\(\int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 (x^3-x^4)+(-44 x+60 x^2-18 x^3+2 x^4+e^4 (2 x^2-2 x^3)) \log (-1+x)+(-16+24 x-9 x^2+x^3+e^4 (x-x^2)) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+(-32 x^2+48 x^3-18 x^4+2 x^5) \log (-1+x)+(-16 x+24 x^2-9 x^3+x^4) \log ^2(-1+x)} \, dx\) [2804]

Optimal result

Integrand size = 174, antiderivative size = 28 \[ \int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 \left (x^3-x^4\right )+\left (-44 x+60 x^2-18 x^3+2 x^4+e^4 \left (2 x^2-2 x^3\right )\right ) \log (-1+x)+\left (-16+24 x-9 x^2+x^3+e^4 \left (x-x^2\right )\right ) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+\left (-32 x^2+48 x^3-18 x^4+2 x^5\right ) \log (-1+x)+\left (-16 x+24 x^2-9 x^3+x^4\right ) \log ^2(-1+x)} \, dx=5+\frac {e^4+\frac {-8-x}{x+\log (-1+x)}}{-4+x}+\log (x) \] Output:

ln(x)+(exp(4)+(-x-8)/(x+ln(-1+x)))/(-4+x)+5
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 \left (x^3-x^4\right )+\left (-44 x+60 x^2-18 x^3+2 x^4+e^4 \left (2 x^2-2 x^3\right )\right ) \log (-1+x)+\left (-16+24 x-9 x^2+x^3+e^4 \left (x-x^2\right )\right ) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+\left (-32 x^2+48 x^3-18 x^4+2 x^5\right ) \log (-1+x)+\left (-16 x+24 x^2-9 x^3+x^4\right ) \log ^2(-1+x)} \, dx=\frac {e^4}{-4+x}+\frac {-8-x}{(-4+x) (x+\log (-1+x))}+\log (x) \] Input:

Integrate[(-60*x^2 + 40*x^3 - 8*x^4 + x^5 + E^4*(x^3 - x^4) + (-44*x + 60* 
x^2 - 18*x^3 + 2*x^4 + E^4*(2*x^2 - 2*x^3))*Log[-1 + x] + (-16 + 24*x - 9* 
x^2 + x^3 + E^4*(x - x^2))*Log[-1 + x]^2)/(-16*x^3 + 24*x^4 - 9*x^5 + x^6 
+ (-32*x^2 + 48*x^3 - 18*x^4 + 2*x^5)*Log[-1 + x] + (-16*x + 24*x^2 - 9*x^ 
3 + x^4)*Log[-1 + x]^2),x]
 

Output:

E^4/(-4 + x) + (-8 - x)/((-4 + x)*(x + Log[-1 + x])) + Log[x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {x^5-8 x^4+40 x^3-60 x^2+e^4 \left (x^3-x^4\right )+\left (x^3-9 x^2+e^4 \left (x-x^2\right )+24 x-16\right ) \log ^2(x-1)+\left (2 x^4-18 x^3+60 x^2+e^4 \left (2 x^2-2 x^3\right )-44 x\right ) \log (x-1)}{x^6-9 x^5+24 x^4-16 x^3+\left (x^4-9 x^3+24 x^2-16 x\right ) \log ^2(x-1)+\left (2 x^5-18 x^4+48 x^3-32 x^2\right ) \log (x-1)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-(x-1) \left (x^2-\left (8+e^4\right ) x+16\right ) \log ^2(x-1)-2 (x-1) \left (x^2-\left (8+e^4\right ) x+22\right ) x \log (x-1)-\left (\left (x^3-\left (8+e^4\right ) x^2+\left (40+e^4\right ) x-60\right ) x^2\right )}{(1-x) (4-x)^2 x (x+\log (x-1))^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x^2-\left (8+e^4\right ) x+16}{(4-x)^2 x}+\frac {x (x+8)}{(x-4) (x-1) (x+\log (x-1))^2}+\frac {12}{(x-4)^2 (x+\log (x-1))}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {1}{(x+\log (x-1))^2}dx+16 \int \frac {1}{(x-4) (x+\log (x-1))^2}dx-3 \int \frac {1}{(x-1) (x+\log (x-1))^2}dx+12 \int \frac {1}{(x-4)^2 (x+\log (x-1))}dx-\frac {e^4}{4-x}+\log (x)\)

Input:

Int[(-60*x^2 + 40*x^3 - 8*x^4 + x^5 + E^4*(x^3 - x^4) + (-44*x + 60*x^2 - 
18*x^3 + 2*x^4 + E^4*(2*x^2 - 2*x^3))*Log[-1 + x] + (-16 + 24*x - 9*x^2 + 
x^3 + E^4*(x - x^2))*Log[-1 + x]^2)/(-16*x^3 + 24*x^4 - 9*x^5 + x^6 + (-32 
*x^2 + 48*x^3 - 18*x^4 + 2*x^5)*Log[-1 + x] + (-16*x + 24*x^2 - 9*x^3 + x^ 
4)*Log[-1 + x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 8.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32

method	result	size
risch	\(\frac {x \ln \left (x \right )-4 \ln \left (x \right )+{\mathrm e}^{4}}{x -4}-\frac {x +8}{\left (x -4\right ) \left (x +\ln \left (-1+x \right )\right )}\)	\(37\)
norman	\(\frac {-8+{\mathrm e}^{4} \ln \left (-1+x \right )+\left ({\mathrm e}^{4}-1\right ) x}{\ln \left (-1+x \right ) x +x^{2}-4 \ln \left (-1+x \right )-4 x}+\ln \left (x \right )\)	\(41\)
derivativedivides	\(-\frac {-{\mathrm e}^{4} \ln \left (-1+x \right )+\left (1-{\mathrm e}^{4}\right ) \left (-1+x \right )+9-{\mathrm e}^{4}}{\left (x -4\right ) \left (x +\ln \left (-1+x \right )\right )}+\ln \left (x \right )\)	\(43\)
default	\(-\frac {-{\mathrm e}^{4} \ln \left (-1+x \right )+\left (1-{\mathrm e}^{4}\right ) \left (-1+x \right )+9-{\mathrm e}^{4}}{\left (x -4\right ) \left (x +\ln \left (-1+x \right )\right )}+\ln \left (x \right )\)	\(43\)
parallelrisch	\(\frac {\ln \left (x \right ) \ln \left (-1+x \right ) x +x^{2} \ln \left (x \right )-4 \ln \left (x \right ) \ln \left (-1+x \right )-4 x \ln \left (x \right )+{\mathrm e}^{4} \ln \left (-1+x \right )+x \,{\mathrm e}^{4}-x -8}{\ln \left (-1+x \right ) x +x^{2}-4 \ln \left (-1+x \right )-4 x}\)	\(66\)

Input:

int((((-x^2+x)*exp(4)+x^3-9*x^2+24*x-16)*ln(-1+x)^2+((-2*x^3+2*x^2)*exp(4) 
+2*x^4-18*x^3+60*x^2-44*x)*ln(-1+x)+(-x^4+x^3)*exp(4)+x^5-8*x^4+40*x^3-60* 
x^2)/((x^4-9*x^3+24*x^2-16*x)*ln(-1+x)^2+(2*x^5-18*x^4+48*x^3-32*x^2)*ln(- 
1+x)+x^6-9*x^5+24*x^4-16*x^3),x,method=_RETURNVERBOSE)
 

Output:

(x*ln(x)-4*ln(x)+exp(4))/(x-4)-(x+8)/(x-4)/(x+ln(-1+x))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 \left (x^3-x^4\right )+\left (-44 x+60 x^2-18 x^3+2 x^4+e^4 \left (2 x^2-2 x^3\right )\right ) \log (-1+x)+\left (-16+24 x-9 x^2+x^3+e^4 \left (x-x^2\right )\right ) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+\left (-32 x^2+48 x^3-18 x^4+2 x^5\right ) \log (-1+x)+\left (-16 x+24 x^2-9 x^3+x^4\right ) \log ^2(-1+x)} \, dx=\frac {x e^{4} + e^{4} \log \left (x - 1\right ) + {\left (x^{2} + {\left (x - 4\right )} \log \left (x - 1\right ) - 4 \, x\right )} \log \left (x\right ) - x - 8}{x^{2} + {\left (x - 4\right )} \log \left (x - 1\right ) - 4 \, x} \] Input:

integrate((((-x^2+x)*exp(4)+x^3-9*x^2+24*x-16)*log(-1+x)^2+((-2*x^3+2*x^2) 
*exp(4)+2*x^4-18*x^3+60*x^2-44*x)*log(-1+x)+(-x^4+x^3)*exp(4)+x^5-8*x^4+40 
*x^3-60*x^2)/((x^4-9*x^3+24*x^2-16*x)*log(-1+x)^2+(2*x^5-18*x^4+48*x^3-32* 
x^2)*log(-1+x)+x^6-9*x^5+24*x^4-16*x^3),x, algorithm="fricas")
 

Output:

(x*e^4 + e^4*log(x - 1) + (x^2 + (x - 4)*log(x - 1) - 4*x)*log(x) - x - 8) 
/(x^2 + (x - 4)*log(x - 1) - 4*x)
 

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 \left (x^3-x^4\right )+\left (-44 x+60 x^2-18 x^3+2 x^4+e^4 \left (2 x^2-2 x^3\right )\right ) \log (-1+x)+\left (-16+24 x-9 x^2+x^3+e^4 \left (x-x^2\right )\right ) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+\left (-32 x^2+48 x^3-18 x^4+2 x^5\right ) \log (-1+x)+\left (-16 x+24 x^2-9 x^3+x^4\right ) \log ^2(-1+x)} \, dx=\frac {- x - 8}{x^{2} - 4 x + \left (x - 4\right ) \log {\left (x - 1 \right )}} + \log {\left (x \right )} + \frac {e^{4}}{x - 4} \] Input:

integrate((((-x**2+x)*exp(4)+x**3-9*x**2+24*x-16)*ln(-1+x)**2+((-2*x**3+2* 
x**2)*exp(4)+2*x**4-18*x**3+60*x**2-44*x)*ln(-1+x)+(-x**4+x**3)*exp(4)+x** 
5-8*x**4+40*x**3-60*x**2)/((x**4-9*x**3+24*x**2-16*x)*ln(-1+x)**2+(2*x**5- 
18*x**4+48*x**3-32*x**2)*ln(-1+x)+x**6-9*x**5+24*x**4-16*x**3),x)
 

Output:

(-x - 8)/(x**2 - 4*x + (x - 4)*log(x - 1)) + log(x) + exp(4)/(x - 4)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 \left (x^3-x^4\right )+\left (-44 x+60 x^2-18 x^3+2 x^4+e^4 \left (2 x^2-2 x^3\right )\right ) \log (-1+x)+\left (-16+24 x-9 x^2+x^3+e^4 \left (x-x^2\right )\right ) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+\left (-32 x^2+48 x^3-18 x^4+2 x^5\right ) \log (-1+x)+\left (-16 x+24 x^2-9 x^3+x^4\right ) \log ^2(-1+x)} \, dx=\frac {x {\left (e^{4} - 1\right )} + e^{4} \log \left (x - 1\right ) - 8}{x^{2} + {\left (x - 4\right )} \log \left (x - 1\right ) - 4 \, x} + \log \left (x\right ) \] Input:

integrate((((-x^2+x)*exp(4)+x^3-9*x^2+24*x-16)*log(-1+x)^2+((-2*x^3+2*x^2) 
*exp(4)+2*x^4-18*x^3+60*x^2-44*x)*log(-1+x)+(-x^4+x^3)*exp(4)+x^5-8*x^4+40 
*x^3-60*x^2)/((x^4-9*x^3+24*x^2-16*x)*log(-1+x)^2+(2*x^5-18*x^4+48*x^3-32* 
x^2)*log(-1+x)+x^6-9*x^5+24*x^4-16*x^3),x, algorithm="maxima")
 

Output:

(x*(e^4 - 1) + e^4*log(x - 1) - 8)/(x^2 + (x - 4)*log(x - 1) - 4*x) + log( 
x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (28) = 56\).

Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 \left (x^3-x^4\right )+\left (-44 x+60 x^2-18 x^3+2 x^4+e^4 \left (2 x^2-2 x^3\right )\right ) \log (-1+x)+\left (-16+24 x-9 x^2+x^3+e^4 \left (x-x^2\right )\right ) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+\left (-32 x^2+48 x^3-18 x^4+2 x^5\right ) \log (-1+x)+\left (-16 x+24 x^2-9 x^3+x^4\right ) \log ^2(-1+x)} \, dx=\frac {x^{2} \log \left (x\right ) + x \log \left (x - 1\right ) \log \left (x\right ) + x e^{4} + e^{4} \log \left (x - 1\right ) - 4 \, x \log \left (x\right ) - 4 \, \log \left (x - 1\right ) \log \left (x\right ) - x - 8}{x^{2} + x \log \left (x - 1\right ) - 4 \, x - 4 \, \log \left (x - 1\right )} \] Input:

integrate((((-x^2+x)*exp(4)+x^3-9*x^2+24*x-16)*log(-1+x)^2+((-2*x^3+2*x^2) 
*exp(4)+2*x^4-18*x^3+60*x^2-44*x)*log(-1+x)+(-x^4+x^3)*exp(4)+x^5-8*x^4+40 
*x^3-60*x^2)/((x^4-9*x^3+24*x^2-16*x)*log(-1+x)^2+(2*x^5-18*x^4+48*x^3-32* 
x^2)*log(-1+x)+x^6-9*x^5+24*x^4-16*x^3),x, algorithm="giac")
 

