\(\int \frac {e^{2 x} (4+16 x)+e^{2 x} (x+2 x^2) \log ^2(3 x+6 x^2)+(e^{2 x} (-4-16 x-16 x^2) \log (3 x+6 x^2)+e^{2 x} (4+17 x+20 x^2+4 x^3) \log ^2(3 x+6 x^2)) \log (\frac {-4+(4+x) \log (3 x+6 x^2)}{\log (3 x+6 x^2)})}{(-4-8 x) \log (3 x+6 x^2)+(4+9 x+2 x^2) \log ^2(3 x+6 x^2)} \, dx\) [2810]

Optimal result

Integrand size = 171, antiderivative size = 24 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=e^{2 x} x \log \left (4+x-\frac {4}{\log (3 x (1+2 x))}\right ) \] Input:

Integrate[(E^(2*x)*(4 + 16*x) + E^(2*x)*(x + 2*x^2)*Log[3*x + 6*x^2]^2 + ( 
E^(2*x)*(-4 - 16*x - 16*x^2)*Log[3*x + 6*x^2] + E^(2*x)*(4 + 17*x + 20*x^2 
 + 4*x^3)*Log[3*x + 6*x^2]^2)*Log[(-4 + (4 + x)*Log[3*x + 6*x^2])/Log[3*x 
+ 6*x^2]])/((-4 - 8*x)*Log[3*x + 6*x^2] + (4 + 9*x + 2*x^2)*Log[3*x + 6*x^ 
2]^2),x]
 

Output:

E^(2*x)*x*Log[4 + x - 4/Log[3*x*(1 + 2*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.

\(\displaystyle \int \frac {e^{2 x} \left (2 x^2+x\right ) \log ^2\left (6 x^2+3 x\right )+\left (e^{2 x} \left (-16 x^2-16 x-4\right ) \log \left (6 x^2+3 x\right )+e^{2 x} \left (4 x^3+20 x^2+17 x+4\right ) \log ^2\left (6 x^2+3 x\right )\right ) \log \left (\frac {(x+4) \log \left (6 x^2+3 x\right )-4}{\log \left (6 x^2+3 x\right )}\right )+e^{2 x} (16 x+4)}{\left (2 x^2+9 x+4\right ) \log ^2\left (6 x^2+3 x\right )+(-8 x-4) \log \left (6 x^2+3 x\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-e^{2 x} \left (2 x^2+x\right ) \log ^2\left (6 x^2+3 x\right )-\left (e^{2 x} \left (-16 x^2-16 x-4\right ) \log \left (6 x^2+3 x\right )+e^{2 x} \left (4 x^3+20 x^2+17 x+4\right ) \log ^2\left (6 x^2+3 x\right )\right ) \log \left (\frac {(x+4) \log \left (6 x^2+3 x\right )-4}{\log \left (6 x^2+3 x\right )}\right )-e^{2 x} (16 x+4)}{(2 x+1) \log (3 x (2 x+1)) (-x \log (3 x (2 x+1))-4 \log (3 x (2 x+1))+4)}dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {e^{2 x}}{\log (3 x (2 x+1))}dx+\int \frac {e^{2 x}}{(2 x+1) \log (3 x (2 x+1))}dx+\frac {23}{2} \int \frac {e^{2 x}}{x \log (3 x (2 x+1))+4 \log (3 x (2 x+1))-4}dx+2 \int \frac {e^{2 x} x}{x \log (3 x (2 x+1))+4 \log (3 x (2 x+1))-4}dx-16 \int \frac {e^{2 x}}{(x+4) (x \log (3 x (2 x+1))+4 \log (3 x (2 x+1))-4)}dx-\frac {7}{2} \int \frac {e^{2 x}}{(2 x+1) (x \log (3 x (2 x+1))+4 \log (3 x (2 x+1))-4)}dx+\int e^{2 x} \log \left (x-\frac {4}{\log (3 x (2 x+1))}+4\right )dx+2 \int e^{2 x} x \log \left (x-\frac {4}{\log (3 x (2 x+1))}+4\right )dx-\frac {4 \operatorname {ExpIntegralEi}(2 (x+4))}{e^8}+\frac {e^{2 x}}{2}\)

