\(\int \frac {-8 x^2+(4 x^2-4 x^5) \log (x)+(8 x+(-4 x+4 x^2) \log (x)) \log (\frac {3 \log (5) \log ^2(x)}{x})-4 x \log (x) \log ^2(\frac {3 \log (5) \log ^2(x)}{x})}{(9 x^4+6 x^6+x^8) \log (x)+(12 x^3+4 x^5) \log (x) \log (\frac {3 \log (5) \log ^2(x)}{x})+(-2 x^2-2 x^4) \log (x) \log ^2(\frac {3 \log (5) \log ^2(x)}{x})-4 x \log (x) \log ^3(\frac {3 \log (5) \log ^2(x)}{x})+\log (x) \log ^4(\frac {3 \log (5) \log ^2(x)}{x})} \, dx\) [2400]

Optimal result

Integrand size = 179, antiderivative size = 34 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=2+\frac {2}{4+x^2-\frac {\left (-x+\log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}{x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=-\frac {2 x^2}{-3 x^2-x^4-2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \] Input:

Integrate[(-8*x^2 + (4*x^2 - 4*x^5)*Log[x] + (8*x + (-4*x + 4*x^2)*Log[x]) 
*Log[(3*Log[5]*Log[x]^2)/x] - 4*x*Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^2)/((9 
*x^4 + 6*x^6 + x^8)*Log[x] + (12*x^3 + 4*x^5)*Log[x]*Log[(3*Log[5]*Log[x]^ 
2)/x] + (-2*x^2 - 2*x^4)*Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^2 - 4*x*Log[x]* 
Log[(3*Log[5]*Log[x]^2)/x]^3 + Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^4),x]
 

Output:

(-2*x^2)/(-3*x^2 - x^4 - 2*x*Log[(3*Log[5]*Log[x]^2)/x] + Log[(3*Log[5]*Lo 
g[x]^2)/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.

\(\displaystyle \int \frac {-8 x^2+\left (\left (4 x^2-4 x\right ) \log (x)+8 x\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (4 x^2-4 x^5\right ) \log (x)-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (4 x^5+12 x^3\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^4-2 x^2\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (x^8+6 x^6+9 x^4\right ) \log (x)+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx\)

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(-8*x^2 + (4*x^2 - 4*x^5)*Log[x] + (8*x + (-4*x + 4*x^2)*Log[x])*Log[( 
3*Log[5]*Log[x]^2)/x] - 4*x*Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^2)/((9*x^4 + 
 6*x^6 + x^8)*Log[x] + (12*x^3 + 4*x^5)*Log[x]*Log[(3*Log[5]*Log[x]^2)/x] 
+ (-2*x^2 - 2*x^4)*Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^2 - 4*x*Log[x]*Log[(3 
*Log[5]*Log[x]^2)/x]^3 + Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^4),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 39.38 (sec) , antiderivative size = 1723, normalized size of antiderivative = 50.68

\[\text {Expression too large to display}\]

Input:

int((-4*x*ln(x)*ln(3*ln(5)*ln(x)^2/x)^2+((4*x^2-4*x)*ln(x)+8*x)*ln(3*ln(5) 
*ln(x)^2/x)+(-4*x^5+4*x^2)*ln(x)-8*x^2)/(ln(x)*ln(3*ln(5)*ln(x)^2/x)^4-4*x 
*ln(x)*ln(3*ln(5)*ln(x)^2/x)^3+(-2*x^4-2*x^2)*ln(x)*ln(3*ln(5)*ln(x)^2/x)^ 
2+(4*x^5+12*x^3)*ln(x)*ln(3*ln(5)*ln(x)^2/x)+(x^8+6*x^6+9*x^4)*ln(x)),x)
 

Output:

8*x^2/(-16*ln(ln(5))*ln(ln(x))-16*ln(3)*ln(ln(x))+16*ln(x)*ln(ln(x))+8*ln( 
3)*ln(x)+8*x*ln(3)-8*x*ln(x)+8*x*ln(ln(5))-8*ln(3)*ln(ln(5))+8*ln(ln(5))*l 
n(x)-16*ln(ln(x))^2-4*ln(ln(5))^2+16*x*ln(ln(x))-4*ln(3)^2-4*ln(x)^2+4*x^4 
+12*x^2-2*Pi^2*csgn(I*ln(x)^2)^2*csgn(I/x)*csgn(I*ln(x)^2/x)^3-2*Pi^2*csgn 
(I*ln(x)^2)*csgn(I/x)^2*csgn(I*ln(x)^2/x)^3+4*Pi^2*csgn(I*ln(x)^2)*csgn(I/ 
x)*csgn(I*ln(x)^2/x)^4-2*Pi^2*csgn(I*ln(x))^2*csgn(I*ln(x)^2)^2*csgn(I*ln( 
x)^2/x)^2+2*Pi^2*csgn(I*ln(x))^2*csgn(I*ln(x)^2)*csgn(I*ln(x)^2/x)^3+4*Pi^ 
2*csgn(I*ln(x))*csgn(I*ln(x)^2)^3*csgn(I*ln(x)^2/x)^2-4*Pi^2*csgn(I*ln(x)) 
*csgn(I*ln(x)^2)^2*csgn(I*ln(x)^2/x)^3+2*Pi^2*csgn(I*ln(x)^2)^4*csgn(I/x)* 
csgn(I*ln(x)^2/x)-2*Pi^2*csgn(I*ln(x)^2)^3*csgn(I/x)*csgn(I*ln(x)^2/x)^2+P 
i^2*csgn(I*ln(x)^2)^2*csgn(I/x)^2*csgn(I*ln(x)^2/x)^2+8*I*ln(ln(x))*Pi*csg 
n(I*ln(x)^2)^3+8*I*ln(ln(x))*Pi*csgn(I*ln(x)^2/x)^3-4*I*ln(x)*Pi*csgn(I*ln 
(x)^2)^3-4*I*ln(x)*Pi*csgn(I*ln(x)^2/x)^3-4*I*x*Pi*csgn(I*ln(x)^2)^3-4*I*x 
*Pi*csgn(I*ln(x)^2/x)^3+4*I*ln(ln(5))*Pi*csgn(I*ln(x)^2)^3+4*I*ln(ln(5))*P 
i*csgn(I*ln(x)^2/x)^3+4*I*ln(3)*Pi*csgn(I*ln(x)^2)^3+4*I*ln(3)*Pi*csgn(I*l 
n(x)^2/x)^3+Pi^2*csgn(I*ln(x))^4*csgn(I*ln(x)^2)^2-4*Pi^2*csgn(I*ln(x))^3* 
csgn(I*ln(x)^2)^3+6*Pi^2*csgn(I*ln(x))^2*csgn(I*ln(x)^2)^4-4*Pi^2*csgn(I*l 
n(x))*csgn(I*ln(x)^2)^5-2*Pi^2*csgn(I*ln(x)^2)^4*csgn(I*ln(x)^2/x)^2+2*Pi^ 
2*csgn(I*ln(x)^2)^3*csgn(I*ln(x)^2/x)^3+Pi^2*csgn(I*ln(x)^2)^2*csgn(I*ln(x 
)^2/x)^4-2*Pi^2*csgn(I*ln(x)^2)*csgn(I*ln(x)^2/x)^5+Pi^2*csgn(I/x)^2*cs...
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=\frac {2 \, x^{2}}{x^{4} + 3 \, x^{2} + 2 \, x \log \left (\frac {3 \, \log \left (5\right ) \log \left (x\right )^{2}}{x}\right ) - \log \left (\frac {3 \, \log \left (5\right ) \log \left (x\right )^{2}}{x}\right )^{2}} \] Input:

integrate((-4*x*log(x)*log(3*log(5)*log(x)^2/x)^2+((4*x^2-4*x)*log(x)+8*x) 
*log(3*log(5)*log(x)^2/x)+(-4*x^5+4*x^2)*log(x)-8*x^2)/(log(x)*log(3*log(5 
)*log(x)^2/x)^4-4*x*log(x)*log(3*log(5)*log(x)^2/x)^3+(-2*x^4-2*x^2)*log(x 
)*log(3*log(5)*log(x)^2/x)^2+(4*x^5+12*x^3)*log(x)*log(3*log(5)*log(x)^2/x 
)+(x^8+6*x^6+9*x^4)*log(x)),x, algorithm="fricas")
 

Output:

2*x^2/(x^4 + 3*x^2 + 2*x*log(3*log(5)*log(x)^2/x) - log(3*log(5)*log(x)^2/ 
x)^2)
 

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=- \frac {2 x^{2}}{- x^{4} - 3 x^{2} - 2 x \log {\left (\frac {3 \log {\left (5 \right )} \log {\left (x \right )}^{2}}{x} \right )} + \log {\left (\frac {3 \log {\left (5 \right )} \log {\left (x \right )}^{2}}{x} \right )}^{2}} \] Input:

integrate((-4*x*ln(x)*ln(3*ln(5)*ln(x)**2/x)**2+((4*x**2-4*x)*ln(x)+8*x)*l 
n(3*ln(5)*ln(x)**2/x)+(-4*x**5+4*x**2)*ln(x)-8*x**2)/(ln(x)*ln(3*ln(5)*ln( 
x)**2/x)**4-4*x*ln(x)*ln(3*ln(5)*ln(x)**2/x)**3+(-2*x**4-2*x**2)*ln(x)*ln( 
3*ln(5)*ln(x)**2/x)**2+(4*x**5+12*x**3)*ln(x)*ln(3*ln(5)*ln(x)**2/x)+(x**8 
+6*x**6+9*x**4)*ln(x)),x)
 

Output:

-2*x**2/(-x**4 - 3*x**2 - 2*x*log(3*log(5)*log(x)**2/x) + log(3*log(5)*log 
(x)**2/x)**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (34) = 68\).

