\(\int \frac {-8 x+2 e^{2 x} x^3+e^x (-4 x+4 x^3)+(8 x+6 e^{2 x} x^2+e^x (-4+4 x+8 x^2)) \log (3 x)+(6 e^{2 x} x+e^x (4+4 x)) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+(-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2) \log (3 x)+(-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x) \log ^2(3 x)+(-4-e^4+e^{2 x}) \log ^3(3 x)} \, dx\) [378]

Optimal result

Integrand size = 214, antiderivative size = 25 \[ \int \frac {-8 x+2 e^{2 x} x^3+e^x \left (-4 x+4 x^3\right )+\left (8 x+6 e^{2 x} x^2+e^x \left (-4+4 x+8 x^2\right )\right ) \log (3 x)+\left (6 e^{2 x} x+e^x (4+4 x)\right ) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+\left (-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2\right ) \log (3 x)+\left (-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x\right ) \log ^2(3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^3(3 x)} \, dx=\log \left (4+e^4-\left (e^x+\frac {2 x}{x+\log (3 x)}\right )^2\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(25)=50\).

Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.52 \[ \int \frac {-8 x+2 e^{2 x} x^3+e^x \left (-4 x+4 x^3\right )+\left (8 x+6 e^{2 x} x^2+e^x \left (-4+4 x+8 x^2\right )\right ) \log (3 x)+\left (6 e^{2 x} x+e^x (4+4 x)\right ) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+\left (-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2\right ) \log (3 x)+\left (-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x\right ) \log ^2(3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^3(3 x)} \, dx=2 \left (-\log (x+\log (3 x))+\frac {1}{2} \log \left (-e^4 x^2+4 e^x x^2+e^{2 x} x^2-8 x \log (3 x)-2 e^4 x \log (3 x)+4 e^x x \log (3 x)+2 e^{2 x} x \log (3 x)-4 \log ^2(3 x)-e^4 \log ^2(3 x)+e^{2 x} \log ^2(3 x)\right )\right ) \] Input:

Integrate[(-8*x + 2*E^(2*x)*x^3 + E^x*(-4*x + 4*x^3) + (8*x + 6*E^(2*x)*x^ 
2 + E^x*(-4 + 4*x + 8*x^2))*Log[3*x] + (6*E^(2*x)*x + E^x*(4 + 4*x))*Log[3 
*x]^2 + 2*E^(2*x)*Log[3*x]^3)/(-(E^4*x^3) + 4*E^x*x^3 + E^(2*x)*x^3 + (-8* 
x^2 - 3*E^4*x^2 + 8*E^x*x^2 + 3*E^(2*x)*x^2)*Log[3*x] + (-12*x - 3*E^4*x + 
 4*E^x*x + 3*E^(2*x)*x)*Log[3*x]^2 + (-4 - E^4 + E^(2*x))*Log[3*x]^3),x]
 

Output:

2*(-Log[x + Log[3*x]] + Log[-(E^4*x^2) + 4*E^x*x^2 + E^(2*x)*x^2 - 8*x*Log 
[3*x] - 2*E^4*x*Log[3*x] + 4*E^x*x*Log[3*x] + 2*E^(2*x)*x*Log[3*x] - 4*Log 
[3*x]^2 - E^4*Log[3*x]^2 + E^(2*x)*Log[3*x]^2]/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.

\(\displaystyle \int \frac {2 e^{2 x} x^3+e^x \left (4 x^3-4 x\right )+\left (6 e^{2 x} x^2+e^x \left (8 x^2+4 x-4\right )+8 x\right ) \log (3 x)-8 x+2 e^{2 x} \log ^3(3 x)+\left (6 e^{2 x} x+e^x (4 x+4)\right ) \log ^2(3 x)}{4 e^x x^3+e^{2 x} x^3-e^4 x^3+\left (8 e^x x^2+3 e^{2 x} x^2-3 e^4 x^2-8 x^2\right ) \log (3 x)+\left (e^{2 x}-4-e^4\right ) \log ^3(3 x)+\left (4 e^x x+3 e^{2 x} x-3 e^4 x-12 x\right ) \log ^2(3 x)} \, dx\)

