\(\int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+(18+e^2 (18-9 x)+3 x-4 x^2-x^3) \log (\frac {1}{9} (9+9 e^2+6 x+x^2))+(-48-28 x+2 x^2+2 x^3+(-24-2 x+2 x^2) \log (16-8 x+x^2)) \log (2+x+\log (16-8 x+x^2))}{(-72-66 x-11 x^2+4 x^3+x^4+e^2 (-72-18 x+9 x^2)+(-36-15 x+2 x^2+x^3+e^2 (-36+9 x)) \log (16-8 x+x^2)) \log ^2(2+x+\log (16-8 x+x^2))} \, dx\) [1163]

Optimal result

Integrand size = 198, antiderivative size = 29 \[ \int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+\left (18+e^2 (18-9 x)+3 x-4 x^2-x^3\right ) \log \left (\frac {1}{9} \left (9+9 e^2+6 x+x^2\right )\right )+\left (-48-28 x+2 x^2+2 x^3+\left (-24-2 x+2 x^2\right ) \log \left (16-8 x+x^2\right )\right ) \log \left (2+x+\log \left (16-8 x+x^2\right )\right )}{\left (-72-66 x-11 x^2+4 x^3+x^4+e^2 \left (-72-18 x+9 x^2\right )+\left (-36-15 x+2 x^2+x^3+e^2 (-36+9 x)\right ) \log \left (16-8 x+x^2\right )\right ) \log ^2\left (2+x+\log \left (16-8 x+x^2\right )\right )} \, dx=\frac {-4+\log \left (e^2+\left (1+\frac {x}{3}\right )^2\right )}{\log \left (2+x+\log \left ((-4+x)^2\right )\right )} \] Output:

(ln(exp(2)+(1+1/3*x)^2)-4)/ln(x+2+ln((-4+x)^2))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+\left (18+e^2 (18-9 x)+3 x-4 x^2-x^3\right ) \log \left (\frac {1}{9} \left (9+9 e^2+6 x+x^2\right )\right )+\left (-48-28 x+2 x^2+2 x^3+\left (-24-2 x+2 x^2\right ) \log \left (16-8 x+x^2\right )\right ) \log \left (2+x+\log \left (16-8 x+x^2\right )\right )}{\left (-72-66 x-11 x^2+4 x^3+x^4+e^2 \left (-72-18 x+9 x^2\right )+\left (-36-15 x+2 x^2+x^3+e^2 (-36+9 x)\right ) \log \left (16-8 x+x^2\right )\right ) \log ^2\left (2+x+\log \left (16-8 x+x^2\right )\right )} \, dx=\frac {-4+\log \left (e^2+\frac {1}{9} (3+x)^2\right )}{\log \left (2+x+\log \left ((-4+x)^2\right )\right )} \] Input:

Integrate[(-72 - 12*x + 16*x^2 + 4*x^3 + E^2*(-72 + 36*x) + (18 + E^2*(18 
- 9*x) + 3*x - 4*x^2 - x^3)*Log[(9 + 9*E^2 + 6*x + x^2)/9] + (-48 - 28*x + 
 2*x^2 + 2*x^3 + (-24 - 2*x + 2*x^2)*Log[16 - 8*x + x^2])*Log[2 + x + Log[ 
16 - 8*x + x^2]])/((-72 - 66*x - 11*x^2 + 4*x^3 + x^4 + E^2*(-72 - 18*x + 
9*x^2) + (-36 - 15*x + 2*x^2 + x^3 + E^2*(-36 + 9*x))*Log[16 - 8*x + x^2]) 
*Log[2 + x + Log[16 - 8*x + x^2]]^2),x]
 

Output:

(-4 + Log[E^2 + (3 + x)^2/9])/Log[2 + x + Log[(-4 + x)^2]]
 

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.

