Integrand size = 128, antiderivative size = 35 \[ \int \frac {\left (6-2 x-4 x^2\right ) \log ^2(1-x)+\left (x^2+\left (x-x^2\right ) \log (1-x)-4 x \log ^2(1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )+(-2+2 x) \log ^2(1-x) \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )}{(-2+2 x) \log ^2(1-x)} \, dx=x+x \left (-4-x+\log ^2\left (\frac {e^{-\frac {x}{4 \log (1-x)}}}{1-x}\right )\right ) \] Output:
(ln(exp(-1/4*x/ln(1-x))/(1-x))^2-4-x)*x+x
Time = 0.55 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {\left (6-2 x-4 x^2\right ) \log ^2(1-x)+\left (x^2+\left (x-x^2\right ) \log (1-x)-4 x \log ^2(1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )+(-2+2 x) \log ^2(1-x) \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )}{(-2+2 x) \log ^2(1-x)} \, dx=x \left (-3-x+\log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )\right ) \] Input:
Integrate[((6 - 2*x - 4*x^2)*Log[1 - x]^2 + (x^2 + (x - x^2)*Log[1 - x] - 4*x*Log[1 - x]^2)*Log[-(1/(E^(x/(4*Log[1 - x]))*(-1 + x)))] + (-2 + 2*x)*L og[1 - x]^2*Log[-(1/(E^(x/(4*Log[1 - x]))*(-1 + x)))]^2)/((-2 + 2*x)*Log[1 - x]^2),x]
Output:
x*(-3 - x + Log[-(1/(E^(x/(4*Log[1 - x]))*(-1 + 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.
| \(\displaystyle \int \frac {\left (-4 x^2-2 x+6\right ) \log ^2(1-x)+\left (x^2+\left (x-x^2\right ) \log (1-x)-4 x \log ^2(1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right )+(2 x-2) \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right ) \log ^2(1-x)}{(2 x-2) \log ^2(1-x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-2 x+\log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right )-\frac {x \left (-x+4 \log ^2(1-x)+x \log (1-x)-\log (1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right )}{2 (x-1) \log ^2(1-x)}-3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \int \frac {\log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right )}{\log ^2(1-x)}dx+\frac {1}{2} \int \frac {\log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right )}{(x-1) \log ^2(1-x)}dx+\frac {1}{2} \int \frac {x \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right )}{\log ^2(1-x)}dx+\int \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right )dx-2 \int \frac {\log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right )}{x-1}dx-\frac {1}{2} \int \frac {x \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{x-1}\right )}{\log (1-x)}dx+\frac {1}{2} \operatorname {ExpIntegralEi}(2 \log (1-x))-\frac {\operatorname {LogIntegral}(1-x)}{2}-x^2-5 x-\frac {(1-x)^2}{2 \log (1-x)}+\frac {1-x}{\log (1-x)}-2 x \log \left (\frac {e^{-\frac {x}{4 \log (1-x)}}}{1-x}\right )-2 \log (1-x)-\frac {1}{2 \log (1-x)}\) |
Input:
Int[((6 - 2*x - 4*x^2)*Log[1 - x]^2 + (x^2 + (x - x^2)*Log[1 - x] - 4*x*Lo g[1 - x]^2)*Log[-(1/(E^(x/(4*Log[1 - x]))*(-1 + x)))] + (-2 + 2*x)*Log[1 - x]^2*Log[-(1/(E^(x/(4*Log[1 - x]))*(-1 + x)))]^2)/((-2 + 2*x)*Log[1 - x]^ 2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(32)=64\).