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \left (2 e^x x^3-e^4 x^3+4 e^x x^2 \log (3 x)-8 \left (1+\frac {3 e^4}{8}\right ) x^2 \log (3 x)+2 e^x x+4 x-4 \left (1+\frac {e^4}{4}\right ) \log ^3(3 x)+2 e^x x \log ^2(3 x)-12 \left (1+\frac {e^4}{4}\right ) x \log ^2(3 x)-2 e^x \log ^2(3 x)-2 e^x x \log (3 x)-4 x \log (3 x)+2 e^x \log (3 x)\right )}{(x+\log (3 x)) \left (-4 e^x x^2-e^{2 x} x^2+e^4 x^2-e^{2 x} \log ^2(3 x)+4 \left (1+\frac {e^4}{4}\right ) \log ^2(3 x)-4 e^x x \log (3 x)-2 e^{2 x} x \log (3 x)+8 \left (1+\frac {e^4}{4}\right ) x \log (3 x)\right )}+2\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {2 \left (2 e^x x^3-e^4 x^3+4 e^x x^2 \log (3 x)-8 \left (1+\frac {3 e^4}{8}\right ) x^2 \log (3 x)+2 e^x x+4 x-4 \left (1+\frac {e^4}{4}\right ) \log ^3(3 x)+2 e^x x \log ^2(3 x)-12 \left (1+\frac {e^4}{4}\right ) x \log ^2(3 x)-2 e^x \log ^2(3 x)-2 e^x x \log (3 x)-4 x \log (3 x)+2 e^x \log (3 x)\right )}{(x+\log (3 x)) \left (-4 e^x x^2-e^{2 x} x^2+e^4 x^2-e^{2 x} \log ^2(3 x)+4 \left (1+\frac {e^4}{4}\right ) \log ^2(3 x)-4 e^x x \log (3 x)-2 e^{2 x} x \log (3 x)+8 \left (1+\frac {e^4}{4}\right ) x \log (3 x)\right )}+2\right )dx\)

Input:

Int[(-8*x + 2*E^(2*x)*x^3 + E^x*(-4*x + 4*x^3) + (8*x + 6*E^(2*x)*x^2 + E^ 
x*(-4 + 4*x + 8*x^2))*Log[3*x] + (6*E^(2*x)*x + E^x*(4 + 4*x))*Log[3*x]^2 
+ 2*E^(2*x)*Log[3*x]^3)/(-(E^4*x^3) + 4*E^x*x^3 + E^(2*x)*x^3 + (-8*x^2 - 
3*E^4*x^2 + 8*E^x*x^2 + 3*E^(2*x)*x^2)*Log[3*x] + (-12*x - 3*E^4*x + 4*E^x 
*x + 3*E^(2*x)*x)*Log[3*x]^2 + (-4 - E^4 + E^(2*x))*Log[3*x]^3),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(23)=46\).

Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.68

\[\ln \left ({\mathrm e}^{2 x}-{\mathrm e}^{4}-4\right )-2 \ln \left (x +\ln \left (3 x \right )\right )+\ln \left (\ln \left (3 x \right )^{2}+\frac {2 x \left (-{\mathrm e}^{2 x}+{\mathrm e}^{4}-2 \,{\mathrm e}^{x}+4\right ) \ln \left (3 x \right )}{-{\mathrm e}^{2 x}+{\mathrm e}^{4}+4}+\frac {\left (-{\mathrm e}^{2 x}+{\mathrm e}^{4}-4 \,{\mathrm e}^{x}\right ) x^{2}}{-{\mathrm e}^{2 x}+{\mathrm e}^{4}+4}\right )\]

