\(\int \frac {(e^{2 x} (-24+8 x) \log ^2(3)+e^x (96 x-32 x^2+48 x^4-16 x^5) \log ^2(3)+(-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9) \log ^2(3)) \log (x)+(e^{2 x} (-32 x+8 x^2) \log ^2(3)+e^x (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6) \log ^2(3)+(-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9) \log ^2(3)) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx\) [2134]

Optimal result

Integrand size = 196, antiderivative size = 29 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\frac {4 \left (-e^x+2 x+x^4\right )^2 \log ^2(3) \log ^2(x)}{(-3+x)^2} \] Output:

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

Mathematica [A] (verified)

Time = 5.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\frac {4 \left (-e^x+2 x+x^4\right )^2 \log ^2(3) \log ^2(x)}{(-3+x)^2} \] Input:

Integrate[((E^(2*x)*(-24 + 8*x)*Log[3]^2 + E^x*(96*x - 32*x^2 + 48*x^4 - 1 
6*x^5)*Log[3]^2 + (-96*x^2 + 32*x^3 - 96*x^5 + 32*x^6 - 24*x^8 + 8*x^9)*Lo 
g[3]^2)*Log[x] + (E^(2*x)*(-32*x + 8*x^2)*Log[3]^2 + E^x*(48*x + 64*x^2 - 
16*x^3 + 96*x^4 + 8*x^5 - 8*x^6)*Log[3]^2 + (-96*x^2 - 240*x^5 + 48*x^6 - 
96*x^8 + 24*x^9)*Log[3]^2)*Log[x]^2)/(-27*x + 27*x^2 - 9*x^3 + x^4),x]
 

Output:

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

\(\displaystyle \int \frac {\left (e^{2 x} \left (8 x^2-32 x\right ) \log ^2(3)+\left (24 x^9-96 x^8+48 x^6-240 x^5-96 x^2\right ) \log ^2(3)+e^x \left (-8 x^6+8 x^5+96 x^4-16 x^3+64 x^2+48 x\right ) \log ^2(3)\right ) \log ^2(x)+\left (e^x \left (-16 x^5+48 x^4-32 x^2+96 x\right ) \log ^2(3)+\left (8 x^9-24 x^8+32 x^6-96 x^5+32 x^3-96 x^2\right ) \log ^2(3)+e^{2 x} (8 x-24) \log ^2(3)\right ) \log (x)}{x^4-9 x^3+27 x^2-27 x} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (e^{2 x} \left (8 x^2-32 x\right ) \log ^2(3)+\left (24 x^9-96 x^8+48 x^6-240 x^5-96 x^2\right ) \log ^2(3)+e^x \left (-8 x^6+8 x^5+96 x^4-16 x^3+64 x^2+48 x\right ) \log ^2(3)\right ) \log ^2(x)+\left (e^x \left (-16 x^5+48 x^4-32 x^2+96 x\right ) \log ^2(3)+\left (8 x^9-24 x^8+32 x^6-96 x^5+32 x^3-96 x^2\right ) \log ^2(3)+e^{2 x} (8 x-24) \log ^2(3)\right ) \log (x)}{x \left (x^3-9 x^2+27 x-27\right )}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {\left (e^{2 x} \left (8 x^2-32 x\right ) \log ^2(3)+\left (24 x^9-96 x^8+48 x^6-240 x^5-96 x^2\right ) \log ^2(3)+e^x \left (-8 x^6+8 x^5+96 x^4-16 x^3+64 x^2+48 x\right ) \log ^2(3)\right ) \log ^2(x)+\left (e^x \left (-16 x^5+48 x^4-32 x^2+96 x\right ) \log ^2(3)+\left (8 x^9-24 x^8+32 x^6-96 x^5+32 x^3-96 x^2\right ) \log ^2(3)+e^{2 x} (8 x-24) \log ^2(3)\right ) \log (x)}{(x-3)^3 x}dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