Input:

Int[(E^(2*x)*(4 + 16*x) + E^(2*x)*(x + 2*x^2)*Log[3*x + 6*x^2]^2 + (E^(2*x 
)*(-4 - 16*x - 16*x^2)*Log[3*x + 6*x^2] + E^(2*x)*(4 + 17*x + 20*x^2 + 4*x 
^3)*Log[3*x + 6*x^2]^2)*Log[(-4 + (4 + x)*Log[3*x + 6*x^2])/Log[3*x + 6*x^ 
2]])/((-4 - 8*x)*Log[3*x + 6*x^2] + (4 + 9*x + 2*x^2)*Log[3*x + 6*x^2]^2), 
x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 98.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54

method	result	size
parallelrisch	\(\ln \left (\frac {\left (4+x \right ) \ln \left (6 x^{2}+3 x \right )-4}{\ln \left (6 x^{2}+3 x \right )}\right ) {\mathrm e}^{2 x} x\)	\(37\)
risch	\(\text {Expression too large to display}\)	\(2063\)

Input:

int((((4*x^3+20*x^2+17*x+4)*exp(x)^2*ln(6*x^2+3*x)^2+(-16*x^2-16*x-4)*exp( 
x)^2*ln(6*x^2+3*x))*ln(((4+x)*ln(6*x^2+3*x)-4)/ln(6*x^2+3*x))+(2*x^2+x)*ex 
p(x)^2*ln(6*x^2+3*x)^2+(16*x+4)*exp(x)^2)/((2*x^2+9*x+4)*ln(6*x^2+3*x)^2+( 
-8*x-4)*ln(6*x^2+3*x)),x,method=_RETURNVERBOSE)
 

Output:

ln(((4+x)*ln(6*x^2+3*x)-4)/ln(6*x^2+3*x))*exp(x)^2*x
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=x e^{\left (2 \, x\right )} \log \left (\frac {{\left (x + 4\right )} \log \left (6 \, x^{2} + 3 \, x\right ) - 4}{\log \left (6 \, x^{2} + 3 \, x\right )}\right ) \] Input:

integrate((((4*x^3+20*x^2+17*x+4)*exp(x)^2*log(6*x^2+3*x)^2+(-16*x^2-16*x- 
4)*exp(x)^2*log(6*x^2+3*x))*log(((4+x)*log(6*x^2+3*x)-4)/log(6*x^2+3*x))+( 
2*x^2+x)*exp(x)^2*log(6*x^2+3*x)^2+(16*x+4)*exp(x)^2)/((2*x^2+9*x+4)*log(6 
*x^2+3*x)^2+(-8*x-4)*log(6*x^2+3*x)),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((((4*x**3+20*x**2+17*x+4)*exp(x)**2*ln(6*x**2+3*x)**2+(-16*x**2- 
16*x-4)*exp(x)**2*ln(6*x**2+3*x))*ln(((4+x)*ln(6*x**2+3*x)-4)/ln(6*x**2+3* 
x))+(2*x**2+x)*exp(x)**2*ln(6*x**2+3*x)**2+(16*x+4)*exp(x)**2)/((2*x**2+9* 
x+4)*ln(6*x**2+3*x)**2+(-8*x-4)*ln(6*x**2+3*x)),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (23) = 46\).

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.42 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=x e^{\left (2 \, x\right )} \log \left (x {\left (\log \left (3\right ) + \log \left (2 \, x + 1\right )\right )} + {\left (x + 4\right )} \log \left (x\right ) + 4 \, \log \left (3\right ) + 4 \, \log \left (2 \, x + 1\right ) - 4\right ) - x e^{\left (2 \, x\right )} \log \left (\log \left (3\right ) + \log \left (2 \, x + 1\right ) + \log \left (x\right )\right ) \] Input:

integrate((((4*x^3+20*x^2+17*x+4)*exp(x)^2*log(6*x^2+3*x)^2+(-16*x^2-16*x- 
4)*exp(x)^2*log(6*x^2+3*x))*log(((4+x)*log(6*x^2+3*x)-4)/log(6*x^2+3*x))+( 
2*x^2+x)*exp(x)^2*log(6*x^2+3*x)^2+(16*x+4)*exp(x)^2)/((2*x^2+9*x+4)*log(6 
*x^2+3*x)^2+(-8*x-4)*log(6*x^2+3*x)),x, algorithm="maxima")
 