Input:

int((2*exp(x)^2*ln(3*x)^3+(6*x*exp(x)^2+(4+4*x)*exp(x))*ln(3*x)^2+(6*exp(x 
)^2*x^2+(8*x^2+4*x-4)*exp(x)+8*x)*ln(3*x)+2*exp(x)^2*x^3+(4*x^3-4*x)*exp(x 
)-8*x)/((exp(x)^2-exp(4)-4)*ln(3*x)^3+(3*x*exp(x)^2+4*exp(x)*x-3*x*exp(4)- 
12*x)*ln(3*x)^2+(3*exp(x)^2*x^2+8*exp(x)*x^2-3*x^2*exp(4)-8*x^2)*ln(3*x)+e 
xp(x)^2*x^3+4*exp(x)*x^3-x^3*exp(4)),x)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (23) = 46\).

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.04 \[ \int \frac {-8 x+2 e^{2 x} x^3+e^x \left (-4 x+4 x^3\right )+\left (8 x+6 e^{2 x} x^2+e^x \left (-4+4 x+8 x^2\right )\right ) \log (3 x)+\left (6 e^{2 x} x+e^x (4+4 x)\right ) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+\left (-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2\right ) \log (3 x)+\left (-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x\right ) \log ^2(3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^3(3 x)} \, dx=-2 \, \log \left (x + \log \left (3 \, x\right )\right ) + \log \left (\frac {x^{2} e^{4} - x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{x} + {\left (e^{4} - e^{\left (2 \, x\right )} + 4\right )} \log \left (3 \, x\right )^{2} + 2 \, {\left (x e^{4} - x e^{\left (2 \, x\right )} - 2 \, x e^{x} + 4 \, x\right )} \log \left (3 \, x\right )}{e^{4} - e^{\left (2 \, x\right )} + 4}\right ) + \log \left (-e^{4} + e^{\left (2 \, x\right )} - 4\right ) \] Input:

integrate((2*exp(x)^2*log(3*x)^3+(6*x*exp(x)^2+(4+4*x)*exp(x))*log(3*x)^2+ 
(6*exp(x)^2*x^2+(8*x^2+4*x-4)*exp(x)+8*x)*log(3*x)+2*exp(x)^2*x^3+(4*x^3-4 
*x)*exp(x)-8*x)/((exp(x)^2-exp(4)-4)*log(3*x)^3+(3*x*exp(x)^2+4*exp(x)*x-3 
*x*exp(4)-12*x)*log(3*x)^2+(3*exp(x)^2*x^2+8*exp(x)*x^2-3*x^2*exp(4)-8*x^2 
)*log(3*x)+exp(x)^2*x^3+4*exp(x)*x^3-x^3*exp(4)),x, algorithm="fricas")
 

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (20) = 40\).

Time = 1.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.40 \[ \int \frac {-8 x+2 e^{2 x} x^3+e^x \left (-4 x+4 x^3\right )+\left (8 x+6 e^{2 x} x^2+e^x \left (-4+4 x+8 x^2\right )\right ) \log (3 x)+\left (6 e^{2 x} x+e^x (4+4 x)\right ) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+\left (-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2\right ) \log (3 x)+\left (-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x\right ) \log ^2(3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^3(3 x)} \, dx=\log {\left (\frac {4 x e^{x}}{x + \log {\left (3 x \right )}} + e^{2 x} + \frac {- x^{2} e^{4} - 2 x e^{4} \log {\left (3 x \right )} - 8 x \log {\left (3 x \right )} - e^{4} \log {\left (3 x \right )}^{2} - 4 \log {\left (3 x \right )}^{2}}{x^{2} + 2 x \log {\left (3 x \right )} + \log {\left (3 x \right )}^{2}} \right )} \] Input:

integrate((2*exp(x)**2*ln(3*x)**3+(6*x*exp(x)**2+(4+4*x)*exp(x))*ln(3*x)** 
2+(6*exp(x)**2*x**2+(8*x**2+4*x-4)*exp(x)+8*x)*ln(3*x)+2*exp(x)**2*x**3+(4 
*x**3-4*x)*exp(x)-8*x)/((exp(x)**2-exp(4)-4)*ln(3*x)**3+(3*x*exp(x)**2+4*e 
xp(x)*x-3*x*exp(4)-12*x)*ln(3*x)**2+(3*exp(x)**2*x**2+8*exp(x)*x**2-3*x**2 
*exp(4)-8*x**2)*ln(3*x)+exp(x)**2*x**3+4*exp(x)*x**3-x**3*exp(4)),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (23) = 46\).

Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 5.56 \[ \int \frac {-8 x+2 e^{2 x} x^3+e^x \left (-4 x+4 x^3\right )+\left (8 x+6 e^{2 x} x^2+e^x \left (-4+4 x+8 x^2\right )\right ) \log (3 x)+\left (6 e^{2 x} x+e^x (4+4 x)\right ) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+\left (-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2\right ) \log (3 x)+\left (-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x\right ) \log ^2(3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^3(3 x)} \, dx=\log \left (-\frac {x^{2} e^{4} + e^{4} \log \left (3\right )^{2} + {\left (e^{4} + 4\right )} \log \left (x\right )^{2} + 2 \, {\left (e^{4} \log \left (3\right ) + 4 \, \log \left (3\right )\right )} x - {\left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2} + 2 \, {\left (x + \log \left (3\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{2} + x \log \left (3\right ) + x \log \left (x\right )\right )} e^{x} + 4 \, \log \left (3\right )^{2} + 2 \, {\left (x {\left (e^{4} + 4\right )} + e^{4} \log \left (3\right ) + 4 \, \log \left (3\right )\right )} \log \left (x\right )}{x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2} + 2 \, {\left (x + \log \left (3\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2}}\right ) \] Input:

integrate((2*exp(x)^2*log(3*x)^3+(6*x*exp(x)^2+(4+4*x)*exp(x))*log(3*x)^2+ 
(6*exp(x)^2*x^2+(8*x^2+4*x-4)*exp(x)+8*x)*log(3*x)+2*exp(x)^2*x^3+(4*x^3-4 
*x)*exp(x)-8*x)/((exp(x)^2-exp(4)-4)*log(3*x)^3+(3*x*exp(x)^2+4*exp(x)*x-3 
*x*exp(4)-12*x)*log(3*x)^2+(3*exp(x)^2*x^2+8*exp(x)*x^2-3*x^2*exp(4)-8*x^2 
)*log(3*x)+exp(x)^2*x^3+4*exp(x)*x^3-x^3*exp(4)),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (23) = 46\).

Time = 0.64 (sec) , antiderivative size = 184, normalized size of antiderivative = 7.36 \[ \int \frac {-8 x+2 e^{2 x} x^3+e^x \left (-4 x+4 x^3\right )+\left (8 x+6 e^{2 x} x^2+e^x \left (-4+4 x+8 x^2\right )\right ) \log (3 x)+\left (6 e^{2 x} x+e^x (4+4 x)\right ) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+\left (-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2\right ) \log (3 x)+\left (-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x\right ) \log ^2(3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^3(3 x)} \, dx=\log \left (x^{2} e^{4} - x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{x} + 2 \, x e^{4} \log \left (3\right ) - 2 \, x e^{\left (2 \, x\right )} \log \left (3\right ) - 4 \, x e^{x} \log \left (3\right ) + e^{4} \log \left (3\right )^{2} - e^{\left (2 \, x\right )} \log \left (3\right )^{2} + 2 \, x e^{4} \log \left (x\right ) - 2 \, x e^{\left (2 \, x\right )} \log \left (x\right ) - 4 \, x e^{x} \log \left (x\right ) + 2 \, e^{4} \log \left (3\right ) \log \left (x\right ) - 2 \, e^{\left (2 \, x\right )} \log \left (3\right ) \log \left (x\right ) + e^{4} \log \left (x\right )^{2} - e^{\left (2 \, x\right )} \log \left (x\right )^{2} + 8 \, x \log \left (3\right ) + 4 \, \log \left (3\right )^{2} + 8 \, x \log \left (x\right ) + 8 \, \log \left (3\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2}\right ) - 2 \, \log \left (x + \log \left (3\right ) + \log \left (x\right )\right ) + \log \left (e^{4} - e^{\left (2 \, x\right )} + 4\right ) - \log \left (-e^{4} + e^{\left (2 \, x\right )} - 4\right ) \] Input:

integrate((2*exp(x)^2*log(3*x)^3+(6*x*exp(x)^2+(4+4*x)*exp(x))*log(3*x)^2+ 
(6*exp(x)^2*x^2+(8*x^2+4*x-4)*exp(x)+8*x)*log(3*x)+2*exp(x)^2*x^3+(4*x^3-4 
*x)*exp(x)-8*x)/((exp(x)^2-exp(4)-4)*log(3*x)^3+(3*x*exp(x)^2+4*exp(x)*x-3 
*x*exp(4)-12*x)*log(3*x)^2+(3*exp(x)^2*x^2+8*exp(x)*x^2-3*x^2*exp(4)-8*x^2 
)*log(3*x)+exp(x)^2*x^3+4*exp(x)*x^3-x^3*exp(4)),x, algorithm="giac")
 

Output:

log(x^2*e^4 - x^2*e^(2*x) - 4*x^2*e^x + 2*x*e^4*log(3) - 2*x*e^(2*x)*log(3 
) - 4*x*e^x*log(3) + e^4*log(3)^2 - e^(2*x)*log(3)^2 + 2*x*e^4*log(x) - 2* 
x*e^(2*x)*log(x) - 4*x*e^x*log(x) + 2*e^4*log(3)*log(x) - 2*e^(2*x)*log(3) 
*log(x) + e^4*log(x)^2 - e^(2*x)*log(x)^2 + 8*x*log(3) + 4*log(3)^2 + 8*x* 
log(x) + 8*log(3)*log(x) + 4*log(x)^2) - 2*log(x + log(3) + log(x)) + log( 
e^4 - e^(2*x) + 4) - log(-e^4 + e^(2*x) - 4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-8 x+2 e^{2 x} x^3+e^x \left (-4 x+4 x^3\right )+\left (8 x+6 e^{2 x} x^2+e^x \left (-4+4 x+8 x^2\right )\right ) \log (3 x)+\left (6 e^{2 x} x+e^x (4+4 x)\right ) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+\left (-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2\right ) \log (3 x)+\left (-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x\right ) \log ^2(3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^3(3 x)} \, dx=\int \frac {2\,{\ln \left (3\,x\right )}^3\,{\mathrm {e}}^{2\,x}-8\,x+2\,x^3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (4\,x-4\,x^3\right )+{\ln \left (3\,x\right )}^2\,\left (6\,x\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (4\,x+4\right )\right )+\ln \left (3\,x\right )\,\left (8\,x+6\,x^2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (8\,x^2+4\,x-4\right )\right )}{4\,x^3\,{\mathrm {e}}^x-{\ln \left (3\,x\right )}^2\,\left (12\,x-3\,x\,{\mathrm {e}}^{2\,x}+3\,x\,{\mathrm {e}}^4-4\,x\,{\mathrm {e}}^x\right )+\ln \left (3\,x\right )\,\left (8\,x^2\,{\mathrm {e}}^x+3\,x^2\,{\mathrm {e}}^{2\,x}-3\,x^2\,{\mathrm {e}}^4-8\,x^2\right )-{\ln \left (3\,x\right )}^3\,\left ({\mathrm {e}}^4-{\mathrm {e}}^{2\,x}+4\right )+x^3\,{\mathrm {e}}^{2\,x}-x^3\,{\mathrm {e}}^4} \,d x \] Input:

int((2*log(3*x)^3*exp(2*x) - 8*x + 2*x^3*exp(2*x) - exp(x)*(4*x - 4*x^3) + 
 log(3*x)^2*(6*x*exp(2*x) + exp(x)*(4*x + 4)) + log(3*x)*(8*x + 6*x^2*exp( 
2*x) + exp(x)*(4*x + 8*x^2 - 4)))/(4*x^3*exp(x) - log(3*x)^2*(12*x - 3*x*e 
xp(2*x) + 3*x*exp(4) - 4*x*exp(x)) + log(3*x)*(8*x^2*exp(x) + 3*x^2*exp(2* 
x) - 3*x^2*exp(4) - 8*x^2) - log(3*x)^3*(exp(4) - exp(2*x) + 4) + x^3*exp( 
2*x) - x^3*exp(4)),x)
 

