\(\int \frac {69 x-16 x^2+x^3+e^x (-69+16 x-x^2)+(216-123 x+20 x^2-x^3+e^x (-216+123 x-20 x^2+x^3)) \log (\frac {-27+12 x-x^2}{-8+x})}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} (-216+123 x-20 x^2+x^3)+e^x (432 x-246 x^2+40 x^3-2 x^4)} \, dx\) [560]

Optimal result

Integrand size = 140, antiderivative size = 28 \[ \int \frac {69 x-16 x^2+x^3+e^x \left (-69+16 x-x^2\right )+\left (216-123 x+20 x^2-x^3+e^x \left (-216+123 x-20 x^2+x^3\right )\right ) \log \left (\frac {-27+12 x-x^2}{-8+x}\right )}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} \left (-216+123 x-20 x^2+x^3\right )+e^x \left (432 x-246 x^2+40 x^3-2 x^4\right )} \, dx=4-\frac {\log \left (4-\frac {5}{8-x}-x\right )}{e^x-x} \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {69 x-16 x^2+x^3+e^x \left (-69+16 x-x^2\right )+\left (216-123 x+20 x^2-x^3+e^x \left (-216+123 x-20 x^2+x^3\right )\right ) \log \left (\frac {-27+12 x-x^2}{-8+x}\right )}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} \left (-216+123 x-20 x^2+x^3\right )+e^x \left (432 x-246 x^2+40 x^3-2 x^4\right )} \, dx=-\frac {\log \left (-\frac {27-12 x+x^2}{-8+x}\right )}{e^x-x} \] Input:

Integrate[(69*x - 16*x^2 + x^3 + E^x*(-69 + 16*x - x^2) + (216 - 123*x + 2 
0*x^2 - x^3 + E^x*(-216 + 123*x - 20*x^2 + x^3))*Log[(-27 + 12*x - x^2)/(- 
8 + x)])/(-216*x^2 + 123*x^3 - 20*x^4 + x^5 + E^(2*x)*(-216 + 123*x - 20*x 
^2 + x^3) + E^x*(432*x - 246*x^2 + 40*x^3 - 2*x^4)),x]
 

Output:

-(Log[-((27 - 12*x + x^2)/(-8 + x))]/(E^x - 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.

\(\displaystyle \int \frac {x^3-16 x^2+e^x \left (-x^2+16 x-69\right )+\left (-x^3+20 x^2+e^x \left (x^3-20 x^2+123 x-216\right )-123 x+216\right ) \log \left (\frac {-x^2+12 x-27}{x-8}\right )+69 x}{x^5-20 x^4+123 x^3-216 x^2+e^{2 x} \left (x^3-20 x^2+123 x-216\right )+e^x \left (-2 x^4+40 x^3-246 x^2+432 x\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-1\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )-\frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )}{x^3-20 x^2+123 x-216}}{\left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (e^x-x\right ) \left (x^2-16 x+69\right )-\left (e^x-1\right ) \left (x^3-20 x^2+123 x-216\right ) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{(3-x) (8-x) (9-x) \left (e^x-x\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x-1) \log \left (-\frac {x^2-12 x+27}{x-8}\right )}{\left (e^x-x\right )^2}+\frac {-x^2-20 x^2 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+123 x \log \left (-\frac {x^2-12 x+27}{x-8}\right )-216 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+x^3 \log \left (-\frac {x^2-12 x+27}{x-8}\right )+16 x-69}{\left (e^x-x\right ) (x-9) (x-8) (x-3)}\right )dx\)

Input:

Int[(69*x - 16*x^2 + x^3 + E^x*(-69 + 16*x - x^2) + (216 - 123*x + 20*x^2 
- x^3 + E^x*(-216 + 123*x - 20*x^2 + x^3))*Log[(-27 + 12*x - x^2)/(-8 + x) 
])/(-216*x^2 + 123*x^3 - 20*x^4 + x^5 + E^(2*x)*(-216 + 123*x - 20*x^2 + x 
^3) + E^x*(432*x - 246*x^2 + 40*x^3 - 2*x^4)),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.06 (sec) , antiderivative size = 196, normalized size of antiderivative = 7.00

\[\frac {\ln \left (x^{2}-12 x +27\right )}{x -{\mathrm e}^{x}}-\frac {2 i \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )}^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{-8+x}\right ) \operatorname {csgn}\left (i \left (x^{2}-12 x +27\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{-8+x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (x^{2}-12 x +27\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )}^{2}-i \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-12 x +27\right )}{-8+x}\right )}^{3}-2 i \pi +2 \ln \left (-8+x \right )}{2 \left (x -{\mathrm e}^{x}\right )}\]

