\(\int \frac {-11520+13248 x-1335 x^2-186 x^3-3 x^4+(81 x^2-144 x^3+82 x^4-16 x^5+x^6) \log (3)+(-6144+2880 x+204 x^2+(108 x^2-132 x^3+44 x^4-4 x^5) \log (3)) \log (x)+(-768+3 x^2+(54 x^2-40 x^3+6 x^4) \log (3)) \log ^2(x)+(12 x^2-4 x^3) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)}{(81 x^2-144 x^3+82 x^4-16 x^5+x^6) \log (3)+(108 x^2-132 x^3+44 x^4-4 x^5) \log (3) \log (x)+(54 x^2-40 x^3+6 x^4) \log (3) \log ^2(x)+(12 x^2-4 x^3) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)} \, dx\) [257]

Optimal result

Integrand size = 251, antiderivative size = 31 \[ \int \frac {-11520+13248 x-1335 x^2-186 x^3-3 x^4+\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (-6144+2880 x+204 x^2+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3)\right ) \log (x)+\left (-768+3 x^2+\left (54 x^2-40 x^3+6 x^4\right ) \log (3)\right ) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)}{\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3) \log (x)+\left (54 x^2-40 x^3+6 x^4\right ) \log (3) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)} \, dx=x+\frac {3 (16+x)^2}{x \log (3) \left (-2 x+(3-x+\log (x))^2\right )} \] Output:

x+3*(x+16)^2/x/ln(3)/((3-x+ln(x))^2-2*x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {-11520+13248 x-1335 x^2-186 x^3-3 x^4+\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (-6144+2880 x+204 x^2+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3)\right ) \log (x)+\left (-768+3 x^2+\left (54 x^2-40 x^3+6 x^4\right ) \log (3)\right ) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)}{\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3) \log (x)+\left (54 x^2-40 x^3+6 x^4\right ) \log (3) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)} \, dx=\frac {x \log (3)+\frac {3 (16+x)^2}{x \left (9-8 x+x^2-2 (-3+x) \log (x)+\log ^2(x)\right )}}{\log (3)} \] Input:

Integrate[(-11520 + 13248*x - 1335*x^2 - 186*x^3 - 3*x^4 + (81*x^2 - 144*x 
^3 + 82*x^4 - 16*x^5 + x^6)*Log[3] + (-6144 + 2880*x + 204*x^2 + (108*x^2 
- 132*x^3 + 44*x^4 - 4*x^5)*Log[3])*Log[x] + (-768 + 3*x^2 + (54*x^2 - 40* 
x^3 + 6*x^4)*Log[3])*Log[x]^2 + (12*x^2 - 4*x^3)*Log[3]*Log[x]^3 + x^2*Log 
[3]*Log[x]^4)/((81*x^2 - 144*x^3 + 82*x^4 - 16*x^5 + x^6)*Log[3] + (108*x^ 
2 - 132*x^3 + 44*x^4 - 4*x^5)*Log[3]*Log[x] + (54*x^2 - 40*x^3 + 6*x^4)*Lo 
g[3]*Log[x]^2 + (12*x^2 - 4*x^3)*Log[3]*Log[x]^3 + x^2*Log[3]*Log[x]^4),x]
 

Output:

(x*Log[3] + (3*(16 + x)^2)/(x*(9 - 8*x + x^2 - 2*(-3 + x)*Log[x] + Log[x]^ 
2)))/Log[3]
 

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.