Output:

x*e^(2*x)*log(x*(log(3) + log(2*x + 1)) + (x + 4)*log(x) + 4*log(3) + 4*lo 
g(2*x + 1) - 4) - x*e^(2*x)*log(log(3) + log(2*x + 1) + log(x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (23) = 46\).

Time = 0.70 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=x e^{\left (2 \, x\right )} \log \left (x \log \left (6 \, x^{2} + 3 \, x\right ) + 4 \, \log \left (6 \, x^{2} + 3 \, x\right ) - 4\right ) - x e^{\left (2 \, x\right )} \log \left (\log \left (6 \, x^{2} + 3 \, x\right )\right ) \] Input:

integrate((((4*x^3+20*x^2+17*x+4)*exp(x)^2*log(6*x^2+3*x)^2+(-16*x^2-16*x- 
4)*exp(x)^2*log(6*x^2+3*x))*log(((4+x)*log(6*x^2+3*x)-4)/log(6*x^2+3*x))+( 
2*x^2+x)*exp(x)^2*log(6*x^2+3*x)^2+(16*x+4)*exp(x)^2)/((2*x^2+9*x+4)*log(6 
*x^2+3*x)^2+(-8*x-4)*log(6*x^2+3*x)),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=x\,{\mathrm {e}}^{2\,x}\,\ln \left (\frac {\ln \left (6\,x^2+3\,x\right )\,\left (x+4\right )-4}{\ln \left (6\,x^2+3\,x\right )}\right ) \] Input:

int((log((log(3*x + 6*x^2)*(x + 4) - 4)/log(3*x + 6*x^2))*(exp(2*x)*log(3* 
x + 6*x^2)^2*(17*x + 20*x^2 + 4*x^3 + 4) - exp(2*x)*log(3*x + 6*x^2)*(16*x 
 + 16*x^2 + 4)) + exp(2*x)*(16*x + 4) + exp(2*x)*log(3*x + 6*x^2)^2*(x + 2 
*x^2))/(log(3*x + 6*x^2)^2*(9*x + 2*x^2 + 4) - log(3*x + 6*x^2)*(8*x + 4)) 
,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {e^{2 x} (4+16 x)+e^{2 x} \left (x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )+\left (e^{2 x} \left (-4-16 x-16 x^2\right ) \log \left (3 x+6 x^2\right )+e^{2 x} \left (4+17 x+20 x^2+4 x^3\right ) \log ^2\left (3 x+6 x^2\right )\right ) \log \left (\frac {-4+(4+x) \log \left (3 x+6 x^2\right )}{\log \left (3 x+6 x^2\right )}\right )}{(-4-8 x) \log \left (3 x+6 x^2\right )+\left (4+9 x+2 x^2\right ) \log ^2\left (3 x+6 x^2\right )} \, dx=e^{2 x} \mathrm {log}\left (\frac {\mathrm {log}\left (6 x^{2}+3 x \right ) x +4 \,\mathrm {log}\left (6 x^{2}+3 x \right )-4}{\mathrm {log}\left (6 x^{2}+3 x \right )}\right ) x \] Input:

int((((4*x^3+20*x^2+17*x+4)*exp(x)^2*log(6*x^2+3*x)^2+(-16*x^2-16*x-4)*exp 
(x)^2*log(6*x^2+3*x))*log(((4+x)*log(6*x^2+3*x)-4)/log(6*x^2+3*x))+(2*x^2+ 
x)*exp(x)^2*log(6*x^2+3*x)^2+(16*x+4)*exp(x)^2)/((2*x^2+9*x+4)*log(6*x^2+3 
*x)^2+(-8*x-4)*log(6*x^2+3*x)),x)
 

Output:

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