\(\int \frac {e^{-2 x} (4 e^{2 x} x^4 \log ^2(3)+e^x (4 x^3 \log (3)-32 x^2 \log ^2(3)) \log (x)+e^x ((-6 x^2+6 x^3-2 x^4) \log (3)+(-32 x^2+16 x^3) \log ^2(3)) \log ^2(x)+(4 x^2-64 x \log (3)+e^x (-4 x^2+2 x^3) \log (3)+256 \log ^2(3)) \log ^3(x)+(-10 x+2 x^2-2 x^3+(80-16 x+32 x^2) \log (3)-128 x \log ^2(3)) \log ^4(x)+(6-2 x+4 x^2-32 x \log (3)) \log ^5(x)-2 x \log ^6(x))}{x \log ^2(3)} \, dx\) [2778]

Optimal result

Integrand size = 200, antiderivative size = 31 \[ \int \frac {e^{-2 x} \left (4 e^{2 x} x^4 \log ^2(3)+e^x \left (4 x^3 \log (3)-32 x^2 \log ^2(3)\right ) \log (x)+e^x \left (\left (-6 x^2+6 x^3-2 x^4\right ) \log (3)+\left (-32 x^2+16 x^3\right ) \log ^2(3)\right ) \log ^2(x)+\left (4 x^2-64 x \log (3)+e^x \left (-4 x^2+2 x^3\right ) \log (3)+256 \log ^2(3)\right ) \log ^3(x)+\left (-10 x+2 x^2-2 x^3+\left (80-16 x+32 x^2\right ) \log (3)-128 x \log ^2(3)\right ) \log ^4(x)+\left (6-2 x+4 x^2-32 x \log (3)\right ) \log ^5(x)-2 x \log ^6(x)\right )}{x \log ^2(3)} \, dx=\left (-x^2+e^{-x} \log ^2(x) \left (8+\frac {-x+\log (x)}{\log (3)}\right )\right )^2 \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.43 (sec) , antiderivative size = 302, normalized size of antiderivative = 9.74 \[ \int \frac {e^{-2 x} \left (4 e^{2 x} x^4 \log ^2(3)+e^x \left (4 x^3 \log (3)-32 x^2 \log ^2(3)\right ) \log (x)+e^x \left (\left (-6 x^2+6 x^3-2 x^4\right ) \log (3)+\left (-32 x^2+16 x^3\right ) \log ^2(3)\right ) \log ^2(x)+\left (4 x^2-64 x \log (3)+e^x \left (-4 x^2+2 x^3\right ) \log (3)+256 \log ^2(3)\right ) \log ^3(x)+\left (-10 x+2 x^2-2 x^3+\left (80-16 x+32 x^2\right ) \log (3)-128 x \log ^2(3)\right ) \log ^4(x)+\left (6-2 x+4 x^2-32 x \log (3)\right ) \log ^5(x)-2 x \log ^6(x)\right )}{x \log ^2(3)} \, dx=\frac {e^{-2 x} \left (32 e^{2 x} \log ^2(3)-32 e^{2 x} \gamma \log ^2(3)+e^{2 x} x^4 \log ^2(3)-2 e^{2 x} x^2 \log (9)-2 e^x \Gamma (2,-x) \log (9)-2 e^x x \Gamma (2,-x) \log (9)-2 e^{2 x} \operatorname {ExpIntegralEi}(-x) \left (16 \log ^2(3)-\log (81)\right )-2 e^{2 x} \Gamma (0,x) \left (16 \log ^2(3)-\log (81)\right )-2 e^{2 x} \log (81)-2 e^{2 x} \gamma \log (531441)+2 e^{2 x} \gamma \log (43046721)-2 e^x \log (81) \log (x)-2 e^x x \log (81) \log (x)-2 e^x x \log (729) \log (x)+2 e^x x \log (59049) \log (x)-2 e^x \log (531441) \log (x)+2 e^x \log (43046721) \log (x)-16 e^x x^2 \log ^2(3) \log ^2(x)+e^x x^3 \log (9) \log ^2(x)-e^x x^2 \log (9) \log ^3(x)+x^2 \log ^4(x)+64 \log ^2(3) \log ^4(x)-2 x \log (6561) \log ^4(x)-2 x \log ^5(x)+2 \log (6561) \log ^5(x)+\log ^6(x)\right )}{\log ^2(3)} \] Input:

Integrate[(4*E^(2*x)*x^4*Log[3]^2 + E^x*(4*x^3*Log[3] - 32*x^2*Log[3]^2)*L 
og[x] + E^x*((-6*x^2 + 6*x^3 - 2*x^4)*Log[3] + (-32*x^2 + 16*x^3)*Log[3]^2 
)*Log[x]^2 + (4*x^2 - 64*x*Log[3] + E^x*(-4*x^2 + 2*x^3)*Log[3] + 256*Log[ 
3]^2)*Log[x]^3 + (-10*x + 2*x^2 - 2*x^3 + (80 - 16*x + 32*x^2)*Log[3] - 12 
8*x*Log[3]^2)*Log[x]^4 + (6 - 2*x + 4*x^2 - 32*x*Log[3])*Log[x]^5 - 2*x*Lo 
g[x]^6)/(E^(2*x)*x*Log[3]^2),x]
 

Output:

(32*E^(2*x)*Log[3]^2 - 32*E^(2*x)*EulerGamma*Log[3]^2 + E^(2*x)*x^4*Log[3] 
^2 - 2*E^(2*x)*x^2*Log[9] - 2*E^x*Gamma[2, -x]*Log[9] - 2*E^x*x*Gamma[2, - 
x]*Log[9] - 2*E^(2*x)*ExpIntegralEi[-x]*(16*Log[3]^2 - Log[81]) - 2*E^(2*x 
)*Gamma[0, x]*(16*Log[3]^2 - Log[81]) - 2*E^(2*x)*Log[81] - 2*E^(2*x)*Eule 
rGamma*Log[531441] + 2*E^(2*x)*EulerGamma*Log[43046721] - 2*E^x*Log[81]*Lo 
g[x] - 2*E^x*x*Log[81]*Log[x] - 2*E^x*x*Log[729]*Log[x] + 2*E^x*x*Log[5904 
9]*Log[x] - 2*E^x*Log[531441]*Log[x] + 2*E^x*Log[43046721]*Log[x] - 16*E^x 
*x^2*Log[3]^2*Log[x]^2 + E^x*x^3*Log[9]*Log[x]^2 - E^x*x^2*Log[9]*Log[x]^3 
 + x^2*Log[x]^4 + 64*Log[3]^2*Log[x]^4 - 2*x*Log[6561]*Log[x]^4 - 2*x*Log[ 
x]^5 + 2*Log[6561]*Log[x]^5 + Log[x]^6)/(E^(2*x)*Log[3]^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.