\(\displaystyle \int \frac {4 x^3+16 x^2+\left (-x^3-4 x^2+3 x+e^2 (18-9 x)+18\right ) \log \left (\frac {1}{9} \left (x^2+6 x+9 e^2+9\right )\right )+\left (2 x^3+2 x^2+\left (2 x^2-2 x-24\right ) \log \left (x^2-8 x+16\right )-28 x-48\right ) \log \left (\log \left (x^2-8 x+16\right )+x+2\right )-12 x+e^2 (36 x-72)-72}{\left (x^4+4 x^3-11 x^2+e^2 \left (9 x^2-18 x-72\right )+\left (x^3+2 x^2-15 x+e^2 (9 x-36)-36\right ) \log \left (x^2-8 x+16\right )-66 x-72\right ) \log ^2\left (\log \left (x^2-8 x+16\right )+x+2\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-4 x^3-16 x^2-\left (-x^3-4 x^2+3 x+e^2 (18-9 x)+18\right ) \log \left (\frac {1}{9} \left (x^2+6 x+9 e^2+9\right )\right )-\left (2 x^3+2 x^2+\left (2 x^2-2 x-24\right ) \log \left (x^2-8 x+16\right )-28 x-48\right ) \log \left (\log \left (x^2-8 x+16\right )+x+2\right )+12 x-e^2 (36 x-72)+72}{(4-x) \left (x^2+6 x+9 \left (1+e^2\right )\right ) \left (x+\log \left ((x-4)^2\right )+2\right ) \log ^2\left (\log \left (x^2-8 x+16\right )+x+2\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {2 (x+3)}{\left (x^2+6 x+9 \left (1+e^2\right )\right ) \log \left (x+\log \left ((x-4)^2\right )+2\right )}-\frac {(x-2) \left (\log \left (\frac {1}{9} (x+3)^2+e^2\right )-4\right )}{(x-4) \left (x+\log \left ((x-4)^2\right )+2\right ) \log ^2\left (x+\log \left ((x-4)^2\right )+2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {\log \left (\frac {x^2}{9}+\frac {2 x}{3}+e^2+1\right )}{\left (x+\log \left ((x-4)^2\right )+2\right ) \log ^2\left (x+\log \left ((x-4)^2\right )+2\right )}dx-2 \int \frac {\log \left (\frac {x^2}{9}+\frac {2 x}{3}+e^2+1\right )}{(x-4) \left (x+\log \left ((x-4)^2\right )+2\right ) \log ^2\left (x+\log \left ((x-4)^2\right )+2\right )}dx+4 \int \frac {1}{\left (x+\log \left ((x-4)^2\right )+2\right ) \log ^2\left (x+\log \left ((x-4)^2\right )+2\right )}dx+8 \int \frac {1}{(x-4) \left (x+\log \left ((x-4)^2\right )+2\right ) \log ^2\left (x+\log \left ((x-4)^2\right )+2\right )}dx+\frac {2 i \int \frac {1}{(-2 x+6 i e-6) \log \left (x+\log \left ((x-4)^2\right )+2\right )}dx}{e}+\frac {2 (e+i) \int \frac {1}{(2 x-6 i e+6) \log \left (x+\log \left ((x-4)^2\right )+2\right )}dx}{e}-\frac {2 (-e+i) \int \frac {1}{(2 x+6 i e+6) \log \left (x+\log \left ((x-4)^2\right )+2\right )}dx}{e}+\frac {2 i \int \frac {1}{(2 x+6 i e+6) \log \left (x+\log \left ((x-4)^2\right )+2\right )}dx}{e}\)

Input:

Int[(-72 - 12*x + 16*x^2 + 4*x^3 + E^2*(-72 + 36*x) + (18 + E^2*(18 - 9*x) 
 + 3*x - 4*x^2 - x^3)*Log[(9 + 9*E^2 + 6*x + x^2)/9] + (-48 - 28*x + 2*x^2 
 + 2*x^3 + (-24 - 2*x + 2*x^2)*Log[16 - 8*x + x^2])*Log[2 + x + Log[16 - 8 
*x + x^2]])/((-72 - 66*x - 11*x^2 + 4*x^3 + x^4 + E^2*(-72 - 18*x + 9*x^2) 
 + (-36 - 15*x + 2*x^2 + x^3 + E^2*(-36 + 9*x))*Log[16 - 8*x + x^2])*Log[2 
 + x + Log[16 - 8*x + x^2]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 171.90 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21

method	result	size
parallelrisch	\(\frac {-8+2 \ln \left ({\mathrm e}^{2}+\frac {x^{2}}{9}+\frac {2 x}{3}+1\right )}{2 \ln \left (\ln \left (x^{2}-8 x +16\right )+2+x \right )}\)	\(35\)
risch	\(\frac {\ln \left ({\mathrm e}^{2}+\frac {x^{2}}{9}+\frac {2 x}{3}+1\right )-4}{\ln \left (2 \ln \left (x -4\right )-\frac {i \pi \,\operatorname {csgn}\left (i \left (x -4\right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (x -4\right )^{2}\right )+\operatorname {csgn}\left (i \left (x -4\right )\right )\right )}^{2}}{2}+2+x \right )}\)	\(63\)