\(\displaystyle 8 \log ^2(3) \int \left (\frac {3 \left (x^3+2\right ) \log ^2(x) x^5}{(x-3)^3}-\frac {12 \left (x^3+2\right ) \log ^2(x) x^4}{(x-3)^3}-\frac {6 \left (x^3+2\right ) \log ^2(x) x}{(x-3)^3}+\frac {\left (x^3+2\right )^2 \log (x) x}{(x-3)^2}-\frac {e^x \log (x) \left (\log (x) x^5-\log (x) x^4+2 x^4-12 \log (x) x^3-6 x^3+2 \log (x) x^2-8 \log (x) x+4 x-6 \log (x)-12\right )}{(x-3)^3}+\frac {e^{2 x} \log (x) \left (\log (x) x^2-4 \log (x) x+x-3\right )}{(x-3)^3 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 8 \log ^2(3) \left (112 e^3 \int \frac {\operatorname {ExpIntegralEi}(x-3)}{x}dx-\frac {5}{9} e^6 \int \frac {\operatorname {ExpIntegralEi}(2 x-6)}{x}dx-\int e^x x^2 \log ^2(x)dx-33 \int e^x \log ^2(x)dx+174 \int \frac {e^x \log ^2(x)}{(x-3)^3}dx-\int \frac {e^{2 x} \log ^2(x)}{(x-3)^3}dx+23 \int \frac {e^x \log ^2(x)}{(x-3)^2}dx+\int \frac {e^{2 x} \log ^2(x)}{(x-3)^2}dx-110 \int \frac {e^x \log ^2(x)}{x-3}dx-8 \int e^x x \log ^2(x)dx-\frac {2}{9} x \, _3F_3(1,1,1;2,2,2;2 x)-\frac {1}{9} \log (x) (\operatorname {ExpIntegralE}(1,-2 x)+\operatorname {ExpIntegralEi}(2 x))+\frac {1}{9} e^6 \operatorname {ExpIntegralEi}(-2 (3-x))-\frac {58}{3} e^3 \operatorname {ExpIntegralEi}(x-3)+\frac {88 \operatorname {ExpIntegralEi}(x)}{3}-\frac {\operatorname {ExpIntegralEi}(2 x)}{9}+\frac {5}{9} e^6 \operatorname {ExpIntegralEi}(-2 (3-x)) \log (x)-112 e^3 \operatorname {ExpIntegralEi}(x-3) \log (x)+\frac {1}{9} \operatorname {ExpIntegralEi}(2 x) \log (x)+841 \operatorname {PolyLog}\left (2,1-\frac {x}{3}\right )-841 \operatorname {PolyLog}\left (2,\frac {3}{x}\right )+\frac {1}{2} x^6 \log ^2(x)+3 x^5 \log ^2(x)+\frac {27}{2} x^4 \log ^2(x)+56 x^3 \log ^2(x)+\frac {429}{2} x^2 \log ^2(x)+2 e^x-\frac {3190 x \log ^2(x)}{3-x}+783 x \log ^2(x)-\frac {1}{18} \log ^2(-2 x)+\frac {7569 \log ^2(x)}{2 (3-x)^2}-2 e^x x \log (x)-841 \log (3) \log (x-3)-10 e^x \log (x)+841 \log \left (1-\frac {3}{x}\right ) \log (x)-\frac {58 e^x \log (x)}{3-x}+\frac {e^{2 x} \log (x)}{3 (3-x)}-\frac {1}{9} \gamma \log (x)\right )\)

Input:

Int[((E^(2*x)*(-24 + 8*x)*Log[3]^2 + E^x*(96*x - 32*x^2 + 48*x^4 - 16*x^5) 
*Log[3]^2 + (-96*x^2 + 32*x^3 - 96*x^5 + 32*x^6 - 24*x^8 + 8*x^9)*Log[3]^2 
)*Log[x] + (E^(2*x)*(-32*x + 8*x^2)*Log[3]^2 + E^x*(48*x + 64*x^2 - 16*x^3 
 + 96*x^4 + 8*x^5 - 8*x^6)*Log[3]^2 + (-96*x^2 - 240*x^5 + 48*x^6 - 96*x^8 
 + 24*x^9)*Log[3]^2)*Log[x]^2)/(-27*x + 27*x^2 - 9*x^3 + x^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59