\(\displaystyle \int \frac {e^{-2 x} \left (4 e^{2 x} x^4 \log ^2(3)+\left (4 x^2-2 x-32 x \log (3)+6\right ) \log ^5(x)+e^x \left (4 x^3 \log (3)-32 x^2 \log ^2(3)\right ) \log (x)+\left (-2 x^3+2 x^2+\left (32 x^2-16 x+80\right ) \log (3)-10 x-128 x \log ^2(3)\right ) \log ^4(x)+\left (4 x^2+e^x \left (2 x^3-4 x^2\right ) \log (3)-64 x \log (3)+256 \log ^2(3)\right ) \log ^3(x)+e^x \left (\left (16 x^3-32 x^2\right ) \log ^2(3)+\left (-2 x^4+6 x^3-6 x^2\right ) \log (3)\right ) \log ^2(x)-2 x \log ^6(x)\right )}{x \log ^2(3)} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {2 e^{-2 x} \left (-x \log ^6(x)+\left (2 x^2-16 \log (3) x-x+3\right ) \log ^5(x)-\left (x^3-x^2+64 \log ^2(3) x+5 x-8 \left (2 x^2-x+5\right ) \log (3)\right ) \log ^4(x)+\left (2 x^2-32 \log (3) x+128 \log ^2(3)-e^x \left (2 x^2-x^3\right ) \log (3)\right ) \log ^3(x)-e^x \left (8 \log ^2(3) \left (2 x^2-x^3\right )+\left (x^4-3 x^3+3 x^2\right ) \log (3)\right ) \log ^2(x)+2 e^x \left (x^3 \log (3)-8 x^2 \log ^2(3)\right ) \log (x)+2 e^{2 x} x^4 \log ^2(3)\right )}{x}dx}{\log ^2(3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {e^{-2 x} \left (-x \log ^6(x)+\left (2 x^2-16 \log (3) x-x+3\right ) \log ^5(x)-\left (x^3-x^2+64 \log ^2(3) x+5 x-8 \left (2 x^2-x+5\right ) \log (3)\right ) \log ^4(x)+\left (2 x^2-32 \log (3) x+128 \log ^2(3)-e^x \left (2 x^2-x^3\right ) \log (3)\right ) \log ^3(x)-e^x \left (8 \log ^2(3) \left (2 x^2-x^3\right )+\left (x^4-3 x^3+3 x^2\right ) \log (3)\right ) \log ^2(x)+2 e^x \left (x^3 \log (3)-8 x^2 \log ^2(3)\right ) \log (x)+2 e^{2 x} x^4 \log ^2(3)\right )}{x}dx}{\log ^2(3)}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {2 \int \left (2 \log ^2(3) x^3+e^{-x} \log (3) \log (x) \left (-\log (x) x^2+\log ^2(x) x+3 \left (1+\frac {8 \log (3)}{3}\right ) \log (x) x+2 x-2 \log ^2(x)-3 \left (1+\frac {16 \log (3)}{3}\right ) \log (x)-16 \log (3)\right ) x+\frac {e^{-2 x} \log ^3(x) \left (-\log (x) x^3+2 \log ^2(x) x^2+(1+16 \log (3)) \log (x) x^2+2 x^2-\log ^3(x) x-(1+16 \log (3)) \log ^2(x) x-5 \left (1+\frac {8}{5} \log (3) (1+8 \log (3))\right ) \log (x) x-32 \log (3) x+3 \log ^2(x)+40 \log (3) \log (x)+128 \log ^2(3)\right )}{x}\right )dx}{\log ^2(3)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\log (3) \int e^{-x} x^3 \log ^2(x)dx-\int e^{-2 x} x^2 \log ^4(x)dx+\log (3) \int e^{-x} x^2 \log ^3(x)dx+\log (3) (3+8 \log (3)) \int e^{-x} x^2 \log ^2(x)dx-\int e^{-2 x} \log ^6(x)dx-(1+16 \log (3)) \int e^{-2 x} \log ^5(x)dx+3 \int \frac {e^{-2 x} \log ^5(x)}{x}dx+2 \int e^{-2 x} x \log ^5(x)dx-(5+8 \log (3) (1+8 \log (3))) \int e^{-2 x} \log ^4(x)dx+40 \log (3) \int \frac {e^{-2 x} \log ^4(x)}{x}dx+(1+16 \log (3)) \int e^{-2 x} x \log ^4(x)dx-32 \log (3) \int e^{-2 x} \log ^3(x)dx+2 \int e^{-2 x} x \log ^3(x)dx-2 \log (3) \int e^{-x} x \log ^3(x)dx-\log (3) (3+16 \log (3)) \int e^{-x} x \log ^2(x)dx+128 \log ^2(3) \int \frac {e^{-2 x} \log ^3(x)}{x}dx+4 \log (3) (1-\log (81)) \operatorname {ExpIntegralEi}(-x)+\frac {1}{2} x^4 \log ^2(3)-2 e^{-x} x^2 \log (3) \log (x)+16 e^{-x} x \log ^2(3) \log (x)+16 e^{-x} \log ^2(3) \log (x)-4 e^{-x} x \log (3) \log (x)-2 e^{-x} x \log (3)-4 e^{-x} \log (3) \log (x)-2 e^{-x} \log (3)-4 e^{-x} (1-4 \log (3)) \log (3)\right )}{\log ^2(3)}\)

Input:

Int[(4*E^(2*x)*x^4*Log[3]^2 + E^x*(4*x^3*Log[3] - 32*x^2*Log[3]^2)*Log[x] 
+ E^x*((-6*x^2 + 6*x^3 - 2*x^4)*Log[3] + (-32*x^2 + 16*x^3)*Log[3]^2)*Log[ 
x]^2 + (4*x^2 - 64*x*Log[3] + E^x*(-4*x^2 + 2*x^3)*Log[3] + 256*Log[3]^2)* 
Log[x]^3 + (-10*x + 2*x^2 - 2*x^3 + (80 - 16*x + 32*x^2)*Log[3] - 128*x*Lo 
g[3]^2)*Log[x]^4 + (6 - 2*x + 4*x^2 - 32*x*Log[3])*Log[x]^5 - 2*x*Log[x]^6 
)/(E^(2*x)*x*Log[3]^2),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs. \(2(30)=60\).

Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.55

\[\frac {{\mathrm e}^{-2 x} \ln \left (x \right )^{6}}{\ln \left (3\right )^{2}}+\frac {2 \left (8 \ln \left (3\right )-x \right ) {\mathrm e}^{-2 x} \ln \left (x \right )^{5}}{\ln \left (3\right )^{2}}+\frac {\left (64 \ln \left (3\right )^{2}-16 x \ln \left (3\right )+x^{2}\right ) {\mathrm e}^{-2 x} \ln \left (x \right )^{4}}{\ln \left (3\right )^{2}}-\frac {2 x^{2} {\mathrm e}^{-x} \ln \left (x \right )^{3}}{\ln \left (3\right )}-\frac {2 x^{2} \left (8 \ln \left (3\right )-x \right ) {\mathrm e}^{-x} \ln \left (x \right )^{2}}{\ln \left (3\right )}+x^{4}\]

Input:

int((-2*x*ln(x)^6+(-32*x*ln(3)+4*x^2-2*x+6)*ln(x)^5+(-128*x*ln(3)^2+(32*x^ 
2-16*x+80)*ln(3)-2*x^3+2*x^2-10*x)*ln(x)^4+((2*x^3-4*x^2)*ln(3)*exp(x)+256 
*ln(3)^2-64*x*ln(3)+4*x^2)*ln(x)^3+((16*x^3-32*x^2)*ln(3)^2+(-2*x^4+6*x^3- 
6*x^2)*ln(3))*exp(x)*ln(x)^2+(-32*x^2*ln(3)^2+4*x^3*ln(3))*exp(x)*ln(x)+4* 
x^4*ln(3)^2*exp(x)^2)/x/ln(3)^2/exp(x)^2,x)
 

Output:

1/ln(3)^2*exp(-2*x)*ln(x)^6+2/ln(3)^2*(8*ln(3)-x)*exp(-2*x)*ln(x)^5+1/ln(3 
)^2*(64*ln(3)^2-16*x*ln(3)+x^2)*exp(-2*x)*ln(x)^4-2/ln(3)*x^2*exp(-x)*ln(x 
)^3-2/ln(3)*x^2*(8*ln(3)-x)*exp(-x)*ln(x)^2+x^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (28) = 56\).

Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int \frac {e^{-2 x} \left (4 e^{2 x} x^4 \log ^2(3)+e^x \left (4 x^3 \log (3)-32 x^2 \log ^2(3)\right ) \log (x)+e^x \left (\left (-6 x^2+6 x^3-2 x^4\right ) \log (3)+\left (-32 x^2+16 x^3\right ) \log ^2(3)\right ) \log ^2(x)+\left (4 x^2-64 x \log (3)+e^x \left (-4 x^2+2 x^3\right ) \log (3)+256 \log ^2(3)\right ) \log ^3(x)+\left (-10 x+2 x^2-2 x^3+\left (80-16 x+32 x^2\right ) \log (3)-128 x \log ^2(3)\right ) \log ^4(x)+\left (6-2 x+4 x^2-32 x \log (3)\right ) \log ^5(x)-2 x \log ^6(x)\right )}{x \log ^2(3)} \, dx=\frac {{\left (x^{4} e^{\left (2 \, x\right )} \log \left (3\right )^{2} - 2 \, x^{2} e^{x} \log \left (3\right ) \log \left (x\right )^{3} - 2 \, {\left (x - 8 \, \log \left (3\right )\right )} \log \left (x\right )^{5} + \log \left (x\right )^{6} + {\left (x^{2} - 16 \, x \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right )} \log \left (x\right )^{4} + 2 \, {\left (x^{3} \log \left (3\right ) - 8 \, x^{2} \log \left (3\right )^{2}\right )} e^{x} \log \left (x\right )^{2}\right )} e^{\left (-2 \, x\right )}}{\log \left (3\right )^{2}} \] Input:

integrate((-2*x*log(x)^6+(-32*x*log(3)+4*x^2-2*x+6)*log(x)^5+(-128*x*log(3 
)^2+(32*x^2-16*x+80)*log(3)-2*x^3+2*x^2-10*x)*log(x)^4+((2*x^3-4*x^2)*log( 
3)*exp(x)+256*log(3)^2-64*x*log(3)+4*x^2)*log(x)^3+((16*x^3-32*x^2)*log(3) 
^2+(-2*x^4+6*x^3-6*x^2)*log(3))*exp(x)*log(x)^2+(-32*x^2*log(3)^2+4*x^3*lo 
g(3))*exp(x)*log(x)+4*x^4*log(3)^2*exp(x)^2)/x/log(3)^2/exp(x)^2,x, algori 
thm="fricas")
 

