\(\int \frac {-32+(32-2 x) \log (x)+(2 x \log (x)-32 \log (x) \log (\frac {x}{\log (x)})) \log (-x+16 \log (\frac {x}{\log (x)}))}{-x^3 \log (x)+16 x^2 \log (x) \log (\frac {x}{\log (x)})+(-2 x^2 \log (x)+32 x \log (x) \log (\frac {x}{\log (x)})) \log (-x+16 \log (\frac {x}{\log (x)}))+(-x \log (x)+16 \log (x) \log (\frac {x}{\log (x)})) \log ^2(-x+16 \log (\frac {x}{\log (x)}))} \, dx\) [1406]

Optimal result

Integrand size = 136, antiderivative size = 25 \[ \int \frac {-32+(32-2 x) \log (x)+\left (2 x \log (x)-32 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )}{-x^3 \log (x)+16 x^2 \log (x) \log \left (\frac {x}{\log (x)}\right )+\left (-2 x^2 \log (x)+32 x \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )+\left (-x \log (x)+16 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log ^2\left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \, dx=-5+\log (5)-\frac {2 x}{x+\log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \] Output:

ln(5)-2*x/(x+ln(16*ln(x/ln(x))-x))-5
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {-32+(32-2 x) \log (x)+\left (2 x \log (x)-32 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )}{-x^3 \log (x)+16 x^2 \log (x) \log \left (\frac {x}{\log (x)}\right )+\left (-2 x^2 \log (x)+32 x \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )+\left (-x \log (x)+16 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log ^2\left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \, dx=-\frac {2 x}{x+\log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \] Input:

Integrate[(-32 + (32 - 2*x)*Log[x] + (2*x*Log[x] - 32*Log[x]*Log[x/Log[x]] 
)*Log[-x + 16*Log[x/Log[x]]])/(-(x^3*Log[x]) + 16*x^2*Log[x]*Log[x/Log[x]] 
 + (-2*x^2*Log[x] + 32*x*Log[x]*Log[x/Log[x]])*Log[-x + 16*Log[x/Log[x]]] 
+ (-(x*Log[x]) + 16*Log[x]*Log[x/Log[x]])*Log[-x + 16*Log[x/Log[x]]]^2),x]
 

Output:

(-2*x)/(x + Log[-x + 16*Log[x/Log[x]]])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {7239, 7262, 17}

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.

\(\displaystyle \int \frac {(32-2 x) \log (x)+\left (2 x \log (x)-32 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (16 \log \left (\frac {x}{\log (x)}\right )-x\right )-32}{x^3 (-\log (x))+16 x^2 \log (x) \log \left (\frac {x}{\log (x)}\right )+\left (32 x \log (x) \log \left (\frac {x}{\log (x)}\right )-2 x^2 \log (x)\right ) \log \left (16 \log \left (\frac {x}{\log (x)}\right )-x\right )+\left (16 \log (x) \log \left (\frac {x}{\log (x)}\right )-x \log (x)\right ) \log ^2\left (16 \log \left (\frac {x}{\log (x)}\right )-x\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \log (x) \left (x-\left (x-16 \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (16 \log \left (\frac {x}{\log (x)}\right )-x\right )-16\right )+32}{\log (x) \left (x-16 \log \left (\frac {x}{\log (x)}\right )\right ) \left (x+\log \left (16 \log \left (\frac {x}{\log (x)}\right )-x\right )\right )^2}dx\)

\(\Big \downarrow \) 7262

\(\displaystyle -2 \int \frac {1}{\left (\frac {x}{\log \left (16 \log \left (\frac {x}{\log (x)}\right )-x\right )}+1\right )^2}d\frac {x}{\log \left (16 \log \left (\frac {x}{\log (x)}\right )-x\right )}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {2}{\frac {x}{\log \left (16 \log \left (\frac {x}{\log (x)}\right )-x\right )}+1}\)

Input:

Int[(-32 + (32 - 2*x)*Log[x] + (2*x*Log[x] - 32*Log[x]*Log[x/Log[x]])*Log[ 
-x + 16*Log[x/Log[x]]])/(-(x^3*Log[x]) + 16*x^2*Log[x]*Log[x/Log[x]] + (-2 
*x^2*Log[x] + 32*x*Log[x]*Log[x/Log[x]])*Log[-x + 16*Log[x/Log[x]]] + (-(x 
*Log[x]) + 16*Log[x]*Log[x/Log[x]])*Log[-x + 16*Log[x/Log[x]]]^2),x]
 