Input:

int((((x^3-20*x^2+123*x-216)*exp(x)-x^3+20*x^2-123*x+216)*ln((-x^2+12*x-27 
)/(-8+x))+(-x^2+16*x-69)*exp(x)+x^3-16*x^2+69*x)/((x^3-20*x^2+123*x-216)*e 
xp(x)^2+(-2*x^4+40*x^3-246*x^2+432*x)*exp(x)+x^5-20*x^4+123*x^3-216*x^2),x 
)
 

Output:

1/(x-exp(x))*ln(x^2-12*x+27)-1/2*(2*I*Pi*csgn(I/(-8+x)*(x^2-12*x+27))^2+I* 
Pi*csgn(I/(-8+x))*csgn(I*(x^2-12*x+27))*csgn(I/(-8+x)*(x^2-12*x+27))-I*Pi* 
csgn(I/(-8+x))*csgn(I/(-8+x)*(x^2-12*x+27))^2-I*Pi*csgn(I*(x^2-12*x+27))*c 
sgn(I/(-8+x)*(x^2-12*x+27))^2-I*Pi*csgn(I/(-8+x)*(x^2-12*x+27))^3-2*I*Pi+2 
*ln(-8+x))/(x-exp(x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {69 x-16 x^2+x^3+e^x \left (-69+16 x-x^2\right )+\left (216-123 x+20 x^2-x^3+e^x \left (-216+123 x-20 x^2+x^3\right )\right ) \log \left (\frac {-27+12 x-x^2}{-8+x}\right )}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} \left (-216+123 x-20 x^2+x^3\right )+e^x \left (432 x-246 x^2+40 x^3-2 x^4\right )} \, dx=\frac {\log \left (-\frac {x^{2} - 12 \, x + 27}{x - 8}\right )}{x - e^{x}} \] Input:

integrate((((x^3-20*x^2+123*x-216)*exp(x)-x^3+20*x^2-123*x+216)*log((-x^2+ 
12*x-27)/(-8+x))+(-x^2+16*x-69)*exp(x)+x^3-16*x^2+69*x)/((x^3-20*x^2+123*x 
-216)*exp(x)^2+(-2*x^4+40*x^3-246*x^2+432*x)*exp(x)+x^5-20*x^4+123*x^3-216 
*x^2),x, algorithm="fricas")
 

Output:

log(-(x^2 - 12*x + 27)/(x - 8))/(x - e^x)
 

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {69 x-16 x^2+x^3+e^x \left (-69+16 x-x^2\right )+\left (216-123 x+20 x^2-x^3+e^x \left (-216+123 x-20 x^2+x^3\right )\right ) \log \left (\frac {-27+12 x-x^2}{-8+x}\right )}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} \left (-216+123 x-20 x^2+x^3\right )+e^x \left (432 x-246 x^2+40 x^3-2 x^4\right )} \, dx=- \frac {\log {\left (\frac {- x^{2} + 12 x - 27}{x - 8} \right )}}{- x + e^{x}} \] Input:

integrate((((x**3-20*x**2+123*x-216)*exp(x)-x**3+20*x**2-123*x+216)*ln((-x 
**2+12*x-27)/(-8+x))+(-x**2+16*x-69)*exp(x)+x**3-16*x**2+69*x)/((x**3-20*x 
**2+123*x-216)*exp(x)**2+(-2*x**4+40*x**3-246*x**2+432*x)*exp(x)+x**5-20*x 
**4+123*x**3-216*x**2),x)
 

Output:

-log((-x**2 + 12*x - 27)/(x - 8))/(-x + exp(x))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {69 x-16 x^2+x^3+e^x \left (-69+16 x-x^2\right )+\left (216-123 x+20 x^2-x^3+e^x \left (-216+123 x-20 x^2+x^3\right )\right ) \log \left (\frac {-27+12 x-x^2}{-8+x}\right )}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} \left (-216+123 x-20 x^2+x^3\right )+e^x \left (432 x-246 x^2+40 x^3-2 x^4\right )} \, dx=\frac {\log \left (x - 3\right ) - \log \left (x - 8\right ) + \log \left (-x + 9\right )}{x - e^{x}} \] Input:

integrate((((x^3-20*x^2+123*x-216)*exp(x)-x^3+20*x^2-123*x+216)*log((-x^2+ 
12*x-27)/(-8+x))+(-x^2+16*x-69)*exp(x)+x^3-16*x^2+69*x)/((x^3-20*x^2+123*x 
-216)*exp(x)^2+(-2*x^4+40*x^3-246*x^2+432*x)*exp(x)+x^5-20*x^4+123*x^3-216 
*x^2),x, algorithm="maxima")
 

Output:

(log(x - 3) - log(x - 8) + log(-x + 9))/(x - e^x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {69 x-16 x^2+x^3+e^x \left (-69+16 x-x^2\right )+\left (216-123 x+20 x^2-x^3+e^x \left (-216+123 x-20 x^2+x^3\right )\right ) \log \left (\frac {-27+12 x-x^2}{-8+x}\right )}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} \left (-216+123 x-20 x^2+x^3\right )+e^x \left (432 x-246 x^2+40 x^3-2 x^4\right )} \, dx=\frac {\log \left (-\frac {x^{2} - 12 \, x + 27}{x - 8}\right )}{x - e^{x}} \] Input:

integrate((((x^3-20*x^2+123*x-216)*exp(x)-x^3+20*x^2-123*x+216)*log((-x^2+ 
12*x-27)/(-8+x))+(-x^2+16*x-69)*exp(x)+x^3-16*x^2+69*x)/((x^3-20*x^2+123*x 
-216)*exp(x)^2+(-2*x^4+40*x^3-246*x^2+432*x)*exp(x)+x^5-20*x^4+123*x^3-216 
*x^2),x, algorithm="giac")
 

Output:

log(-(x^2 - 12*x + 27)/(x - 8))/(x - e^x)
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {69 x-16 x^2+x^3+e^x \left (-69+16 x-x^2\right )+\left (216-123 x+20 x^2-x^3+e^x \left (-216+123 x-20 x^2+x^3\right )\right ) \log \left (\frac {-27+12 x-x^2}{-8+x}\right )}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} \left (-216+123 x-20 x^2+x^3\right )+e^x \left (432 x-246 x^2+40 x^3-2 x^4\right )} \, dx=\frac {\ln \left (-\frac {x^2-12\,x+27}{x-8}\right )}{x-{\mathrm {e}}^x} \] Input:

int((69*x - exp(x)*(x^2 - 16*x + 69) + log(-(x^2 - 12*x + 27)/(x - 8))*(ex 
p(x)*(123*x - 20*x^2 + x^3 - 216) - 123*x + 20*x^2 - x^3 + 216) - 16*x^2 + 
 x^3)/(exp(x)*(432*x - 246*x^2 + 40*x^3 - 2*x^4) - 216*x^2 + 123*x^3 - 20* 
x^4 + x^5 + exp(2*x)*(123*x - 20*x^2 + x^3 - 216)),x)
 

Output:

log(-(x^2 - 12*x + 27)/(x - 8))/(x - exp(x))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {69 x-16 x^2+x^3+e^x \left (-69+16 x-x^2\right )+\left (216-123 x+20 x^2-x^3+e^x \left (-216+123 x-20 x^2+x^3\right )\right ) \log \left (\frac {-27+12 x-x^2}{-8+x}\right )}{-216 x^2+123 x^3-20 x^4+x^5+e^{2 x} \left (-216+123 x-20 x^2+x^3\right )+e^x \left (432 x-246 x^2+40 x^3-2 x^4\right )} \, dx=-\frac {\mathrm {log}\left (\frac {-x^{2}+12 x -27}{x -8}\right )}{e^{x}-x} \] Input:

int((((x^3-20*x^2+123*x-216)*exp(x)-x^3+20*x^2-123*x+216)*log((-x^2+12*x-2 
7)/(-8+x))+(-x^2+16*x-69)*exp(x)+x^3-16*x^2+69*x)/((x^3-20*x^2+123*x-216)* 
exp(x)^2+(-2*x^4+40*x^3-246*x^2+432*x)*exp(x)+x^5-20*x^4+123*x^3-216*x^2), 
x)
 

Output:

( - log(( - x**2 + 12*x - 27)/(x - 8)))/(e**x - x)