Output:

(x^4*e^(2*x)*log(3)^2 - 2*x^2*e^x*log(3)*log(x)^3 - 2*(x - 8*log(3))*log(x 
)^5 + log(x)^6 + (x^2 - 16*x*log(3) + 64*log(3)^2)*log(x)^4 + 2*(x^3*log(3 
) - 8*x^2*log(3)^2)*e^x*log(x)^2)*e^(-2*x)/log(3)^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (22) = 44\).

Time = 0.75 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.23 \[ \int \frac {e^{-2 x} \left (4 e^{2 x} x^4 \log ^2(3)+e^x \left (4 x^3 \log (3)-32 x^2 \log ^2(3)\right ) \log (x)+e^x \left (\left (-6 x^2+6 x^3-2 x^4\right ) \log (3)+\left (-32 x^2+16 x^3\right ) \log ^2(3)\right ) \log ^2(x)+\left (4 x^2-64 x \log (3)+e^x \left (-4 x^2+2 x^3\right ) \log (3)+256 \log ^2(3)\right ) \log ^3(x)+\left (-10 x+2 x^2-2 x^3+\left (80-16 x+32 x^2\right ) \log (3)-128 x \log ^2(3)\right ) \log ^4(x)+\left (6-2 x+4 x^2-32 x \log (3)\right ) \log ^5(x)-2 x \log ^6(x)\right )}{x \log ^2(3)} \, dx=x^{4} + \frac {\left (2 x^{3} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} - 2 x^{2} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{3} - 16 x^{2} \log {\left (3 \right )}^{3} \log {\left (x \right )}^{2}\right ) e^{- x} + \left (x^{2} \log {\left (3 \right )} \log {\left (x \right )}^{4} - 2 x \log {\left (3 \right )} \log {\left (x \right )}^{5} - 16 x \log {\left (3 \right )}^{2} \log {\left (x \right )}^{4} + \log {\left (3 \right )} \log {\left (x \right )}^{6} + 16 \log {\left (3 \right )}^{2} \log {\left (x \right )}^{5} + 64 \log {\left (3 \right )}^{3} \log {\left (x \right )}^{4}\right ) e^{- 2 x}}{\log {\left (3 \right )}^{3}} \] Input:

integrate((-2*x*ln(x)**6+(-32*x*ln(3)+4*x**2-2*x+6)*ln(x)**5+(-128*x*ln(3) 
**2+(32*x**2-16*x+80)*ln(3)-2*x**3+2*x**2-10*x)*ln(x)**4+((2*x**3-4*x**2)* 
ln(3)*exp(x)+256*ln(3)**2-64*x*ln(3)+4*x**2)*ln(x)**3+((16*x**3-32*x**2)*l 
n(3)**2+(-2*x**4+6*x**3-6*x**2)*ln(3))*exp(x)*ln(x)**2+(-32*x**2*ln(3)**2+ 
4*x**3*ln(3))*exp(x)*ln(x)+4*x**4*ln(3)**2*exp(x)**2)/x/ln(3)**2/exp(x)**2 
,x)
 

Output:

x**4 + ((2*x**3*log(3)**2*log(x)**2 - 2*x**2*log(3)**2*log(x)**3 - 16*x**2 
*log(3)**3*log(x)**2)*exp(-x) + (x**2*log(3)*log(x)**4 - 2*x*log(3)*log(x) 
**5 - 16*x*log(3)**2*log(x)**4 + log(3)*log(x)**6 + 16*log(3)**2*log(x)**5 
 + 64*log(3)**3*log(x)**4)*exp(-2*x))/log(3)**3
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (28) = 56\).