\(\displaystyle \int \frac {-3 x^4-186 x^3-1335 x^2+x^2 \log (3) \log ^4(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+\left (3 x^2+\left (6 x^4-40 x^3+54 x^2\right ) \log (3)-768\right ) \log ^2(x)+\left (204 x^2+\left (-4 x^5+44 x^4-132 x^3+108 x^2\right ) \log (3)+2880 x-6144\right ) \log (x)+\left (x^6-16 x^5+82 x^4-144 x^3+81 x^2\right ) \log (3)+13248 x-11520}{x^2 \log (3) \log ^4(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+\left (6 x^4-40 x^3+54 x^2\right ) \log (3) \log ^2(x)+\left (-4 x^5+44 x^4-132 x^3+108 x^2\right ) \log (3) \log (x)+\left (x^6-16 x^5+82 x^4-144 x^3+81 x^2\right ) \log (3)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-3 x^4-186 x^3-1335 x^2+x^2 \log (3) \log ^4(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+\left (3 x^2+\left (6 x^4-40 x^3+54 x^2\right ) \log (3)-768\right ) \log ^2(x)+\left (204 x^2+\left (-4 x^5+44 x^4-132 x^3+108 x^2\right ) \log (3)+2880 x-6144\right ) \log (x)+\left (x^6-16 x^5+82 x^4-144 x^3+81 x^2\right ) \log (3)+13248 x-11520}{x^2 \log (3) \left (x^2-8 x+\log ^2(x)-2 x \log (x)+6 \log (x)+9\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int -\frac {3 x^4+186 x^3-\log (3) \log ^4(x) x^2+1335 x^2-13248 x-4 \left (3 x^2-x^3\right ) \log (3) \log ^3(x)+\left (-3 x^2-2 \left (3 x^4-20 x^3+27 x^2\right ) \log (3)+768\right ) \log ^2(x)+4 \left (-51 x^2-720 x-\left (-x^5+11 x^4-33 x^3+27 x^2\right ) \log (3)+1536\right ) \log (x)-\left (x^6-16 x^5+82 x^4-144 x^3+81 x^2\right ) \log (3)+11520}{x^2 \left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx}{\log (3)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {3 x^4+186 x^3-\log (3) \log ^4(x) x^2+1335 x^2-13248 x-4 \left (3 x^2-x^3\right ) \log (3) \log ^3(x)+\left (-3 x^2-2 \left (3 x^4-20 x^3+27 x^2\right ) \log (3)+768\right ) \log ^2(x)+4 \left (-51 x^2-720 x-\left (-x^5+11 x^4-33 x^3+27 x^2\right ) \log (3)+1536\right ) \log (x)-\left (x^6-16 x^5+82 x^4-144 x^3+81 x^2\right ) \log (3)+11520}{x^2 \left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx}{\log (3)}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {6 \left (x^2-\log (x) x-5 x+\log (x)+3\right ) (x+16)^2}{x^2 \left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}-\frac {3 \left (x^2-256\right )}{x^2 \left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )}-\log (3)\right )dx}{\log (3)}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1554 \int \frac {1}{\left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx+4608 \int \frac {1}{x^2 \left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx-7680 \int \frac {1}{x \left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx-30 \int \frac {x}{\left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx+6 \int \frac {x^2}{\left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx+6 \int \frac {\log (x)}{\left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx+1536 \int \frac {\log (x)}{x^2 \left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx-1536 \int \frac {\log (x)}{x \left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx-6 \int \frac {x \log (x)}{\left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )^2}dx-3 \int \frac {1}{x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9}dx+768 \int \frac {1}{x^2 \left (x^2-2 \log (x) x-8 x+\log ^2(x)+6 \log (x)+9\right )}dx-\frac {96}{x^2-8 x+\log ^2(x)-2 x \log (x)+6 \log (x)+9}+x (-\log (3))}{\log (3)}\)

Input:

Int[(-11520 + 13248*x - 1335*x^2 - 186*x^3 - 3*x^4 + (81*x^2 - 144*x^3 + 8 
2*x^4 - 16*x^5 + x^6)*Log[3] + (-6144 + 2880*x + 204*x^2 + (108*x^2 - 132* 
x^3 + 44*x^4 - 4*x^5)*Log[3])*Log[x] + (-768 + 3*x^2 + (54*x^2 - 40*x^3 + 
6*x^4)*Log[3])*Log[x]^2 + (12*x^2 - 4*x^3)*Log[3]*Log[x]^3 + x^2*Log[3]*Lo 
g[x]^4)/((81*x^2 - 144*x^3 + 82*x^4 - 16*x^5 + x^6)*Log[3] + (108*x^2 - 13 
2*x^3 + 44*x^4 - 4*x^5)*Log[3]*Log[x] + (54*x^2 - 40*x^3 + 6*x^4)*Log[3]*L 
og[x]^2 + (12*x^2 - 4*x^3)*Log[3]*Log[x]^3 + x^2*Log[3]*Log[x]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39

\[x +\frac {3 x^{2}+96 x +768}{\ln \left (3\right ) x \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+6 \ln \left (x \right )-8 x +9\right )}\]