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.68 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=\frac {2 \, x^{2}}{x^{4} + 3 \, x^{2} + 2 \, x {\left (\log \left (3\right ) + \log \left (\log \left (5\right )\right )\right )} - \log \left (3\right )^{2} - 2 \, {\left (x - \log \left (3\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (x\right ) - \log \left (x\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (5\right )\right ) - \log \left (\log \left (5\right )\right )^{2} + 4 \, {\left (x - \log \left (3\right ) + \log \left (x\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (\log \left (x\right )\right ) - 4 \, \log \left (\log \left (x\right )\right )^{2}} \] Input:

integrate((-4*x*log(x)*log(3*log(5)*log(x)^2/x)^2+((4*x^2-4*x)*log(x)+8*x) 
*log(3*log(5)*log(x)^2/x)+(-4*x^5+4*x^2)*log(x)-8*x^2)/(log(x)*log(3*log(5 
)*log(x)^2/x)^4-4*x*log(x)*log(3*log(5)*log(x)^2/x)^3+(-2*x^4-2*x^2)*log(x 
)*log(3*log(5)*log(x)^2/x)^2+(4*x^5+12*x^3)*log(x)*log(3*log(5)*log(x)^2/x 
)+(x^8+6*x^6+9*x^4)*log(x)),x, algorithm="maxima")
 

Output:

2*x^2/(x^4 + 3*x^2 + 2*x*(log(3) + log(log(5))) - log(3)^2 - 2*(x - log(3) 
 - log(log(5)))*log(x) - log(x)^2 - 2*log(3)*log(log(5)) - log(log(5))^2 + 
 4*(x - log(3) + log(x) - log(log(5)))*log(log(x)) - 4*log(log(x))^2)
 

Giac [A] (verification not implemented)

Time = 1.95 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=\frac {2 \, x^{2}}{x^{4} + 3 \, x^{2} + 2 \, x \log \left (3 \, \log \left (5\right ) \log \left (x\right )^{2}\right ) - \log \left (3 \, \log \left (5\right ) \log \left (x\right )^{2}\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, \log \left (3 \, \log \left (5\right ) \log \left (x\right )^{2}\right ) \log \left (x\right ) - \log \left (x\right )^{2}} \] Input:

integrate((-4*x*log(x)*log(3*log(5)*log(x)^2/x)^2+((4*x^2-4*x)*log(x)+8*x) 
*log(3*log(5)*log(x)^2/x)+(-4*x^5+4*x^2)*log(x)-8*x^2)/(log(x)*log(3*log(5 
)*log(x)^2/x)^4-4*x*log(x)*log(3*log(5)*log(x)^2/x)^3+(-2*x^4-2*x^2)*log(x 
)*log(3*log(5)*log(x)^2/x)^2+(4*x^5+12*x^3)*log(x)*log(3*log(5)*log(x)^2/x 
)+(x^8+6*x^6+9*x^4)*log(x)),x, algorithm="giac")
 

Output:

2*x^2/(x^4 + 3*x^2 + 2*x*log(3*log(5)*log(x)^2) - log(3*log(5)*log(x)^2)^2 
 - 2*x*log(x) + 2*log(3*log(5)*log(x)^2)*log(x) - log(x)^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=\int \frac {\ln \left (x\right )\,\left (4\,x^2-4\,x^5\right )+\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )\,\left (8\,x-\ln \left (x\right )\,\left (4\,x-4\,x^2\right )\right )-8\,x^2-4\,x\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^2\,\ln \left (x\right )}{\ln \left (x\right )\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^4-4\,x\,\ln \left (x\right )\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^3-\ln \left (x\right )\,\left (2\,x^4+2\,x^2\right )\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^2+\ln \left (x\right )\,\left (4\,x^5+12\,x^3\right )\,\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )+\ln \left (x\right )\,\left (x^8+6\,x^6+9\,x^4\right )} \,d x \] Input:

int((log(x)*(4*x^2 - 4*x^5) + log((3*log(5)*log(x)^2)/x)*(8*x - log(x)*(4* 
x - 4*x^2)) - 8*x^2 - 4*x*log((3*log(5)*log(x)^2)/x)^2*log(x))/(log(x)*(9* 
x^4 + 6*x^6 + x^8) + log((3*log(5)*log(x)^2)/x)^4*log(x) + log((3*log(5)*l 
og(x)^2)/x)*log(x)*(12*x^3 + 4*x^5) - 4*x*log((3*log(5)*log(x)^2)/x)^3*log 
(x) - log((3*log(5)*log(x)^2)/x)^2*log(x)*(2*x^2 + 2*x^4)),x)
 