Time = 0.17 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.19 \[ \int \frac {e^{-2 x} \left (4 e^{2 x} x^4 \log ^2(3)+e^x \left (4 x^3 \log (3)-32 x^2 \log ^2(3)\right ) \log (x)+e^x \left (\left (-6 x^2+6 x^3-2 x^4\right ) \log (3)+\left (-32 x^2+16 x^3\right ) \log ^2(3)\right ) \log ^2(x)+\left (4 x^2-64 x \log (3)+e^x \left (-4 x^2+2 x^3\right ) \log (3)+256 \log ^2(3)\right ) \log ^3(x)+\left (-10 x+2 x^2-2 x^3+\left (80-16 x+32 x^2\right ) \log (3)-128 x \log ^2(3)\right ) \log ^4(x)+\left (6-2 x+4 x^2-32 x \log (3)\right ) \log ^5(x)-2 x \log ^6(x)\right )}{x \log ^2(3)} \, dx=\frac {x^{4} \log \left (3\right )^{2} - 2 \, {\left (x^{2} \log \left (3\right ) \log \left (x\right )^{3} - {\left (x^{3} \log \left (3\right ) - 8 \, x^{2} \log \left (3\right )^{2}\right )} \log \left (x\right )^{2}\right )} e^{\left (-x\right )} - {\left (2 \, {\left (x - 8 \, \log \left (3\right )\right )} \log \left (x\right )^{5} - \log \left (x\right )^{6} - {\left (x^{2} - 16 \, x \log \left (3\right ) + 64 \, \log \left (3\right )^{2}\right )} \log \left (x\right )^{4}\right )} e^{\left (-2 \, x\right )}}{\log \left (3\right )^{2}} \] Input:

integrate((-2*x*log(x)^6+(-32*x*log(3)+4*x^2-2*x+6)*log(x)^5+(-128*x*log(3 
)^2+(32*x^2-16*x+80)*log(3)-2*x^3+2*x^2-10*x)*log(x)^4+((2*x^3-4*x^2)*log( 
3)*exp(x)+256*log(3)^2-64*x*log(3)+4*x^2)*log(x)^3+((16*x^3-32*x^2)*log(3) 
^2+(-2*x^4+6*x^3-6*x^2)*log(3))*exp(x)*log(x)^2+(-32*x^2*log(3)^2+4*x^3*lo 
g(3))*exp(x)*log(x)+4*x^4*log(3)^2*exp(x)^2)/x/log(3)^2/exp(x)^2,x, algori 
thm="maxima")
 

Output:

(x^4*log(3)^2 - 2*(x^2*log(3)*log(x)^3 - (x^3*log(3) - 8*x^2*log(3)^2)*log 
(x)^2)*e^(-x) - (2*(x - 8*log(3))*log(x)^5 - log(x)^6 - (x^2 - 16*x*log(3) 
 + 64*log(3)^2)*log(x)^4)*e^(-2*x))/log(3)^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (28) = 56\).

Time = 0.17 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52 \[ \int \frac {e^{-2 x} \left (4 e^{2 x} x^4 \log ^2(3)+e^x \left (4 x^3 \log (3)-32 x^2 \log ^2(3)\right ) \log (x)+e^x \left (\left (-6 x^2+6 x^3-2 x^4\right ) \log (3)+\left (-32 x^2+16 x^3\right ) \log ^2(3)\right ) \log ^2(x)+\left (4 x^2-64 x \log (3)+e^x \left (-4 x^2+2 x^3\right ) \log (3)+256 \log ^2(3)\right ) \log ^3(x)+\left (-10 x+2 x^2-2 x^3+\left (80-16 x+32 x^2\right ) \log (3)-128 x \log ^2(3)\right ) \log ^4(x)+\left (6-2 x+4 x^2-32 x \log (3)\right ) \log ^5(x)-2 x \log ^6(x)\right )}{x \log ^2(3)} \, dx=\frac {{\left (x^{4} e^{\left (2 \, x\right )} \log \left (3\right )^{2} + 2 \, x^{3} e^{x} \log \left (3\right ) \log \left (x\right )^{2} - 16 \, x^{2} e^{x} \log \left (3\right )^{2} \log \left (x\right )^{2} - 2 \, x^{2} e^{x} \log \left (3\right ) \log \left (x\right )^{3} + x^{2} \log \left (x\right )^{4} - 16 \, x \log \left (3\right ) \log \left (x\right )^{4} + 64 \, \log \left (3\right )^{2} \log \left (x\right )^{4} - 2 \, x \log \left (x\right )^{5} + 16 \, \log \left (3\right ) \log \left (x\right )^{5} + \log \left (x\right )^{6}\right )} e^{\left (-2 \, x\right )}}{\log \left (3\right )^{2}} \] Input:

integrate((-2*x*log(x)^6+(-32*x*log(3)+4*x^2-2*x+6)*log(x)^5+(-128*x*log(3 
)^2+(32*x^2-16*x+80)*log(3)-2*x^3+2*x^2-10*x)*log(x)^4+((2*x^3-4*x^2)*log( 
3)*exp(x)+256*log(3)^2-64*x*log(3)+4*x^2)*log(x)^3+((16*x^3-32*x^2)*log(3) 
^2+(-2*x^4+6*x^3-6*x^2)*log(3))*exp(x)*log(x)^2+(-32*x^2*log(3)^2+4*x^3*lo 
g(3))*exp(x)*log(x)+4*x^4*log(3)^2*exp(x)^2)/x/log(3)^2/exp(x)^2,x, algori 
thm="giac")
 

Output:

(x^4*e^(2*x)*log(3)^2 + 2*x^3*e^x*log(3)*log(x)^2 - 16*x^2*e^x*log(3)^2*lo 
g(x)^2 - 2*x^2*e^x*log(3)*log(x)^3 + x^2*log(x)^4 - 16*x*log(3)*log(x)^4 + 
 64*log(3)^2*log(x)^4 - 2*x*log(x)^5 + 16*log(3)*log(x)^5 + log(x)^6)*e^(- 
2*x)/log(3)^2
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-2 x} \left (4 e^{2 x} x^4 \log ^2(3)+e^x \left (4 x^3 \log (3)-32 x^2 \log ^2(3)\right ) \log (x)+e^x \left (\left (-6 x^2+6 x^3-2 x^4\right ) \log (3)+\left (-32 x^2+16 x^3\right ) \log ^2(3)\right ) \log ^2(x)+\left (4 x^2-64 x \log (3)+e^x \left (-4 x^2+2 x^3\right ) \log (3)+256 \log ^2(3)\right ) \log ^3(x)+\left (-10 x+2 x^2-2 x^3+\left (80-16 x+32 x^2\right ) \log (3)-128 x \log ^2(3)\right ) \log ^4(x)+\left (6-2 x+4 x^2-32 x \log (3)\right ) \log ^5(x)-2 x \log ^6(x)\right )}{x \log ^2(3)} \, dx=\int -\frac {{\mathrm {e}}^{-2\,x}\,\left (2\,x\,{\ln \left (x\right )}^6+{\ln \left (x\right )}^3\,\left (64\,x\,\ln \left (3\right )-256\,{\ln \left (3\right )}^2-4\,x^2+{\mathrm {e}}^x\,\ln \left (3\right )\,\left (4\,x^2-2\,x^3\right )\right )+{\ln \left (x\right )}^4\,\left (10\,x-\ln \left (3\right )\,\left (32\,x^2-16\,x+80\right )+128\,x\,{\ln \left (3\right )}^2-2\,x^2+2\,x^3\right )+{\ln \left (x\right )}^5\,\left (2\,x+32\,x\,\ln \left (3\right )-4\,x^2-6\right )+{\mathrm {e}}^x\,\ln \left (x\right )\,\left (32\,x^2\,{\ln \left (3\right )}^2-4\,x^3\,\ln \left (3\right )\right )-4\,x^4\,{\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2+{\mathrm {e}}^x\,{\ln \left (x\right )}^2\,\left (\ln \left (3\right )\,\left (2\,x^4-6\,x^3+6\,x^2\right )+{\ln \left (3\right )}^2\,\left (32\,x^2-16\,x^3\right )\right )\right )}{x\,{\ln \left (3\right )}^2} \,d x \] Input:

int(-(exp(-2*x)*(2*x*log(x)^6 + log(x)^3*(64*x*log(3) - 256*log(3)^2 - 4*x 
^2 + exp(x)*log(3)*(4*x^2 - 2*x^3)) + log(x)^4*(10*x - log(3)*(32*x^2 - 16 
*x + 80) + 128*x*log(3)^2 - 2*x^2 + 2*x^3) + log(x)^5*(2*x + 32*x*log(3) - 
 4*x^2 - 6) + exp(x)*log(x)*(32*x^2*log(3)^2 - 4*x^3*log(3)) - 4*x^4*exp(2 
*x)*log(3)^2 + exp(x)*log(x)^2*(log(3)*(6*x^2 - 6*x^3 + 2*x^4) + log(3)^2* 
(32*x^2 - 16*x^3))))/(x*log(3)^2),x)
 