Time = 0.90 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.26
| method | result | size |
| parallelrisch | \(-\frac {96 \ln \left (1-x \right )^{3} x^{2}+288 \ln \left (1-x \right )^{3} x +144 \ln \left (1-x \right )^{3}-96 \ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )^{2} \ln \left (1-x \right )^{3} x}{96 \ln \left (1-x \right )^{3}}\) | \(79\) |
| default | \(-3 x -\frac {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) x^{2}}{2 \ln \left (1-x \right )}-\frac {1}{2}-\ln \left (1-x \right )^{2}-2 \left (1-x \right ) \ln \left (1-x \right )-\frac {x^{2}}{2}-2 \left (1-x \right ) \left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right )+\left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right ) \left (-\frac {1-x}{\ln \left (1-x \right )}-\operatorname {expIntegral}_{1}\left (-\ln \left (1-x \right )\right )\right )+\frac {\ln \left (-1+x \right )}{2 \ln \left (1-x \right )}-\left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right ) \operatorname {expIntegral}_{1}\left (-2 \ln \left (1-x \right )\right )-\frac {\left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right ) \left (-\frac {\left (1-x \right )^{2}}{\ln \left (1-x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (1-x \right )\right )\right )}{2}+2 \ln \left (-1+x \right ) \ln \left (1-x \right )-2 \left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \left (x +\ln \left (-1+x \right )\right )+\frac {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \operatorname {expIntegral}_{1}\left (-\ln \left (1-x \right )\right )}{2}-\frac {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \operatorname {expIntegral}_{1}\left (-2 \ln \left (1-x \right )\right )}{2}-\frac {\left (1-x \right )^{3}}{16 \ln \left (1-x \right )^{2}}+\frac {3 \left (1-x \right )^{2}}{16 \ln \left (1-x \right )^{2}}-\frac {3 \left (1-x \right )}{16 \ln \left (1-x \right )^{2}}+\left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right ) \operatorname {expIntegral}_{1}\left (-\ln \left (1-x \right )\right )+x {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right )}^{2}+\ln \left (-1+x \right )^{2} \left (-1+x \right )-2 \ln \left (-1+x \right ) \left (-1+x \right )-\frac {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \left (\operatorname {expIntegral}_{1}\left (-\ln \left (1-x \right )\right )-\operatorname {expIntegral}_{1}\left (-2 \ln \left (1-x \right )\right )\right )}{2}+2 \left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \left (\left (1-x \right ) \ln \left (1-x \right )-1+x \right )+\frac {1}{16 \ln \left (1-x \right )^{2}}+2 \left (1-x \right ) \left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right )\) | \(693\) |
| parts | \(-3 x -\frac {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) x^{2}}{2 \ln \left (1-x \right )}-\frac {1}{2}-\ln \left (1-x \right )^{2}-2 \left (1-x \right ) \ln \left (1-x \right )-\frac {x^{2}}{2}-2 \left (1-x \right ) \left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right )+\left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right ) \left (-\frac {1-x}{\ln \left (1-x \right )}-\operatorname {expIntegral}_{1}\left (-\ln \left (1-x \right )\right )\right )+\frac {\ln \left (-1+x \right )}{2 \ln \left (1-x \right )}-\left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right ) \operatorname {expIntegral}_{1}\left (-2 \ln \left (1-x \right )\right )-\frac {\left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right ) \left (-\frac {\left (1-x \right )^{2}}{\ln \left (1-x \right )}-2 \,\operatorname {expIntegral}_{1}\left (-2 \ln \left (1-x \right )\right )\right )}{2}+2 \ln \left (-1+x \right ) \ln \left (1-x \right )-2 \left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \left (x +\ln \left (-1+x \right )\right )+\frac {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \operatorname {expIntegral}_{1}\left (-\ln \left (1-x \right )\right )}{2}-\frac {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \operatorname {expIntegral}_{1}\left (-2 \ln \left (1-x \right )\right )}{2}-\frac {\left (1-x \right )^{3}}{16 \ln \left (1-x \right )^{2}}+\frac {3 \left (1-x \right )^{2}}{16 \ln \left (1-x \right )^{2}}-\frac {3 \left (1-x \right )}{16 \ln \left (1-x \right )^{2}}+\left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right ) \operatorname {expIntegral}_{1}\left (-\ln \left (1-x \right )\right )+x {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right )}^{2}+\ln \left (-1+x \right )^{2} \left (-1+x \right )-2 \ln \left (-1+x \right ) \left (-1+x \right )-\frac {\left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \left (\operatorname {expIntegral}_{1}\left (-\ln \left (1-x \right )\right )-\operatorname {expIntegral}_{1}\left (-2 \ln \left (1-x \right )\right )\right )}{2}+2 \left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \left (\left (1-x \right ) \ln \left (1-x \right )-1+x \right )+\frac {1}{16 \ln \left (1-x \right )^{2}}+2 \left (1-x \right ) \left (\ln \left (-\frac {{\mathrm e}^{-\frac {x}{4 \ln \left (1-x \right )}}}{-1+x}\right )+\frac {x}{4 \ln \left (1-x \right )}+\ln \left (-1+x \right )\right ) \left (\ln \left (-1+x \right )-\ln \left (1-x \right )\right )\) | \(693\) |
Input:
int(((-2+2*x)*ln(1-x)^2*ln(-exp(-1/4*x/ln(1-x))/(-1+x))^2+(-4*x*ln(1-x)^2+ (-x^2+x)*ln(1-x)+x^2)*ln(-exp(-1/4*x/ln(1-x))/(-1+x))+(-4*x^2-2*x+6)*ln(1- x)^2)/(-2+2*x)/ln(1-x)^2,x,method=_RETURNVERBOSE)
Output:
-1/96*(96*ln(1-x)^3*x^2+288*ln(1-x)^3*x+144*ln(1-x)^3-96*ln(-exp(-1/4*x/ln (1-x))/(-1+x))^2*ln(1-x)^3*x)/ln(1-x)^3
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int \frac {\left (6-2 x-4 x^2\right ) \log ^2(1-x)+\left (x^2+\left (x-x^2\right ) \log (1-x)-4 x \log ^2(1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )+(-2+2 x) \log ^2(1-x) \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )}{(-2+2 x) \log ^2(1-x)} \, dx=\frac {16 \, x \log \left (-x + 1\right )^{4} + x^{3} - 8 \, {\left (x^{2} + 6 \, x\right )} \log \left (-x + 1\right )^{2}}{16 \, \log \left (-x + 1\right )^{2}} \] Input:
integrate(((2*x-2)*log(1-x)^2*log(-exp(-1/4*x/log(1-x))/(-1+x))^2+(-4*x*lo g(1-x)^2+(-x^2+x)*log(1-x)+x^2)*log(-exp(-1/4*x/log(1-x))/(-1+x))+(-4*x^2- 2*x+6)*log(1-x)^2)/(2*x-2)/log(1-x)^2,x, algorithm="fricas")
Output:
1/16*(16*x*log(-x + 1)^4 + x^3 - 8*(x^2 + 6*x)*log(-x + 1)^2)/log(-x + 1)^ 2
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {\left (6-2 x-4 x^2\right ) \log ^2(1-x)+\left (x^2+\left (x-x^2\right ) \log (1-x)-4 x \log ^2(1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )+(-2+2 x) \log ^2(1-x) \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )}{(-2+2 x) \log ^2(1-x)} \, dx=\frac {x^{3}}{16 \log {\left (1 - x \right )}^{2}} - \frac {x^{2}}{2} + x \log {\left (1 - x \right )}^{2} - 3 x \] Input:
integrate(((2*x-2)*ln(1-x)**2*ln(-exp(-1/4*x/ln(1-x))/(-1+x))**2+(-4*x*ln( 1-x)**2+(-x**2+x)*ln(1-x)+x**2)*ln(-exp(-1/4*x/ln(1-x))/(-1+x))+(-4*x**2-2 *x+6)*ln(1-x)**2)/(2*x-2)/ln(1-x)**2,x)
Output:
x**3/(16*log(1 - x)**2) - x**2/2 + x*log(1 - x)**2 - 3*x
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (31) = 62\).
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.06 \[ \int \frac {\left (6-2 x-4 x^2\right ) \log ^2(1-x)+\left (x^2+\left (x-x^2\right ) \log (1-x)-4 x \log ^2(1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )+(-2+2 x) \log ^2(1-x) \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )}{(-2+2 x) \log ^2(1-x)} \, dx=x \log \left (-x + 1\right )^{2} + 2 \, x \log \left (-x + 1\right ) \log \left (e^{\left (\frac {x}{4 \, \log \left (-x + 1\right )}\right )}\right ) + x \log \left (e^{\left (\frac {x}{4 \, \log \left (-x + 1\right )}\right )}\right )^{2} - x^{2} - 3 \, x - 3 \, \log \left (x - 1\right ) + 3 \, \log \left (-x + 1\right ) \] Input:
integrate(((2*x-2)*log(1-x)^2*log(-exp(-1/4*x/log(1-x))/(-1+x))^2+(-4*x*lo g(1-x)^2+(-x^2+x)*log(1-x)+x^2)*log(-exp(-1/4*x/log(1-x))/(-1+x))+(-4*x^2- 2*x+6)*log(1-x)^2)/(2*x-2)/log(1-x)^2,x, algorithm="maxima")
Output:
x*log(-x + 1)^2 + 2*x*log(-x + 1)*log(e^(1/4*x/log(-x + 1))) + x*log(e^(1/ 4*x/log(-x + 1)))^2 - x^2 - 3*x - 3*log(x - 1) + 3*log(-x + 1)
Result contains complex when optimal does not.