Output:

int((log(x)*(4*x^2 - 4*x^5) + log((3*log(5)*log(x)^2)/x)*(8*x - log(x)*(4* 
x - 4*x^2)) - 8*x^2 - 4*x*log((3*log(5)*log(x)^2)/x)^2*log(x))/(log(x)*(9* 
x^4 + 6*x^6 + x^8) + log((3*log(5)*log(x)^2)/x)^4*log(x) + log((3*log(5)*l 
og(x)^2)/x)*log(x)*(12*x^3 + 4*x^5) - 4*x*log((3*log(5)*log(x)^2)/x)^3*log 
(x) - log((3*log(5)*log(x)^2)/x)^2*log(x)*(2*x^2 + 2*x^4)), x)
 

Reduce [F]

\[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx =\text {Too large to display} \] Input:

int((-4*x*log(x)*log(3*log(5)*log(x)^2/x)^2+((4*x^2-4*x)*log(x)+8*x)*log(3 
*log(5)*log(x)^2/x)+(-4*x^5+4*x^2)*log(x)-8*x^2)/(log(x)*log(3*log(5)*log( 
x)^2/x)^4-4*x*log(x)*log(3*log(5)*log(x)^2/x)^3+(-2*x^4-2*x^2)*log(x)*log( 
3*log(5)*log(x)^2/x)^2+(4*x^5+12*x^3)*log(x)*log(3*log(5)*log(x)^2/x)+(x^8 
+6*x^6+9*x^4)*log(x)),x)
 

Output:

4*( - int(x**5/(log((3*log(x)**2*log(5))/x)**4 - 4*log((3*log(x)**2*log(5) 
)/x)**3*x - 2*log((3*log(x)**2*log(5))/x)**2*x**4 - 2*log((3*log(x)**2*log 
(5))/x)**2*x**2 + 4*log((3*log(x)**2*log(5))/x)*x**5 + 12*log((3*log(x)**2 
*log(5))/x)*x**3 + x**8 + 6*x**6 + 9*x**4),x) + int(x**2/(log((3*log(x)**2 
*log(5))/x)**4 - 4*log((3*log(x)**2*log(5))/x)**3*x - 2*log((3*log(x)**2*l 
og(5))/x)**2*x**4 - 2*log((3*log(x)**2*log(5))/x)**2*x**2 + 4*log((3*log(x 
)**2*log(5))/x)*x**5 + 12*log((3*log(x)**2*log(5))/x)*x**3 + x**8 + 6*x**6 
 + 9*x**4),x) - 2*int(x**2/(log((3*log(x)**2*log(5))/x)**4*log(x) - 4*log( 
(3*log(x)**2*log(5))/x)**3*log(x)*x - 2*log((3*log(x)**2*log(5))/x)**2*log 
(x)*x**4 - 2*log((3*log(x)**2*log(5))/x)**2*log(x)*x**2 + 4*log((3*log(x)* 
*2*log(5))/x)*log(x)*x**5 + 12*log((3*log(x)**2*log(5))/x)*log(x)*x**3 + l 
og(x)*x**8 + 6*log(x)*x**6 + 9*log(x)*x**4),x) - int((log((3*log(x)**2*log 
(5))/x)**2*x)/(log((3*log(x)**2*log(5))/x)**4 - 4*log((3*log(x)**2*log(5)) 
/x)**3*x - 2*log((3*log(x)**2*log(5))/x)**2*x**4 - 2*log((3*log(x)**2*log( 
5))/x)**2*x**2 + 4*log((3*log(x)**2*log(5))/x)*x**5 + 12*log((3*log(x)**2* 
log(5))/x)*x**3 + x**8 + 6*x**6 + 9*x**4),x) + int((log((3*log(x)**2*log(5 
))/x)*x**2)/(log((3*log(x)**2*log(5))/x)**4 - 4*log((3*log(x)**2*log(5))/x 
)**3*x - 2*log((3*log(x)**2*log(5))/x)**2*x**4 - 2*log((3*log(x)**2*log(5) 
)/x)**2*x**2 + 4*log((3*log(x)**2*log(5))/x)*x**5 + 12*log((3*log(x)**2*lo 
g(5))/x)*x**3 + x**8 + 6*x**6 + 9*x**4),x) - int((log((3*log(x)**2*log(...