Output:

2/(1 + x/Log[-x + 16*Log[x/Log[x]]])
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7262

Int[(u_)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c 
= Simplify[u/(p*w*D[v, x] - q*v*D[w, x])]}, Simp[c*p   Subst[Int[(b + a*x^p 
)^m, x], x, v*w^(m*q + 1)], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p, q}, x] 
 && EqQ[p + q*(m*p + 1), 0] && IntegerQ[p] && IntegerQ[m]
 

Maple [A] (verified)

Time = 7.60 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88

method	result	size
parallelrisch	\(-\frac {2 x}{x +\ln \left (16 \ln \left (\frac {x}{\ln \left (x \right )}\right )-x \right )}\)	\(22\)
risch	\(-\frac {2 x}{x +\ln \left (-16 \ln \left (\ln \left (x \right )\right )+16 \ln \left (x \right )-8 i \pi \,\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (i x \right )\right )-x \right )}\)	\(72\)
default	\(-\frac {32 x}{\left (8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) x -8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x -8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x +8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3} x +x^{2} \ln \left (x \right )+16 x \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-16 x \ln \left (x \right )^{2}+x \ln \left (x \right )-16 \ln \left (x \right )+16\right ) \left (x +\ln \left (-16 \ln \left (\ln \left (x \right )\right )+16 \ln \left (x \right )-8 i \pi \,\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (i x \right )\right )-x \right )\right )}-\frac {2 \ln \left (x \right ) x \left (8 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) x -8 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x -8 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x +8 i \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3} x +x^{2}+16 x \ln \left (\ln \left (x \right )\right )-16 x \ln \left (x \right )+x -16\right )}{\left (8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) x -8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x -8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x +8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3} x +x^{2} \ln \left (x \right )+16 x \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-16 x \ln \left (x \right )^{2}+x \ln \left (x \right )-16 \ln \left (x \right )+16\right ) \left (x +\ln \left (-16 \ln \left (\ln \left (x \right )\right )+16 \ln \left (x \right )-8 i \pi \,\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (i x \right )\right )-x \right )\right )}\)	\(509\)
parts	\(-\frac {32 x}{\left (8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) x -8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x -8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x +8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3} x +x^{2} \ln \left (x \right )+16 x \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-16 x \ln \left (x \right )^{2}+x \ln \left (x \right )-16 \ln \left (x \right )+16\right ) \left (x +\ln \left (-16 \ln \left (\ln \left (x \right )\right )+16 \ln \left (x \right )-8 i \pi \,\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (i x \right )\right )-x \right )\right )}-\frac {2 \ln \left (x \right ) x \left (8 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) x -8 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x -8 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x +8 i \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3} x +x^{2}+16 x \ln \left (\ln \left (x \right )\right )-16 x \ln \left (x \right )+x -16\right )}{\left (8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) x -8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x -8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{2} x +8 i \pi \ln \left (x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )^{3} x +x^{2} \ln \left (x \right )+16 x \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-16 x \ln \left (x \right )^{2}+x \ln \left (x \right )-16 \ln \left (x \right )+16\right ) \left (x +\ln \left (-16 \ln \left (\ln \left (x \right )\right )+16 \ln \left (x \right )-8 i \pi \,\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )}\right )+\operatorname {csgn}\left (i x \right )\right )-x \right )\right )}\)	\(509\)

Input:

int(((-32*ln(x)*ln(x/ln(x))+2*x*ln(x))*ln(16*ln(x/ln(x))-x)+(-2*x+32)*ln(x 
)-32)/((16*ln(x)*ln(x/ln(x))-x*ln(x))*ln(16*ln(x/ln(x))-x)^2+(32*x*ln(x)*l 
n(x/ln(x))-2*x^2*ln(x))*ln(16*ln(x/ln(x))-x)+16*x^2*ln(x)*ln(x/ln(x))-x^3* 
ln(x)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {-32+(32-2 x) \log (x)+\left (2 x \log (x)-32 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )}{-x^3 \log (x)+16 x^2 \log (x) \log \left (\frac {x}{\log (x)}\right )+\left (-2 x^2 \log (x)+32 x \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )+\left (-x \log (x)+16 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log ^2\left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \, dx=-\frac {2 \, x}{x + \log \left (-x + 16 \, \log \left (\frac {x}{\log \left (x\right )}\right )\right )} \] Input:

integrate(((-32*log(x)*log(x/log(x))+2*x*log(x))*log(16*log(x/log(x))-x)+( 
-2*x+32)*log(x)-32)/((16*log(x)*log(x/log(x))-x*log(x))*log(16*log(x/log(x 
))-x)^2+(32*x*log(x)*log(x/log(x))-2*x^2*log(x))*log(16*log(x/log(x))-x)+1 
6*x^2*log(x)*log(x/log(x))-x^3*log(x)),x, algorithm="fricas")
 