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

Input:

int((((8*x^2-32*x)*ln(3)^2*exp(x)^2+(-8*x^6+8*x^5+96*x^4-16*x^3+64*x^2+48* 
x)*ln(3)^2*exp(x)+(24*x^9-96*x^8+48*x^6-240*x^5-96*x^2)*ln(3)^2)*ln(x)^2+( 
(8*x-24)*ln(3)^2*exp(x)^2+(-16*x^5+48*x^4-32*x^2+96*x)*ln(3)^2*exp(x)+(8*x 
^9-24*x^8+32*x^6-96*x^5+32*x^3-96*x^2)*ln(3)^2)*ln(x))/(x^4-9*x^3+27*x^2-2 
7*x),x)
 

Output:

4*(x^8+4*x^5-2*exp(x)*x^4+4*x^2-4*exp(x)*x+exp(2*x))*ln(3)^2/(-3+x)^2*ln(x 
)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (28) = 56\).

Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=-\frac {4 \, {\left (2 \, {\left (x^{4} + 2 \, x\right )} e^{x} \log \left (3\right )^{2} - {\left (x^{8} + 4 \, x^{5} + 4 \, x^{2}\right )} \log \left (3\right )^{2} - e^{\left (2 \, x\right )} \log \left (3\right )^{2}\right )} \log \left (x\right )^{2}}{x^{2} - 6 \, x + 9} \] Input:

integrate((((8*x^2-32*x)*log(3)^2*exp(x)^2+(-8*x^6+8*x^5+96*x^4-16*x^3+64* 
x^2+48*x)*log(3)^2*exp(x)+(24*x^9-96*x^8+48*x^6-240*x^5-96*x^2)*log(3)^2)* 
log(x)^2+((8*x-24)*log(3)^2*exp(x)^2+(-16*x^5+48*x^4-32*x^2+96*x)*log(3)^2 
*exp(x)+(8*x^9-24*x^8+32*x^6-96*x^5+32*x^3-96*x^2)*log(3)^2)*log(x))/(x^4- 
9*x^3+27*x^2-27*x),x, algorithm="fricas")
 

Output:

-4*(2*(x^4 + 2*x)*e^x*log(3)^2 - (x^8 + 4*x^5 + 4*x^2)*log(3)^2 - e^(2*x)* 
log(3)^2)*log(x)^2/(x^2 - 6*x + 9)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (27) = 54\).

Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 6.93 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\frac {\left (4 x^{2} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} - 24 x \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} + 36 \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2}\right ) e^{2 x} + \left (- 8 x^{6} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} + 48 x^{5} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} - 72 x^{4} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} - 16 x^{3} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} + 96 x^{2} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} - 144 x \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2}\right ) e^{x}}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81} + \frac {\left (4 x^{8} \log {\left (3 \right )}^{2} + 16 x^{5} \log {\left (3 \right )}^{2} + 16 x^{2} \log {\left (3 \right )}^{2}\right ) \log {\left (x \right )}^{2}}{x^{2} - 6 x + 9} \] Input:

integrate((((8*x**2-32*x)*ln(3)**2*exp(x)**2+(-8*x**6+8*x**5+96*x**4-16*x* 
*3+64*x**2+48*x)*ln(3)**2*exp(x)+(24*x**9-96*x**8+48*x**6-240*x**5-96*x**2 
)*ln(3)**2)*ln(x)**2+((8*x-24)*ln(3)**2*exp(x)**2+(-16*x**5+48*x**4-32*x** 
2+96*x)*ln(3)**2*exp(x)+(8*x**9-24*x**8+32*x**6-96*x**5+32*x**3-96*x**2)*l 
n(3)**2)*ln(x))/(x**4-9*x**3+27*x**2-27*x),x)
 

Output:

((4*x**2*log(3)**2*log(x)**2 - 24*x*log(3)**2*log(x)**2 + 36*log(3)**2*log 
(x)**2)*exp(2*x) + (-8*x**6*log(3)**2*log(x)**2 + 48*x**5*log(3)**2*log(x) 
**2 - 72*x**4*log(3)**2*log(x)**2 - 16*x**3*log(3)**2*log(x)**2 + 96*x**2* 
log(3)**2*log(x)**2 - 144*x*log(3)**2*log(x)**2)*exp(x))/(x**4 - 12*x**3 + 
 54*x**2 - 108*x + 81) + (4*x**8*log(3)**2 + 16*x**5*log(3)**2 + 16*x**2*l 
og(3)**2)*log(x)**2/(x**2 - 6*x + 9)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (28) = 56\).