Time = 1.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60 \[ \int \frac {\left (6-2 x-4 x^2\right ) \log ^2(1-x)+\left (x^2+\left (x-x^2\right ) \log (1-x)-4 x \log ^2(1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )+(-2+2 x) \log ^2(1-x) \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )}{(-2+2 x) \log ^2(1-x)} \, dx=-2 i \, \pi x \log \left (x - 1\right ) + x \log \left (x - 1\right )^{2} - {\left (\pi ^{2} + 3\right )} x - \frac {1}{2} \, x^{2} - \frac {x^{3}}{16 \, {\left (\pi ^{2} + 2 i \, \pi \log \left (x - 1\right ) - \log \left (x - 1\right )^{2}\right )}} \] Input:
integrate(((2*x-2)*log(1-x)^2*log(-exp(-1/4*x/log(1-x))/(-1+x))^2+(-4*x*lo g(1-x)^2+(-x^2+x)*log(1-x)+x^2)*log(-exp(-1/4*x/log(1-x))/(-1+x))+(-4*x^2- 2*x+6)*log(1-x)^2)/(2*x-2)/log(1-x)^2,x, algorithm="giac")
Output:
-2*I*pi*x*log(x - 1) + x*log(x - 1)^2 - (pi^2 + 3)*x - 1/2*x^2 - 1/16*x^3/ (pi^2 + 2*I*pi*log(x - 1) - log(x - 1)^2)
Time = 4.54 (sec) , antiderivative size = 157, normalized size of antiderivative = 4.49 \[ \int \frac {\left (6-2 x-4 x^2\right ) \log ^2(1-x)+\left (x^2+\left (x-x^2\right ) \log (1-x)-4 x \log ^2(1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )+(-2+2 x) \log ^2(1-x) \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )}{(-2+2 x) \log ^2(1-x)} \, dx=\frac {{\ln \left (1-x\right )}^2}{2}-\frac {23\,x}{8}-\frac {\ln \left (x-1\right )\,\ln \left (\frac {1}{{\mathrm {e}}^{\frac {x}{4\,\ln \left (1-x\right )}}-x\,{\mathrm {e}}^{\frac {x}{4\,\ln \left (1-x\right )}}}\right )}{2}-\frac {\ln \left (x-1\right )\,\ln \left (1-x\right )}{2}+\frac {\ln \left (\frac {1}{{\mathrm {e}}^{\frac {x}{4\,\ln \left (1-x\right )}}-x\,{\mathrm {e}}^{\frac {x}{4\,\ln \left (1-x\right )}}}\right )\,\ln \left (1-x\right )}{2}+x\,{\ln \left (\frac {1}{{\mathrm {e}}^{\frac {x}{4\,\ln \left (1-x\right )}}-x\,{\mathrm {e}}^{\frac {x}{4\,\ln \left (1-x\right )}}}\right )}^2-x^2-\frac {x\,\ln \left (x-1\right )}{8\,\ln \left (1-x\right )} \] Input:
int((log(-exp(-x/(4*log(1 - x)))/(x - 1))*(log(1 - x)*(x - x^2) + x^2 - 4* x*log(1 - x)^2) - log(1 - x)^2*(2*x + 4*x^2 - 6) + log(-exp(-x/(4*log(1 - x)))/(x - 1))^2*log(1 - x)^2*(2*x - 2))/(log(1 - x)^2*(2*x - 2)),x)
Output:
log(1 - x)^2/2 - (23*x)/8 - (log(x - 1)*log(1/(exp(x/(4*log(1 - x))) - x*e xp(x/(4*log(1 - x))))))/2 - (log(x - 1)*log(1 - x))/2 + (log(1/(exp(x/(4*l og(1 - x))) - x*exp(x/(4*log(1 - x)))))*log(1 - x))/2 + x*log(1/(exp(x/(4* log(1 - x))) - x*exp(x/(4*log(1 - x)))))^2 - x^2 - (x*log(x - 1))/(8*log(1 - x))
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \frac {\left (6-2 x-4 x^2\right ) \log ^2(1-x)+\left (x^2+\left (x-x^2\right ) \log (1-x)-4 x \log ^2(1-x)\right ) \log \left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )+(-2+2 x) \log ^2(1-x) \log ^2\left (-\frac {e^{-\frac {x}{4 \log (1-x)}}}{-1+x}\right )}{(-2+2 x) \log ^2(1-x)} \, dx=x \left (\mathrm {log}\left (-\frac {1}{e^{\frac {x}{4 \,\mathrm {log}\left (1-x \right )}} x -e^{\frac {x}{4 \,\mathrm {log}\left (1-x \right )}}}\right )^{2}-x -3\right ) \] Input:
int(((2*x-2)*log(1-x)^2*log(-exp(-1/4*x/log(1-x))/(-1+x))^2+(-4*x*log(1-x) ^2+(-x^2+x)*log(1-x)+x^2)*log(-exp(-1/4*x/log(1-x))/(-1+x))+(-4*x^2-2*x+6) *log(1-x)^2)/(2*x-2)/log(1-x)^2,x)
Output:
x*(log(( - 1)/(e**(x/(4*log( - x + 1)))*x - e**(x/(4*log( - x + 1)))))**2 - x - 3)