, dx=\ln \left (x^2-2\,x+1\right )\,\left (4\,\ln \left (81\right )-16\,\ln \left (3\right )+6\,x^2\,{\ln \left (3\right )}^2+4\,x\,\ln \left (3\right )+3\,x\,\ln \left (81\right )-8\,x^2\,\ln \left (3\right )+4\,x\,{\ln \left (3\right )}^3+2\,x^2\,\ln \left (81\right )+x^3\,\ln \left (81\right )+8\,{\ln \left (3\right )}^2+{\ln \left (3\right )}^4+8\,x^2+x^4+22\right )-\frac {5\,\ln \left (x^2-2\,x+1\right )\,\left (4\,\ln \left (3\right )-\ln \left (81\right )\right )}{x-1}-\frac {\ln \left (x^2-2\,x+1\right )\,\left (4\,\ln \left (3\right )-\ln \left (81\right )\right )}{{\left (x-1\right )}^2} \] Input:
int((log(x^2 - 2*x + 1)*(log(3)^3*(4*x - 4) - 16*x + log(3)*(16*x - 12*x^2 + 12*x^3 - 16) - log(3)^2*(12*x - 12*x^2) + 16*x^2 - 4*x^3 + 4*x^4) + log (3)*(32*x + 8*x^3) + 8*x*log(3)^3 + log(3)^2*(12*x^2 + 16) + 2*log(3)^4 + 16*x^2 + 2*x^4 + 44)/(x - 1),x)
Output:
log(x^2 - 2*x + 1)*(4*log(81) - 16*log(3) + 6*x^2*log(3)^2 + 4*x*log(3) + 3*x*log(81) - 8*x^2*log(3) + 4*x*log(3)^3 + 2*x^2*log(81) + x^3*log(81) + 8*log(3)^2 + log(3)^4 + 8*x^2 + x^4 + 22) - (5*log(x^2 - 2*x + 1)*(4*log(3 ) - log(81)))/(x - 1) - (log(x^2 - 2*x + 1)*(4*log(3) - log(81)))/(x - 1)^ 2
Time = 0.19 (sec) , antiderivative size = 190, normalized size of antiderivative = 9.05 \[ \int \frac {44+16 x^2+2 x^4+\left (32 x+8 x^3\right ) \log (3)+\left (16+12 x^2\right ) \log ^2(3)+8 x \log ^3(3)+2 \log ^4(3)+\left (-16 x+16 x^2-4 x^3+4 x^4+\left (-16+16 x-12 x^2+12 x^3\right ) \log (3)+\left (-12 x+12 x^2\right ) \log ^2(3)+(-4+4 x) \log ^3(3)\right ) \log \left (1-2 x+x^2\right )}{-1+x} \, dx=4 \,\mathrm {log}\left (x^{2}-2 x +1\right ) \mathrm {log}\left (3\right )^{3} x -4 \,\mathrm {log}\left (x^{2}-2 x +1\right ) \mathrm {log}\left (3\right )^{3}+6 \,\mathrm {log}\left (x^{2}-2 x +1\right ) \mathrm {log}\left (3\right )^{2} x^{2}-6 \,\mathrm {log}\left (x^{2}-2 x +1\right ) \mathrm {log}\left (3\right )^{2}+4 \,\mathrm {log}\left (x^{2}-2 x +1\right ) \mathrm {log}\left (3\right ) x^{3}+16 \,\mathrm {log}\left (x^{2}-2 x +1\right ) \mathrm {log}\left (3\right ) x -20 \,\mathrm {log}\left (x^{2}-2 x +1\right ) \mathrm {log}\left (3\right )+\mathrm {log}\left (x^{2}-2 x +1\right ) x^{4}+8 \,\mathrm {log}\left (x^{2}-2 x +1\right ) x^{2}-9 \,\mathrm {log}\left (x^{2}-2 x +1\right )+2 \,\mathrm {log}\left (x -1\right ) \mathrm {log}\left (3\right )^{4}+8 \,\mathrm {log}\left (x -1\right ) \mathrm {log}\left (3\right )^{3}+28 \,\mathrm {log}\left (x -1\right ) \mathrm {log}\left (3\right )^{2}+40 \,\mathrm {log}\left (x -1\right ) \mathrm {log}\left (3\right )+62 \,\mathrm {log}\left (x -1\right ) \] Input:
int((((-4+4*x)*log(3)^3+(12*x^2-12*x)*log(3)^2+(12*x^3-12*x^2+16*x-16)*log (3)+4*x^4-4*x^3+16*x^2-16*x)*log(x^2-2*x+1)+2*log(3)^4+8*x*log(3)^3+(12*x^ 2+16)*log(3)^2+(8*x^3+32*x)*log(3)+2*x^4+16*x^2+44)/(-1+x),x)
Output:
4*log(x**2 - 2*x + 1)*log(3)**3*x - 4*log(x**2 - 2*x + 1)*log(3)**3 + 6*lo g(x**2 - 2*x + 1)*log(3)**2*x**2 - 6*log(x**2 - 2*x + 1)*log(3)**2 + 4*log (x**2 - 2*x + 1)*log(3)*x**3 + 16*log(x**2 - 2*x + 1)*log(3)*x - 20*log(x* *2 - 2*x + 1)*log(3) + log(x**2 - 2*x + 1)*x**4 + 8*log(x**2 - 2*x + 1)*x* *2 - 9*log(x**2 - 2*x + 1) + 2*log(x - 1)*log(3)**4 + 8*log(x - 1)*log(3)* *3 + 28*log(x - 1)*log(3)**2 + 40*log(x - 1)*log(3) + 62*log(x - 1)