Input:

int((((2*x^2-2*x-24)*ln(x^2-8*x+16)+2*x^3+2*x^2-28*x-48)*ln(ln(x^2-8*x+16) 
+2+x)+((-9*x+18)*exp(2)-x^3-4*x^2+3*x+18)*ln(exp(2)+1/9*x^2+2/3*x+1)+(36*x 
-72)*exp(2)+4*x^3+16*x^2-12*x-72)/(((9*x-36)*exp(2)+x^3+2*x^2-15*x-36)*ln( 
x^2-8*x+16)+(9*x^2-18*x-72)*exp(2)+x^4+4*x^3-11*x^2-66*x-72)/ln(ln(x^2-8*x 
+16)+2+x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*(-8+2*ln(exp(2)+1/9*x^2+2/3*x+1))/ln(ln(x^2-8*x+16)+2+x)
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+\left (18+e^2 (18-9 x)+3 x-4 x^2-x^3\right ) \log \left (\frac {1}{9} \left (9+9 e^2+6 x+x^2\right )\right )+\left (-48-28 x+2 x^2+2 x^3+\left (-24-2 x+2 x^2\right ) \log \left (16-8 x+x^2\right )\right ) \log \left (2+x+\log \left (16-8 x+x^2\right )\right )}{\left (-72-66 x-11 x^2+4 x^3+x^4+e^2 \left (-72-18 x+9 x^2\right )+\left (-36-15 x+2 x^2+x^3+e^2 (-36+9 x)\right ) \log \left (16-8 x+x^2\right )\right ) \log ^2\left (2+x+\log \left (16-8 x+x^2\right )\right )} \, dx=\frac {\log \left (\frac {1}{9} \, x^{2} + \frac {2}{3} \, x + e^{2} + 1\right ) - 4}{\log \left (x + \log \left (x^{2} - 8 \, x + 16\right ) + 2\right )} \] Input:

integrate((((2*x^2-2*x-24)*log(x^2-8*x+16)+2*x^3+2*x^2-28*x-48)*log(log(x^ 
2-8*x+16)+2+x)+((-9*x+18)*exp(2)-x^3-4*x^2+3*x+18)*log(exp(2)+1/9*x^2+2/3* 
x+1)+(36*x-72)*exp(2)+4*x^3+16*x^2-12*x-72)/(((9*x-36)*exp(2)+x^3+2*x^2-15 
*x-36)*log(x^2-8*x+16)+(9*x^2-18*x-72)*exp(2)+x^4+4*x^3-11*x^2-66*x-72)/lo 
g(log(x^2-8*x+16)+2+x)^2,x, algorithm="fricas")
 

Output:

(log(1/9*x^2 + 2/3*x + e^2 + 1) - 4)/log(x + log(x^2 - 8*x + 16) + 2)
 

Sympy [A] (verification not implemented)

Time = 1.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+\left (18+e^2 (18-9 x)+3 x-4 x^2-x^3\right ) \log \left (\frac {1}{9} \left (9+9 e^2+6 x+x^2\right )\right )+\left (-48-28 x+2 x^2+2 x^3+\left (-24-2 x+2 x^2\right ) \log \left (16-8 x+x^2\right )\right ) \log \left (2+x+\log \left (16-8 x+x^2\right )\right )}{\left (-72-66 x-11 x^2+4 x^3+x^4+e^2 \left (-72-18 x+9 x^2\right )+\left (-36-15 x+2 x^2+x^3+e^2 (-36+9 x)\right ) \log \left (16-8 x+x^2\right )\right ) \log ^2\left (2+x+\log \left (16-8 x+x^2\right )\right )} \, dx=\frac {\log {\left (\frac {x^{2}}{9} + \frac {2 x}{3} + 1 + e^{2} \right )} - 4}{\log {\left (x + \log {\left (x^{2} - 8 x + 16 \right )} + 2 \right )}} \] Input:

integrate((((2*x**2-2*x-24)*ln(x**2-8*x+16)+2*x**3+2*x**2-28*x-48)*ln(ln(x 
**2-8*x+16)+2+x)+((-9*x+18)*exp(2)-x**3-4*x**2+3*x+18)*ln(exp(2)+1/9*x**2+ 
2/3*x+1)+(36*x-72)*exp(2)+4*x**3+16*x**2-12*x-72)/(((9*x-36)*exp(2)+x**3+2 
*x**2-15*x-36)*ln(x**2-8*x+16)+(9*x**2-18*x-72)*exp(2)+x**4+4*x**3-11*x**2 
-66*x-72)/ln(ln(x**2-8*x+16)+2+x)**2,x)
 