Time = 0.18 (sec) , antiderivative size = 160, normalized size of antiderivative = 5.52 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=16 \, {\left (\frac {3 \, {\left (2 \, x - 3\right )} \log \left (x\right )}{x^{2} - 6 \, x + 9} + \frac {3}{x - 3} - \log \left (x - 3\right ) + \log \left (x\right )\right )} \log \left (3\right )^{2} + 16 \, \log \left (3\right )^{2} \log \left (x - 3\right ) - \frac {4 \, {\left (4 \, x^{2} \log \left (3\right )^{2} \log \left (x\right ) - e^{\left (2 \, x\right )} \log \left (3\right )^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{4} \log \left (3\right )^{2} + 2 \, x \log \left (3\right )^{2}\right )} e^{x} \log \left (x\right )^{2} + 12 \, x \log \left (3\right )^{2} - {\left (x^{8} \log \left (3\right )^{2} + 4 \, x^{5} \log \left (3\right )^{2} + 4 \, x^{2} \log \left (3\right )^{2}\right )} \log \left (x\right )^{2} - 36 \, \log \left (3\right )^{2}\right )}}{x^{2} - 6 \, x + 9} \] Input:

integrate((((8*x^2-32*x)*log(3)^2*exp(x)^2+(-8*x^6+8*x^5+96*x^4-16*x^3+64* 
x^2+48*x)*log(3)^2*exp(x)+(24*x^9-96*x^8+48*x^6-240*x^5-96*x^2)*log(3)^2)* 
log(x)^2+((8*x-24)*log(3)^2*exp(x)^2+(-16*x^5+48*x^4-32*x^2+96*x)*log(3)^2 
*exp(x)+(8*x^9-24*x^8+32*x^6-96*x^5+32*x^3-96*x^2)*log(3)^2)*log(x))/(x^4- 
9*x^3+27*x^2-27*x),x, algorithm="maxima")
 

Output:

16*(3*(2*x - 3)*log(x)/(x^2 - 6*x + 9) + 3/(x - 3) - log(x - 3) + log(x))* 
log(3)^2 + 16*log(3)^2*log(x - 3) - 4*(4*x^2*log(3)^2*log(x) - e^(2*x)*log 
(3)^2*log(x)^2 + 2*(x^4*log(3)^2 + 2*x*log(3)^2)*e^x*log(x)^2 + 12*x*log(3 
)^2 - (x^8*log(3)^2 + 4*x^5*log(3)^2 + 4*x^2*log(3)^2)*log(x)^2 - 36*log(3 
)^2)/(x^2 - 6*x + 9)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (28) = 56\).

Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.17 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\frac {4 \, {\left (x^{8} \log \left (3\right )^{2} \log \left (x\right )^{2} + 4 \, x^{5} \log \left (3\right )^{2} \log \left (x\right )^{2} - 2 \, x^{4} e^{x} \log \left (3\right )^{2} \log \left (x\right )^{2} + 4 \, x^{2} \log \left (3\right )^{2} \log \left (x\right )^{2} - 4 \, x e^{x} \log \left (3\right )^{2} \log \left (x\right )^{2} + e^{\left (2 \, x\right )} \log \left (3\right )^{2} \log \left (x\right )^{2}\right )}}{x^{2} - 6 \, x + 9} \] Input:

integrate((((8*x^2-32*x)*log(3)^2*exp(x)^2+(-8*x^6+8*x^5+96*x^4-16*x^3+64* 
x^2+48*x)*log(3)^2*exp(x)+(24*x^9-96*x^8+48*x^6-240*x^5-96*x^2)*log(3)^2)* 
log(x)^2+((8*x-24)*log(3)^2*exp(x)^2+(-16*x^5+48*x^4-32*x^2+96*x)*log(3)^2 
*exp(x)+(8*x^9-24*x^8+32*x^6-96*x^5+32*x^3-96*x^2)*log(3)^2)*log(x))/(x^4- 
9*x^3+27*x^2-27*x),x, algorithm="giac")
 