Output:

(x^2*log(x) + x*log(x - 1)*log(x) + x*e^4 + e^4*log(x - 1) - 4*x*log(x) - 
4*log(x - 1)*log(x) - x - 8)/(x^2 + x*log(x - 1) - 4*x - 4*log(x - 1))
 

Mupad [B] (verification not implemented)

Time = 3.58 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.79 \[ \int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 \left (x^3-x^4\right )+\left (-44 x+60 x^2-18 x^3+2 x^4+e^4 \left (2 x^2-2 x^3\right )\right ) \log (-1+x)+\left (-16+24 x-9 x^2+x^3+e^4 \left (x-x^2\right )\right ) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+\left (-32 x^2+48 x^3-18 x^4+2 x^5\right ) \log (-1+x)+\left (-16 x+24 x^2-9 x^3+x^4\right ) \log ^2(-1+x)} \, dx=\ln \left (x\right )-\frac {\frac {x^2+16\,x-44}{{\left (x-4\right )}^2}+\frac {12\,\ln \left (x-1\right )\,\left (x-1\right )}{x\,{\left (x-4\right )}^2}}{x+\ln \left (x-1\right )}-\frac {-{\mathrm {e}}^4\,x^2+\left (4\,{\mathrm {e}}^4-12\right )\,x+12}{x^3-8\,x^2+16\,x} \] Input:

int(-(log(x - 1)*(exp(4)*(2*x^2 - 2*x^3) - 44*x + 60*x^2 - 18*x^3 + 2*x^4) 
 + log(x - 1)^2*(24*x + exp(4)*(x - x^2) - 9*x^2 + x^3 - 16) + exp(4)*(x^3 
 - x^4) - 60*x^2 + 40*x^3 - 8*x^4 + x^5)/(16*x^3 - 24*x^4 + 9*x^5 - x^6 + 
log(x - 1)^2*(16*x - 24*x^2 + 9*x^3 - x^4) + log(x - 1)*(32*x^2 - 48*x^3 + 
 18*x^4 - 2*x^5)),x)
 

Output:

log(x) - ((16*x + x^2 - 44)/(x - 4)^2 + (12*log(x - 1)*(x - 1))/(x*(x - 4) 
^2))/(x + log(x - 1)) - (x*(4*exp(4) - 12) - x^2*exp(4) + 12)/(16*x - 8*x^ 
2 + x^3)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {-60 x^2+40 x^3-8 x^4+x^5+e^4 \left (x^3-x^4\right )+\left (-44 x+60 x^2-18 x^3+2 x^4+e^4 \left (2 x^2-2 x^3\right )\right ) \log (-1+x)+\left (-16+24 x-9 x^2+x^3+e^4 \left (x-x^2\right )\right ) \log ^2(-1+x)}{-16 x^3+24 x^4-9 x^5+x^6+\left (-32 x^2+48 x^3-18 x^4+2 x^5\right ) \log (-1+x)+\left (-16 x+24 x^2-9 x^3+x^4\right ) \log ^2(-1+x)} \, dx=\frac {\mathrm {log}\left (x -1\right ) \mathrm {log}\left (x \right ) x -4 \,\mathrm {log}\left (x -1\right ) \mathrm {log}\left (x \right )+\mathrm {log}\left (x -1\right ) e^{4}+\mathrm {log}\left (x \right ) x^{2}-4 \,\mathrm {log}\left (x \right ) x +e^{4} x -x -8}{\mathrm {log}\left (x -1\right ) x -4 \,\mathrm {log}\left (x -1\right )+x^{2}-4 x} \] Input:

int((((-x^2+x)*exp(4)+x^3-9*x^2+24*x-16)*log(-1+x)^2+((-2*x^3+2*x^2)*exp(4 
)+2*x^4-18*x^3+60*x^2-44*x)*log(-1+x)+(-x^4+x^3)*exp(4)+x^5-8*x^4+40*x^3-6 
0*x^2)/((x^4-9*x^3+24*x^2-16*x)*log(-1+x)^2+(2*x^5-18*x^4+48*x^3-32*x^2)*l 
og(-1+x)+x^6-9*x^5+24*x^4-16*x^3),x)
 

Output:

(log(x - 1)*log(x)*x - 4*log(x - 1)*log(x) + log(x - 1)*e**4 + log(x)*x**2 
 - 4*log(x)*x + e**4*x - x - 8)/(log(x - 1)*x - 4*log(x - 1) + x**2 - 4*x)