\(\int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log (4 x^2+x^3)+(-8-2 x+(-8-2 x) \log (x)+(20 x^2+5 x^3) \log ^2(x)) \log ^2(4 x^2+x^3)}{(4 x+x^2) \log ^2(x) \log (4 x^2+x^3)+((8 x+2 x^2) \log (x)+(8 x^2+22 x^3+5 x^4) \log ^2(x)) \log ^2(4 x^2+x^3)} \, dx\) [8]

Optimal result

Integrand size = 146, antiderivative size = 33 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log \left (5 x+\frac {2 x+\frac {2+\frac {\log (x)}{\log \left (x^2 (4+x)\right )}}{\log (x)}}{x}\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(33)=66\).

Time = 55.73 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.27 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log (2+5 x)-\log (x (2+5 x))-\log (\log (x))-\log \left (\log \left (x^2 (4+x)\right )\right )+\log \left (\log (x)+2 \log \left (x^2 (4+x)\right )+2 x \log (x) \log \left (x^2 (4+x)\right )+5 x^2 \log (x) \log \left (x^2 (4+x)\right )\right ) \] Input:

Integrate[((-8 - 3*x)*Log[x]^2 + (-4 - x)*Log[x]^2*Log[4*x^2 + x^3] + (-8 
- 2*x + (-8 - 2*x)*Log[x] + (20*x^2 + 5*x^3)*Log[x]^2)*Log[4*x^2 + x^3]^2) 
/((4*x + x^2)*Log[x]^2*Log[4*x^2 + x^3] + ((8*x + 2*x^2)*Log[x] + (8*x^2 + 
 22*x^3 + 5*x^4)*Log[x]^2)*Log[4*x^2 + x^3]^2),x]
 

Output:

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

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.

\(\displaystyle \int \frac {(-x-4) \log \left (x^3+4 x^2\right ) \log ^2(x)+\left (\left (5 x^3+20 x^2\right ) \log ^2(x)-2 x+(-2 x-8) \log (x)-8\right ) \log ^2\left (x^3+4 x^2\right )+(-3 x-8) \log ^2(x)}{\left (x^2+4 x\right ) \log \left (x^3+4 x^2\right ) \log ^2(x)+\left (\left (2 x^2+8 x\right ) \log (x)+\left (5 x^4+22 x^3+8 x^2\right ) \log ^2(x)\right ) \log ^2\left (x^3+4 x^2\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {(-x-4) \log \left (x^3+4 x^2\right ) \log ^2(x)+\left (\left (5 x^3+20 x^2\right ) \log ^2(x)-2 x+(-2 x-8) \log (x)-8\right ) \log ^2\left (x^3+4 x^2\right )+(-3 x-8) \log ^2(x)}{x (x+4) \log (x) \log \left (x^2 (x+4)\right ) \left (5 x^2 \log (x) \log \left (x^2 (x+4)\right )+2 x \log (x) \log \left (x^2 (x+4)\right )+2 \log \left (x^2 (x+4)\right )+\log (x)\right )}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

Input:

Int[((-8 - 3*x)*Log[x]^2 + (-4 - x)*Log[x]^2*Log[4*x^2 + x^3] + (-8 - 2*x 
+ (-8 - 2*x)*Log[x] + (20*x^2 + 5*x^3)*Log[x]^2)*Log[4*x^2 + x^3]^2)/((4*x 
 + x^2)*Log[x]^2*Log[4*x^2 + x^3] + ((8*x + 2*x^2)*Log[x] + (8*x^2 + 22*x^ 
3 + 5*x^4)*Log[x]^2)*Log[4*x^2 + x^3]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 0.80 (sec) , antiderivative size = 675, normalized size of antiderivative = 20.45

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

Input:

int((((5*x^3+20*x^2)*ln(x)^2+(-2*x-8)*ln(x)-2*x-8)*ln(x^3+4*x^2)^2+(-4-x)* 
ln(x)^2*ln(x^3+4*x^2)+(-3*x-8)*ln(x)^2)/(((5*x^4+22*x^3+8*x^2)*ln(x)^2+(2* 
x^2+8*x)*ln(x))*ln(x^3+4*x^2)^2+(x^2+4*x)*ln(x)^2*ln(x^3+4*x^2)),x)
 

Output:

-ln(x)+ln(x^2+2/5*x+2/5/ln(x))-ln(ln(x)-1/4*I*(Pi*csgn(I*(4+x))*csgn(I*x^2 
)*csgn(I*x^2*(4+x))-Pi*csgn(I*(4+x))*csgn(I*x^2*(4+x))^2+Pi*csgn(I*x)^2*cs 
gn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-Pi*csgn(I*x^2)*csg 
n(I*x^2*(4+x))^2+Pi*csgn(I*x^2*(4+x))^3+2*I*ln(4+x)))+ln(ln(x)-1/4*I*(2*Pi 
*csgn(I*(4+x))*csgn(I*x^2)*csgn(I*x^2*(4+x))-2*Pi*csgn(I*(4+x))*csgn(I*x^2 
*(4+x))^2+2*Pi*csgn(I*x)^2*csgn(I*x^2)-4*Pi*csgn(I*x)*csgn(I*x^2)^2-2*Pi*c 
sgn(I*x^2)*csgn(I*x^2*(4+x))^2-5*ln(x)*x^2*Pi*csgn(I*(4+x))*csgn(I*x^2*(4+ 
x))^2+5*ln(x)*x^2*Pi*csgn(I*x)^2*csgn(I*x^2)-10*ln(x)*x^2*Pi*csgn(I*x)*csg 
n(I*x^2)^2-5*ln(x)*x^2*Pi*csgn(I*x^2)*csgn(I*x^2*(4+x))^2-2*x*ln(x)*Pi*csg 
n(I*(4+x))*csgn(I*x^2*(4+x))^2+2*x*ln(x)*Pi*csgn(I*x)^2*csgn(I*x^2)-4*x*ln 
(x)*Pi*csgn(I*x)*csgn(I*x^2)^2-2*x*ln(x)*Pi*csgn(I*x^2)*csgn(I*x^2*(4+x))^ 
2+2*Pi*csgn(I*x^2)^3+2*Pi*csgn(I*x^2*(4+x))^3+4*I*ln(4+x)+2*I*ln(x)+2*x*ln 
(x)*Pi*csgn(I*(4+x))*csgn(I*x^2)*csgn(I*x^2*(4+x))+5*ln(x)*x^2*Pi*csgn(I*( 
4+x))*csgn(I*x^2)*csgn(I*x^2*(4+x))+5*ln(x)*x^2*Pi*csgn(I*x^2)^3+5*ln(x)*x 
^2*Pi*csgn(I*x^2*(4+x))^3+10*I*ln(x)*x^2*ln(4+x)+4*I*x*ln(x)*ln(4+x)+2*x*l 
n(x)*Pi*csgn(I*x^2)^3+2*x*ln(x)*Pi*csgn(I*x^2*(4+x))^3)/(5*x^2*ln(x)+2*x*l 
n(x)+2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (33) = 66\).

Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.97 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log \left (5 \, x + 2\right ) + \log \left (\frac {{\left ({\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2\right )} \log \left (x^{3} + 4 \, x^{2}\right ) + \log \left (x\right )}{{\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2}\right ) + \log \left (\frac {{\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2}{5 \, x^{2} + 2 \, x}\right ) - \log \left (\log \left (x^{3} + 4 \, x^{2}\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

integrate((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^ 
2+(-4-x)*log(x)^2*log(x^3+4*x^2)+(-3*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2) 
*log(x)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^3+ 
4*x^2)),x, algorithm="fricas")
 

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:

integrate((((5*x**3+20*x**2)*ln(x)**2+(-2*x-8)*ln(x)-2*x-8)*ln(x**3+4*x**2 
)**2+(-4-x)*ln(x)**2*ln(x**3+4*x**2)+(-3*x-8)*ln(x)**2)/(((5*x**4+22*x**3+ 
8*x**2)*ln(x)**2+(2*x**2+8*x)*ln(x))*ln(x**3+4*x**2)**2+(x**2+4*x)*ln(x)** 
2*ln(x**3+4*x**2)),x)
 

Output:

Exception raised: PolynomialError >> 1/(25*_t0**2*x**6 + 120*_t0**2*x**5 + 
 84*_t0**2*x**4 + 16*_t0**2*x**3 + 20*_t0*x**4 + 88*_t0*x**3 + 32*_t0*x**2 
 + 4*x**2 + 16*x) contains an element of the set of generators.
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (33) = 66\).

Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.27 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log \left (5 \, x + 2\right ) + \log \left (\frac {2 \, {\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right )^{2} + {\left ({\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2\right )} \log \left (x + 4\right ) + 5 \, \log \left (x\right )}{{\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2}\right ) + \log \left (\frac {{\left (5 \, x^{2} + 2 \, x\right )} \log \left (x\right ) + 2}{5 \, x^{2} + 2 \, x}\right ) - \log \left (\log \left (x + 4\right ) + 2 \, \log \left (x\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

integrate((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^ 
2+(-4-x)*log(x)^2*log(x^3+4*x^2)+(-3*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2) 
*log(x)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^3+ 
4*x^2)),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (33) = 66\).

Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\log \left (5 \, x^{2} \log \left (x + 4\right ) \log \left (x\right ) + 10 \, x^{2} \log \left (x\right )^{2} + 2 \, x \log \left (x + 4\right ) \log \left (x\right ) + 4 \, x \log \left (x\right )^{2} + 2 \, \log \left (x + 4\right ) + 5 \, \log \left (x\right )\right ) - \log \left (x\right ) - \log \left (\log \left (x + 4\right ) + 2 \, \log \left (x\right )\right ) - \log \left (\log \left (x\right )\right ) \] Input:

integrate((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^ 
2+(-4-x)*log(x)^2*log(x^3+4*x^2)+(-3*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2) 
*log(x)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^3+ 
4*x^2)),x, algorithm="giac")
 

Output:

log(5*x^2*log(x + 4)*log(x) + 10*x^2*log(x)^2 + 2*x*log(x + 4)*log(x) + 4* 
x*log(x)^2 + 2*log(x + 4) + 5*log(x)) - log(x) - log(log(x + 4) + 2*log(x) 
) - log(log(x))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=\int -\frac {{\ln \left (x^3+4\,x^2\right )}^2\,\left (\left (-5\,x^3-20\,x^2\right )\,{\ln \left (x\right )}^2+\left (2\,x+8\right )\,\ln \left (x\right )+2\,x+8\right )+{\ln \left (x\right )}^2\,\left (3\,x+8\right )+\ln \left (x^3+4\,x^2\right )\,{\ln \left (x\right )}^2\,\left (x+4\right )}{{\ln \left (x^3+4\,x^2\right )}^2\,\left (\left (5\,x^4+22\,x^3+8\,x^2\right )\,{\ln \left (x\right )}^2+\left (2\,x^2+8\,x\right )\,\ln \left (x\right )\right )+\ln \left (x^3+4\,x^2\right )\,{\ln \left (x\right )}^2\,\left (x^2+4\,x\right )} \,d x \] Input:

int(-(log(4*x^2 + x^3)^2*(2*x - log(x)^2*(20*x^2 + 5*x^3) + log(x)*(2*x + 
8) + 8) + log(x)^2*(3*x + 8) + log(4*x^2 + x^3)*log(x)^2*(x + 4))/(log(4*x 
^2 + x^3)^2*(log(x)^2*(8*x^2 + 22*x^3 + 5*x^4) + log(x)*(8*x + 2*x^2)) + l 
og(4*x^2 + x^3)*log(x)^2*(4*x + x^2)),x)
 

Output:

int(-(log(4*x^2 + x^3)^2*(2*x - log(x)^2*(20*x^2 + 5*x^3) + log(x)*(2*x + 
8) + 8) + log(x)^2*(3*x + 8) + log(4*x^2 + x^3)*log(x)^2*(x + 4))/(log(4*x 
^2 + x^3)^2*(log(x)^2*(8*x^2 + 22*x^3 + 5*x^4) + log(x)*(8*x + 2*x^2)) + l 
og(4*x^2 + x^3)*log(x)^2*(4*x + x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.15 \[ \int \frac {(-8-3 x) \log ^2(x)+(-4-x) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (-8-2 x+(-8-2 x) \log (x)+\left (20 x^2+5 x^3\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )}{\left (4 x+x^2\right ) \log ^2(x) \log \left (4 x^2+x^3\right )+\left (\left (8 x+2 x^2\right ) \log (x)+\left (8 x^2+22 x^3+5 x^4\right ) \log ^2(x)\right ) \log ^2\left (4 x^2+x^3\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (x^{3}+4 x^{2}\right )\right )-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (5 \,\mathrm {log}\left (x^{3}+4 x^{2}\right ) \mathrm {log}\left (x \right ) x^{2}+2 \,\mathrm {log}\left (x^{3}+4 x^{2}\right ) \mathrm {log}\left (x \right ) x +2 \,\mathrm {log}\left (x^{3}+4 x^{2}\right )+\mathrm {log}\left (x \right )\right )-\mathrm {log}\left (x \right ) \] Input:

int((((5*x^3+20*x^2)*log(x)^2+(-2*x-8)*log(x)-2*x-8)*log(x^3+4*x^2)^2+(-4- 
x)*log(x)^2*log(x^3+4*x^2)+(-3*x-8)*log(x)^2)/(((5*x^4+22*x^3+8*x^2)*log(x 
)^2+(2*x^2+8*x)*log(x))*log(x^3+4*x^2)^2+(x^2+4*x)*log(x)^2*log(x^3+4*x^2) 
),x)
 

Output:

 - log(log(x**3 + 4*x**2)) - log(log(x)) + log(5*log(x**3 + 4*x**2)*log(x) 
*x**2 + 2*log(x**3 + 4*x**2)*log(x)*x + 2*log(x**3 + 4*x**2) + log(x)) - l 
og(x)