Output:

int(-(exp(-2*x)*(2*x*log(x)^6 + log(x)^3*(64*x*log(3) - 256*log(3)^2 - 4*x 
^2 + exp(x)*log(3)*(4*x^2 - 2*x^3)) + log(x)^4*(10*x - log(3)*(32*x^2 - 16 
*x + 80) + 128*x*log(3)^2 - 2*x^2 + 2*x^3) + log(x)^5*(2*x + 32*x*log(3) - 
 4*x^2 - 6) + exp(x)*log(x)*(32*x^2*log(3)^2 - 4*x^3*log(3)) - 4*x^4*exp(2 
*x)*log(3)^2 + exp(x)*log(x)^2*(log(3)*(6*x^2 - 6*x^3 + 2*x^4) + log(3)^2* 
(32*x^2 - 16*x^3))))/(x*log(3)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.74 \[ \int \frac {e^{-2 x} \left (4 e^{2 x} x^4 \log ^2(3)+e^x \left (4 x^3 \log (3)-32 x^2 \log ^2(3)\right ) \log (x)+e^x \left (\left (-6 x^2+6 x^3-2 x^4\right ) \log (3)+\left (-32 x^2+16 x^3\right ) \log ^2(3)\right ) \log ^2(x)+\left (4 x^2-64 x \log (3)+e^x \left (-4 x^2+2 x^3\right ) \log (3)+256 \log ^2(3)\right ) \log ^3(x)+\left (-10 x+2 x^2-2 x^3+\left (80-16 x+32 x^2\right ) \log (3)-128 x \log ^2(3)\right ) \log ^4(x)+\left (6-2 x+4 x^2-32 x \log (3)\right ) \log ^5(x)-2 x \log ^6(x)\right )}{x \log ^2(3)} \, dx=\frac {e^{2 x} \mathrm {log}\left (3\right )^{2} x^{4}-2 e^{x} \mathrm {log}\left (x \right )^{3} \mathrm {log}\left (3\right ) x^{2}-16 e^{x} \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (3\right )^{2} x^{2}+2 e^{x} \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (3\right ) x^{3}+\mathrm {log}\left (x \right )^{6}+16 \mathrm {log}\left (x \right )^{5} \mathrm {log}\left (3\right )-2 \mathrm {log}\left (x \right )^{5} x +64 \mathrm {log}\left (x \right )^{4} \mathrm {log}\left (3\right )^{2}-16 \mathrm {log}\left (x \right )^{4} \mathrm {log}\left (3\right ) x +\mathrm {log}\left (x \right )^{4} x^{2}}{e^{2 x} \mathrm {log}\left (3\right )^{2}} \] Input:

int((-2*x*log(x)^6+(-32*x*log(3)+4*x^2-2*x+6)*log(x)^5+(-128*x*log(3)^2+(3 
2*x^2-16*x+80)*log(3)-2*x^3+2*x^2-10*x)*log(x)^4+((2*x^3-4*x^2)*log(3)*exp 
(x)+256*log(3)^2-64*x*log(3)+4*x^2)*log(x)^3+((16*x^3-32*x^2)*log(3)^2+(-2 
*x^4+6*x^3-6*x^2)*log(3))*exp(x)*log(x)^2+(-32*x^2*log(3)^2+4*x^3*log(3))* 
exp(x)*log(x)+4*x^4*log(3)^2*exp(x)^2)/x/log(3)^2/exp(x)^2,x)
 

Output:

(e**(2*x)*log(3)**2*x**4 - 2*e**x*log(x)**3*log(3)*x**2 - 16*e**x*log(x)** 
2*log(3)**2*x**2 + 2*e**x*log(x)**2*log(3)*x**3 + log(x)**6 + 16*log(x)**5 
*log(3) - 2*log(x)**5*x + 64*log(x)**4*log(3)**2 - 16*log(x)**4*log(3)*x + 
 log(x)**4*x**2)/(e**(2*x)*log(3)**2)