Input:

int((x^2*ln(3)*ln(x)^4+(-4*x^3+12*x^2)*ln(3)*ln(x)^3+((6*x^4-40*x^3+54*x^2 
)*ln(3)+3*x^2-768)*ln(x)^2+((-4*x^5+44*x^4-132*x^3+108*x^2)*ln(3)+204*x^2+ 
2880*x-6144)*ln(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*ln(3)-3*x^4-186*x^3- 
1335*x^2+13248*x-11520)/(x^2*ln(3)*ln(x)^4+(-4*x^3+12*x^2)*ln(3)*ln(x)^3+( 
6*x^4-40*x^3+54*x^2)*ln(3)*ln(x)^2+(-4*x^5+44*x^4-132*x^3+108*x^2)*ln(3)*l 
n(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*ln(3)),x)
 

Output:

x+3/ln(3)*(x^2+32*x+256)/x/(ln(x)^2-2*x*ln(x)+x^2+6*ln(x)-8*x+9)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (31) = 62\).

Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.97 \[ \int \frac {-11520+13248 x-1335 x^2-186 x^3-3 x^4+\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (-6144+2880 x+204 x^2+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3)\right ) \log (x)+\left (-768+3 x^2+\left (54 x^2-40 x^3+6 x^4\right ) \log (3)\right ) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)}{\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3) \log (x)+\left (54 x^2-40 x^3+6 x^4\right ) \log (3) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)} \, dx=\frac {x^{2} \log \left (3\right ) \log \left (x\right )^{2} - 2 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (3\right ) \log \left (x\right ) + 3 \, x^{2} + {\left (x^{4} - 8 \, x^{3} + 9 \, x^{2}\right )} \log \left (3\right ) + 96 \, x + 768}{x \log \left (3\right ) \log \left (x\right )^{2} - 2 \, {\left (x^{2} - 3 \, x\right )} \log \left (3\right ) \log \left (x\right ) + {\left (x^{3} - 8 \, x^{2} + 9 \, x\right )} \log \left (3\right )} \] Input:

integrate((x^2*log(3)*log(x)^4+(-4*x^3+12*x^2)*log(3)*log(x)^3+((6*x^4-40* 
x^3+54*x^2)*log(3)+3*x^2-768)*log(x)^2+((-4*x^5+44*x^4-132*x^3+108*x^2)*lo 
g(3)+204*x^2+2880*x-6144)*log(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*log(3) 
-3*x^4-186*x^3-1335*x^2+13248*x-11520)/(x^2*log(3)*log(x)^4+(-4*x^3+12*x^2 
)*log(3)*log(x)^3+(6*x^4-40*x^3+54*x^2)*log(3)*log(x)^2+(-4*x^5+44*x^4-132 
*x^3+108*x^2)*log(3)*log(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*log(3)),x, 
algorithm="fricas")
 

Output:

(x^2*log(3)*log(x)^2 - 2*(x^3 - 3*x^2)*log(3)*log(x) + 3*x^2 + (x^4 - 8*x^ 
3 + 9*x^2)*log(3) + 96*x + 768)/(x*log(3)*log(x)^2 - 2*(x^2 - 3*x)*log(3)* 
log(x) + (x^3 - 8*x^2 + 9*x)*log(3))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).

Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {-11520+13248 x-1335 x^2-186 x^3-3 x^4+\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (-6144+2880 x+204 x^2+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3)\right ) \log (x)+\left (-768+3 x^2+\left (54 x^2-40 x^3+6 x^4\right ) \log (3)\right ) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)}{\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3) \log (x)+\left (54 x^2-40 x^3+6 x^4\right ) \log (3) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)} \, dx=x + \frac {3 x^{2} + 96 x + 768}{x^{3} \log {\left (3 \right )} - 8 x^{2} \log {\left (3 \right )} + x \log {\left (3 \right )} \log {\left (x \right )}^{2} + 9 x \log {\left (3 \right )} + \left (- 2 x^{2} \log {\left (3 \right )} + 6 x \log {\left (3 \right )}\right ) \log {\left (x \right )}} \] Input:

integrate((x**2*ln(3)*ln(x)**4+(-4*x**3+12*x**2)*ln(3)*ln(x)**3+((6*x**4-4 
0*x**3+54*x**2)*ln(3)+3*x**2-768)*ln(x)**2+((-4*x**5+44*x**4-132*x**3+108* 
x**2)*ln(3)+204*x**2+2880*x-6144)*ln(x)+(x**6-16*x**5+82*x**4-144*x**3+81* 
x**2)*ln(3)-3*x**4-186*x**3-1335*x**2+13248*x-11520)/(x**2*ln(3)*ln(x)**4+ 
(-4*x**3+12*x**2)*ln(3)*ln(x)**3+(6*x**4-40*x**3+54*x**2)*ln(3)*ln(x)**2+( 
-4*x**5+44*x**4-132*x**3+108*x**2)*ln(3)*ln(x)+(x**6-16*x**5+82*x**4-144*x 
**3+81*x**2)*ln(3)),x)
 

Output:

x + (3*x**2 + 96*x + 768)/(x**3*log(3) - 8*x**2*log(3) + x*log(3)*log(x)** 
2 + 9*x*log(3) + (-2*x**2*log(3) + 6*x*log(3))*log(x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (31) = 62\).

Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.32 \[ \int \frac {-11520+13248 x-1335 x^2-186 x^3-3 x^4+\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (-6144+2880 x+204 x^2+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3)\right ) \log (x)+\left (-768+3 x^2+\left (54 x^2-40 x^3+6 x^4\right ) \log (3)\right ) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)}{\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3) \log (x)+\left (54 x^2-40 x^3+6 x^4\right ) \log (3) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)} \, dx=\frac {x^{4} \log \left (3\right ) + x^{2} \log \left (3\right ) \log \left (x\right )^{2} - 8 \, x^{3} \log \left (3\right ) + 3 \, x^{2} {\left (3 \, \log \left (3\right ) + 1\right )} - 2 \, {\left (x^{3} \log \left (3\right ) - 3 \, x^{2} \log \left (3\right )\right )} \log \left (x\right ) + 96 \, x + 768}{x^{3} \log \left (3\right ) + x \log \left (3\right ) \log \left (x\right )^{2} - 8 \, x^{2} \log \left (3\right ) + 9 \, x \log \left (3\right ) - 2 \, {\left (x^{2} \log \left (3\right ) - 3 \, x \log \left (3\right )\right )} \log \left (x\right )} \] Input:

integrate((x^2*log(3)*log(x)^4+(-4*x^3+12*x^2)*log(3)*log(x)^3+((6*x^4-40* 
x^3+54*x^2)*log(3)+3*x^2-768)*log(x)^2+((-4*x^5+44*x^4-132*x^3+108*x^2)*lo 
g(3)+204*x^2+2880*x-6144)*log(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*log(3) 
-3*x^4-186*x^3-1335*x^2+13248*x-11520)/(x^2*log(3)*log(x)^4+(-4*x^3+12*x^2 
)*log(3)*log(x)^3+(6*x^4-40*x^3+54*x^2)*log(3)*log(x)^2+(-4*x^5+44*x^4-132 
*x^3+108*x^2)*log(3)*log(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*log(3)),x, 
algorithm="maxima")
 

Output:

(x^4*log(3) + x^2*log(3)*log(x)^2 - 8*x^3*log(3) + 3*x^2*(3*log(3) + 1) - 
2*(x^3*log(3) - 3*x^2*log(3))*log(x) + 96*x + 768)/(x^3*log(3) + x*log(3)* 
log(x)^2 - 8*x^2*log(3) + 9*x*log(3) - 2*(x^2*log(3) - 3*x*log(3))*log(x))
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84 \[ \int \frac {-11520+13248 x-1335 x^2-186 x^3-3 x^4+\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (-6144+2880 x+204 x^2+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3)\right ) \log (x)+\left (-768+3 x^2+\left (54 x^2-40 x^3+6 x^4\right ) \log (3)\right ) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)}{\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3) \log (x)+\left (54 x^2-40 x^3+6 x^4\right ) \log (3) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)} \, dx=x + \frac {3 \, {\left (x^{2} + 32 \, x + 256\right )}}{x^{3} \log \left (3\right ) - 2 \, x^{2} \log \left (3\right ) \log \left (x\right ) + x \log \left (3\right ) \log \left (x\right )^{2} - 8 \, x^{2} \log \left (3\right ) + 6 \, x \log \left (3\right ) \log \left (x\right ) + 9 \, x \log \left (3\right )} \] Input:

integrate((x^2*log(3)*log(x)^4+(-4*x^3+12*x^2)*log(3)*log(x)^3+((6*x^4-40* 
x^3+54*x^2)*log(3)+3*x^2-768)*log(x)^2+((-4*x^5+44*x^4-132*x^3+108*x^2)*lo 
g(3)+204*x^2+2880*x-6144)*log(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*log(3) 
-3*x^4-186*x^3-1335*x^2+13248*x-11520)/(x^2*log(3)*log(x)^4+(-4*x^3+12*x^2 
)*log(3)*log(x)^3+(6*x^4-40*x^3+54*x^2)*log(3)*log(x)^2+(-4*x^5+44*x^4-132 
*x^3+108*x^2)*log(3)*log(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*log(3)),x, 
algorithm="giac")
 

Output:

x + 3*(x^2 + 32*x + 256)/(x^3*log(3) - 2*x^2*log(3)*log(x) + x*log(3)*log( 
x)^2 - 8*x^2*log(3) + 6*x*log(3)*log(x) + 9*x*log(3))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-11520+13248 x-1335 x^2-186 x^3-3 x^4+\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (-6144+2880 x+204 x^2+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3)\right ) \log (x)+\left (-768+3 x^2+\left (54 x^2-40 x^3+6 x^4\right ) \log (3)\right ) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)}{\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3) \log (x)+\left (54 x^2-40 x^3+6 x^4\right ) \log (3) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)} \, dx=\int \frac {13248\,x+{\ln \left (x\right )}^2\,\left (\ln \left (3\right )\,\left (6\,x^4-40\,x^3+54\,x^2\right )+3\,x^2-768\right )+\ln \left (x\right )\,\left (2880\,x+\ln \left (3\right )\,\left (-4\,x^5+44\,x^4-132\,x^3+108\,x^2\right )+204\,x^2-6144\right )-1335\,x^2-186\,x^3-3\,x^4+\ln \left (3\right )\,\left (x^6-16\,x^5+82\,x^4-144\,x^3+81\,x^2\right )+\ln \left (3\right )\,{\ln \left (x\right )}^3\,\left (12\,x^2-4\,x^3\right )+x^2\,\ln \left (3\right )\,{\ln \left (x\right )}^4-11520}{\ln \left (3\right )\,\left (x^6-16\,x^5+82\,x^4-144\,x^3+81\,x^2\right )+\ln \left (3\right )\,{\ln \left (x\right )}^2\,\left (6\,x^4-40\,x^3+54\,x^2\right )+\ln \left (3\right )\,{\ln \left (x\right )}^3\,\left (12\,x^2-4\,x^3\right )+x^2\,\ln \left (3\right )\,{\ln \left (x\right )}^4+\ln \left (3\right )\,\ln \left (x\right )\,\left (-4\,x^5+44\,x^4-132\,x^3+108\,x^2\right )} \,d x \] Input:

int((13248*x + log(x)^2*(log(3)*(54*x^2 - 40*x^3 + 6*x^4) + 3*x^2 - 768) + 
 log(x)*(2880*x + log(3)*(108*x^2 - 132*x^3 + 44*x^4 - 4*x^5) + 204*x^2 - 
6144) - 1335*x^2 - 186*x^3 - 3*x^4 + log(3)*(81*x^2 - 144*x^3 + 82*x^4 - 1 
6*x^5 + x^6) + log(3)*log(x)^3*(12*x^2 - 4*x^3) + x^2*log(3)*log(x)^4 - 11 
520)/(log(3)*(81*x^2 - 144*x^3 + 82*x^4 - 16*x^5 + x^6) + log(3)*log(x)^2* 
(54*x^2 - 40*x^3 + 6*x^4) + log(3)*log(x)^3*(12*x^2 - 4*x^3) + x^2*log(3)* 
log(x)^4 + log(3)*log(x)*(108*x^2 - 132*x^3 + 44*x^4 - 4*x^5)),x)
 