Output:

(log(x**2/9 + 2*x/3 + 1 + exp(2)) - 4)/log(x + log(x**2 - 8*x + 16) + 2)
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+\left (18+e^2 (18-9 x)+3 x-4 x^2-x^3\right ) \log \left (\frac {1}{9} \left (9+9 e^2+6 x+x^2\right )\right )+\left (-48-28 x+2 x^2+2 x^3+\left (-24-2 x+2 x^2\right ) \log \left (16-8 x+x^2\right )\right ) \log \left (2+x+\log \left (16-8 x+x^2\right )\right )}{\left (-72-66 x-11 x^2+4 x^3+x^4+e^2 \left (-72-18 x+9 x^2\right )+\left (-36-15 x+2 x^2+x^3+e^2 (-36+9 x)\right ) \log \left (16-8 x+x^2\right )\right ) \log ^2\left (2+x+\log \left (16-8 x+x^2\right )\right )} \, dx=-\frac {2 \, \log \left (3\right ) - \log \left (x^{2} + 6 \, x + 9 \, e^{2} + 9\right ) + 4}{\log \left (x + 2 \, \log \left (x - 4\right ) + 2\right )} \] Input:

integrate((((2*x^2-2*x-24)*log(x^2-8*x+16)+2*x^3+2*x^2-28*x-48)*log(log(x^ 
2-8*x+16)+2+x)+((-9*x+18)*exp(2)-x^3-4*x^2+3*x+18)*log(exp(2)+1/9*x^2+2/3* 
x+1)+(36*x-72)*exp(2)+4*x^3+16*x^2-12*x-72)/(((9*x-36)*exp(2)+x^3+2*x^2-15 
*x-36)*log(x^2-8*x+16)+(9*x^2-18*x-72)*exp(2)+x^4+4*x^3-11*x^2-66*x-72)/lo 
g(log(x^2-8*x+16)+2+x)^2,x, algorithm="maxima")
 

Output:

-(2*log(3) - log(x^2 + 6*x + 9*e^2 + 9) + 4)/log(x + 2*log(x - 4) + 2)
 

Giac [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+\left (18+e^2 (18-9 x)+3 x-4 x^2-x^3\right ) \log \left (\frac {1}{9} \left (9+9 e^2+6 x+x^2\right )\right )+\left (-48-28 x+2 x^2+2 x^3+\left (-24-2 x+2 x^2\right ) \log \left (16-8 x+x^2\right )\right ) \log \left (2+x+\log \left (16-8 x+x^2\right )\right )}{\left (-72-66 x-11 x^2+4 x^3+x^4+e^2 \left (-72-18 x+9 x^2\right )+\left (-36-15 x+2 x^2+x^3+e^2 (-36+9 x)\right ) \log \left (16-8 x+x^2\right )\right ) \log ^2\left (2+x+\log \left (16-8 x+x^2\right )\right )} \, dx=-\frac {2 \, \log \left (3\right ) - \log \left (x^{2} + 6 \, x + 9 \, e^{2} + 9\right ) + 4}{\log \left (x + \log \left (x^{2} - 8 \, x + 16\right ) + 2\right )} \] Input:

integrate((((2*x^2-2*x-24)*log(x^2-8*x+16)+2*x^3+2*x^2-28*x-48)*log(log(x^ 
2-8*x+16)+2+x)+((-9*x+18)*exp(2)-x^3-4*x^2+3*x+18)*log(exp(2)+1/9*x^2+2/3* 
x+1)+(36*x-72)*exp(2)+4*x^3+16*x^2-12*x-72)/(((9*x-36)*exp(2)+x^3+2*x^2-15 
*x-36)*log(x^2-8*x+16)+(9*x^2-18*x-72)*exp(2)+x^4+4*x^3-11*x^2-66*x-72)/lo 
g(log(x^2-8*x+16)+2+x)^2,x, algorithm="giac")
 

Output:

-(2*log(3) - log(x^2 + 6*x + 9*e^2 + 9) + 4)/log(x + log(x^2 - 8*x + 16) + 
 2)
 

Mupad [B] (verification not implemented)

Time = 3.50 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+\left (18+e^2 (18-9 x)+3 x-4 x^2-x^3\right ) \log \left (\frac {1}{9} \left (9+9 e^2+6 x+x^2\right )\right )+\left (-48-28 x+2 x^2+2 x^3+\left (-24-2 x+2 x^2\right ) \log \left (16-8 x+x^2\right )\right ) \log \left (2+x+\log \left (16-8 x+x^2\right )\right )}{\left (-72-66 x-11 x^2+4 x^3+x^4+e^2 \left (-72-18 x+9 x^2\right )+\left (-36-15 x+2 x^2+x^3+e^2 (-36+9 x)\right ) \log \left (16-8 x+x^2\right )\right ) \log ^2\left (2+x+\log \left (16-8 x+x^2\right )\right )} \, dx=-\frac {2\,\ln \left (x+\ln \left (x^2-8\,x+16\right )+2\right )-\ln \left (\frac {x^2}{9}+\frac {2\,x}{3}+{\mathrm {e}}^2+1\right )+4}{\ln \left (x+\ln \left (x^2-8\,x+16\right )+2\right )} \] Input:

int((12*x + log(x + log(x^2 - 8*x + 16) + 2)*(28*x + log(x^2 - 8*x + 16)*( 
2*x - 2*x^2 + 24) - 2*x^2 - 2*x^3 + 48) - 16*x^2 - 4*x^3 + log((2*x)/3 + e 
xp(2) + x^2/9 + 1)*(4*x^2 - 3*x + x^3 + exp(2)*(9*x - 18) - 18) - exp(2)*( 
36*x - 72) + 72)/(log(x + log(x^2 - 8*x + 16) + 2)^2*(66*x - log(x^2 - 8*x 
 + 16)*(2*x^2 - 15*x + x^3 + exp(2)*(9*x - 36) - 36) + exp(2)*(18*x - 9*x^ 
2 + 72) + 11*x^2 - 4*x^3 - x^4 + 72)),x)
 

Output:

-(2*log(x + log(x^2 - 8*x + 16) + 2) - log((2*x)/3 + exp(2) + x^2/9 + 1) + 
 4)/log(x + log(x^2 - 8*x + 16) + 2)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {-72-12 x+16 x^2+4 x^3+e^2 (-72+36 x)+\left (18+e^2 (18-9 x)+3 x-4 x^2-x^3\right ) \log \left (\frac {1}{9} \left (9+9 e^2+6 x+x^2\right )\right )+\left (-48-28 x+2 x^2+2 x^3+\left (-24-2 x+2 x^2\right ) \log \left (16-8 x+x^2\right )\right ) \log \left (2+x+\log \left (16-8 x+x^2\right )\right )}{\left (-72-66 x-11 x^2+4 x^3+x^4+e^2 \left (-72-18 x+9 x^2\right )+\left (-36-15 x+2 x^2+x^3+e^2 (-36+9 x)\right ) \log \left (16-8 x+x^2\right )\right ) \log ^2\left (2+x+\log \left (16-8 x+x^2\right )\right )} \, dx=\frac {\mathrm {log}\left (e^{2}+\frac {1}{9} x^{2}+\frac {2}{3} x +1\right )-4}{\mathrm {log}\left (\mathrm {log}\left (x^{2}-8 x +16\right )+x +2\right )} \] Input:

int((((2*x^2-2*x-24)*log(x^2-8*x+16)+2*x^3+2*x^2-28*x-48)*log(log(x^2-8*x+ 
16)+2+x)+((-9*x+18)*exp(2)-x^3-4*x^2+3*x+18)*log(exp(2)+1/9*x^2+2/3*x+1)+( 
36*x-72)*exp(2)+4*x^3+16*x^2-12*x-72)/(((9*x-36)*exp(2)+x^3+2*x^2-15*x-36) 
*log(x^2-8*x+16)+(9*x^2-18*x-72)*exp(2)+x^4+4*x^3-11*x^2-66*x-72)/log(log( 
x^2-8*x+16)+2+x)^2,x)
 

Output:

(log((9*e**2 + x**2 + 6*x + 9)/9) - 4)/log(log(x**2 - 8*x + 16) + x + 2)