Output:

4*(x^8*log(3)^2*log(x)^2 + 4*x^5*log(3)^2*log(x)^2 - 2*x^4*e^x*log(3)^2*lo 
g(x)^2 + 4*x^2*log(3)^2*log(x)^2 - 4*x*e^x*log(3)^2*log(x)^2 + e^(2*x)*log 
(3)^2*log(x)^2)/(x^2 - 6*x + 9)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\int \frac {{\ln \left (x\right )}^2\,\left ({\ln \left (3\right )}^2\,\left (-24\,x^9+96\,x^8-48\,x^6+240\,x^5+96\,x^2\right )-{\mathrm {e}}^x\,{\ln \left (3\right )}^2\,\left (-8\,x^6+8\,x^5+96\,x^4-16\,x^3+64\,x^2+48\,x\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (32\,x-8\,x^2\right )\right )-\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (8\,x-24\right )-{\ln \left (3\right )}^2\,\left (-8\,x^9+24\,x^8-32\,x^6+96\,x^5-32\,x^3+96\,x^2\right )+{\mathrm {e}}^x\,{\ln \left (3\right )}^2\,\left (-16\,x^5+48\,x^4-32\,x^2+96\,x\right )\right )}{-x^4+9\,x^3-27\,x^2+27\,x} \,d x \] Input:

int((log(x)^2*(log(3)^2*(96*x^2 + 240*x^5 - 48*x^6 + 96*x^8 - 24*x^9) - ex 
p(x)*log(3)^2*(48*x + 64*x^2 - 16*x^3 + 96*x^4 + 8*x^5 - 8*x^6) + exp(2*x) 
*log(3)^2*(32*x - 8*x^2)) - log(x)*(exp(2*x)*log(3)^2*(8*x - 24) - log(3)^ 
2*(96*x^2 - 32*x^3 + 96*x^5 - 32*x^6 + 24*x^8 - 8*x^9) + exp(x)*log(3)^2*( 
96*x - 32*x^2 + 48*x^4 - 16*x^5)))/(27*x - 27*x^2 + 9*x^3 - x^4),x)
 

Output:

int((log(x)^2*(log(3)^2*(96*x^2 + 240*x^5 - 48*x^6 + 96*x^8 - 24*x^9) - ex 
p(x)*log(3)^2*(48*x + 64*x^2 - 16*x^3 + 96*x^4 + 8*x^5 - 8*x^6) + exp(2*x) 
*log(3)^2*(32*x - 8*x^2)) - log(x)*(exp(2*x)*log(3)^2*(8*x - 24) - log(3)^ 
2*(96*x^2 - 32*x^3 + 96*x^5 - 32*x^6 + 24*x^8 - 8*x^9) + exp(x)*log(3)^2*( 
96*x - 32*x^2 + 48*x^4 - 16*x^5)))/(27*x - 27*x^2 + 9*x^3 - x^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\frac {4 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (3\right )^{2} \left (e^{2 x}-2 e^{x} x^{4}-4 e^{x} x +x^{8}+4 x^{5}+4 x^{2}\right )}{x^{2}-6 x +9} \] Input:

int((((8*x^2-32*x)*log(3)^2*exp(x)^2+(-8*x^6+8*x^5+96*x^4-16*x^3+64*x^2+48 
*x)*log(3)^2*exp(x)+(24*x^9-96*x^8+48*x^6-240*x^5-96*x^2)*log(3)^2)*log(x) 
^2+((8*x-24)*log(3)^2*exp(x)^2+(-16*x^5+48*x^4-32*x^2+96*x)*log(3)^2*exp(x 
)+(8*x^9-24*x^8+32*x^6-96*x^5+32*x^3-96*x^2)*log(3)^2)*log(x))/(x^4-9*x^3+ 
27*x^2-27*x),x)
 

Output:

(4*log(x)**2*log(3)**2*(e**(2*x) - 2*e**x*x**4 - 4*e**x*x + x**8 + 4*x**5 
+ 4*x**2))/(x**2 - 6*x + 9)