Output:

int((13248*x + log(x)^2*(log(3)*(54*x^2 - 40*x^3 + 6*x^4) + 3*x^2 - 768) + 
 log(x)*(2880*x + log(3)*(108*x^2 - 132*x^3 + 44*x^4 - 4*x^5) + 204*x^2 - 
6144) - 1335*x^2 - 186*x^3 - 3*x^4 + log(3)*(81*x^2 - 144*x^3 + 82*x^4 - 1 
6*x^5 + x^6) + log(3)*log(x)^3*(12*x^2 - 4*x^3) + x^2*log(3)*log(x)^4 - 11 
520)/(log(3)*(81*x^2 - 144*x^3 + 82*x^4 - 16*x^5 + x^6) + log(3)*log(x)^2* 
(54*x^2 - 40*x^3 + 6*x^4) + log(3)*log(x)^3*(12*x^2 - 4*x^3) + x^2*log(3)* 
log(x)^4 + log(3)*log(x)*(108*x^2 - 132*x^3 + 44*x^4 - 4*x^5)), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.03 \[ \int \frac {-11520+13248 x-1335 x^2-186 x^3-3 x^4+\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (-6144+2880 x+204 x^2+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3)\right ) \log (x)+\left (-768+3 x^2+\left (54 x^2-40 x^3+6 x^4\right ) \log (3)\right ) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)}{\left (81 x^2-144 x^3+82 x^4-16 x^5+x^6\right ) \log (3)+\left (108 x^2-132 x^3+44 x^4-4 x^5\right ) \log (3) \log (x)+\left (54 x^2-40 x^3+6 x^4\right ) \log (3) \log ^2(x)+\left (12 x^2-4 x^3\right ) \log (3) \log ^3(x)+x^2 \log (3) \log ^4(x)} \, dx=\frac {8 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (3\right ) x^{2}+9 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (3\right ) x +3 \mathrm {log}\left (x \right )^{2} x -16 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x^{3}+30 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x^{2}+54 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x -6 \,\mathrm {log}\left (x \right ) x^{2}+18 \,\mathrm {log}\left (x \right ) x +8 \,\mathrm {log}\left (3\right ) x^{4}-55 \,\mathrm {log}\left (3\right ) x^{3}+81 \,\mathrm {log}\left (3\right ) x +3 x^{3}+795 x +6144}{8 \,\mathrm {log}\left (3\right ) x \left (\mathrm {log}\left (x \right )^{2}-2 \,\mathrm {log}\left (x \right ) x +6 \,\mathrm {log}\left (x \right )+x^{2}-8 x +9\right )} \] Input:

int((x^2*log(3)*log(x)^4+(-4*x^3+12*x^2)*log(3)*log(x)^3+((6*x^4-40*x^3+54 
*x^2)*log(3)+3*x^2-768)*log(x)^2+((-4*x^5+44*x^4-132*x^3+108*x^2)*log(3)+2 
04*x^2+2880*x-6144)*log(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*log(3)-3*x^4 
-186*x^3-1335*x^2+13248*x-11520)/(x^2*log(3)*log(x)^4+(-4*x^3+12*x^2)*log( 
3)*log(x)^3+(6*x^4-40*x^3+54*x^2)*log(3)*log(x)^2+(-4*x^5+44*x^4-132*x^3+1 
08*x^2)*log(3)*log(x)+(x^6-16*x^5+82*x^4-144*x^3+81*x^2)*log(3)),x)
 

Output:

(8*log(x)**2*log(3)*x**2 + 9*log(x)**2*log(3)*x + 3*log(x)**2*x - 16*log(x 
)*log(3)*x**3 + 30*log(x)*log(3)*x**2 + 54*log(x)*log(3)*x - 6*log(x)*x**2 
 + 18*log(x)*x + 8*log(3)*x**4 - 55*log(3)*x**3 + 81*log(3)*x + 3*x**3 + 7 
95*x + 6144)/(8*log(3)*x*(log(x)**2 - 2*log(x)*x + 6*log(x) + x**2 - 8*x + 
 9))