Output:

int((2*log(3*x)^3*exp(2*x) - 8*x + 2*x^3*exp(2*x) - exp(x)*(4*x - 4*x^3) + 
 log(3*x)^2*(6*x*exp(2*x) + exp(x)*(4*x + 4)) + log(3*x)*(8*x + 6*x^2*exp( 
2*x) + exp(x)*(4*x + 8*x^2 - 4)))/(4*x^3*exp(x) - log(3*x)^2*(12*x - 3*x*e 
xp(2*x) + 3*x*exp(4) - 4*x*exp(x)) + log(3*x)*(8*x^2*exp(x) + 3*x^2*exp(2* 
x) - 3*x^2*exp(4) - 8*x^2) - log(3*x)^3*(exp(4) - exp(2*x) + 4) + x^3*exp( 
2*x) - x^3*exp(4)), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.44 \[ \int \frac {-8 x+2 e^{2 x} x^3+e^x \left (-4 x+4 x^3\right )+\left (8 x+6 e^{2 x} x^2+e^x \left (-4+4 x+8 x^2\right )\right ) \log (3 x)+\left (6 e^{2 x} x+e^x (4+4 x)\right ) \log ^2(3 x)+2 e^{2 x} \log ^3(3 x)}{-e^4 x^3+4 e^x x^3+e^{2 x} x^3+\left (-8 x^2-3 e^4 x^2+8 e^x x^2+3 e^{2 x} x^2\right ) \log (3 x)+\left (-12 x-3 e^4 x+4 e^x x+3 e^{2 x} x\right ) \log ^2(3 x)+\left (-4-e^4+e^{2 x}\right ) \log ^3(3 x)} \, dx=-2 \,\mathrm {log}\left (\mathrm {log}\left (3 x \right )+x \right )+\mathrm {log}\left (-\sqrt {e^{4}+4}\, \mathrm {log}\left (3 x \right )-\sqrt {e^{4}+4}\, x +e^{x} \mathrm {log}\left (3 x \right )+e^{x} x +2 x \right )+\mathrm {log}\left (\sqrt {e^{4}+4}\, \mathrm {log}\left (3 x \right )+\sqrt {e^{4}+4}\, x +e^{x} \mathrm {log}\left (3 x \right )+e^{x} x +2 x \right ) \] Input:

int((2*exp(x)^2*log(3*x)^3+(6*x*exp(x)^2+(4+4*x)*exp(x))*log(3*x)^2+(6*exp 
(x)^2*x^2+(8*x^2+4*x-4)*exp(x)+8*x)*log(3*x)+2*exp(x)^2*x^3+(4*x^3-4*x)*ex 
p(x)-8*x)/((exp(x)^2-exp(4)-4)*log(3*x)^3+(3*x*exp(x)^2+4*exp(x)*x-3*x*exp 
(4)-12*x)*log(3*x)^2+(3*exp(x)^2*x^2+8*exp(x)*x^2-3*x^2*exp(4)-8*x^2)*log( 
3*x)+exp(x)^2*x^3+4*exp(x)*x^3-x^3*exp(4)),x)
 

Output:

 - 2*log(log(3*x) + x) + log( - sqrt(e**4 + 4)*log(3*x) - sqrt(e**4 + 4)*x 
 + e**x*log(3*x) + e**x*x + 2*x) + log(sqrt(e**4 + 4)*log(3*x) + sqrt(e**4 
 + 4)*x + e**x*log(3*x) + e**x*x + 2*x)