Output:

-2*x/(x + log(-x + 16*log(x/log(x))))
 

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {-32+(32-2 x) \log (x)+\left (2 x \log (x)-32 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )}{-x^3 \log (x)+16 x^2 \log (x) \log \left (\frac {x}{\log (x)}\right )+\left (-2 x^2 \log (x)+32 x \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )+\left (-x \log (x)+16 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log ^2\left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \, dx=- \frac {2 x}{x + \log {\left (- x + 16 \log {\left (\frac {x}{\log {\left (x \right )}} \right )} \right )}} \] Input:

integrate(((-32*ln(x)*ln(x/ln(x))+2*x*ln(x))*ln(16*ln(x/ln(x))-x)+(-2*x+32 
)*ln(x)-32)/((16*ln(x)*ln(x/ln(x))-x*ln(x))*ln(16*ln(x/ln(x))-x)**2+(32*x* 
ln(x)*ln(x/ln(x))-2*x**2*ln(x))*ln(16*ln(x/ln(x))-x)+16*x**2*ln(x)*ln(x/ln 
(x))-x**3*ln(x)),x)
 

Output:

-2*x/(x + log(-x + 16*log(x/log(x))))
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {-32+(32-2 x) \log (x)+\left (2 x \log (x)-32 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )}{-x^3 \log (x)+16 x^2 \log (x) \log \left (\frac {x}{\log (x)}\right )+\left (-2 x^2 \log (x)+32 x \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )+\left (-x \log (x)+16 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log ^2\left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \, dx=-\frac {2 \, x}{x + \log \left (-x + 16 \, \log \left (x\right ) - 16 \, \log \left (\log \left (x\right )\right )\right )} \] Input:

integrate(((-32*log(x)*log(x/log(x))+2*x*log(x))*log(16*log(x/log(x))-x)+( 
-2*x+32)*log(x)-32)/((16*log(x)*log(x/log(x))-x*log(x))*log(16*log(x/log(x 
))-x)^2+(32*x*log(x)*log(x/log(x))-2*x^2*log(x))*log(16*log(x/log(x))-x)+1 
6*x^2*log(x)*log(x/log(x))-x^3*log(x)),x, algorithm="maxima")
 

Output:

-2*x/(x + log(-x + 16*log(x) - 16*log(log(x))))
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {-32+(32-2 x) \log (x)+\left (2 x \log (x)-32 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )}{-x^3 \log (x)+16 x^2 \log (x) \log \left (\frac {x}{\log (x)}\right )+\left (-2 x^2 \log (x)+32 x \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )+\left (-x \log (x)+16 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log ^2\left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \, dx=-\frac {2 \, x}{x + \log \left (-x + 16 \, \log \left (x\right ) - 16 \, \log \left (\log \left (x\right )\right )\right )} \] Input:

integrate(((-32*log(x)*log(x/log(x))+2*x*log(x))*log(16*log(x/log(x))-x)+( 
-2*x+32)*log(x)-32)/((16*log(x)*log(x/log(x))-x*log(x))*log(16*log(x/log(x 
))-x)^2+(32*x*log(x)*log(x/log(x))-2*x^2*log(x))*log(16*log(x/log(x))-x)+1 
6*x^2*log(x)*log(x/log(x))-x^3*log(x)),x, algorithm="giac")
 

Output:

-2*x/(x + log(-x + 16*log(x) - 16*log(log(x))))
 

Mupad [B] (verification not implemented)

Time = 3.66 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {-32+(32-2 x) \log (x)+\left (2 x \log (x)-32 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )}{-x^3 \log (x)+16 x^2 \log (x) \log \left (\frac {x}{\log (x)}\right )+\left (-2 x^2 \log (x)+32 x \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )+\left (-x \log (x)+16 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log ^2\left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \, dx=-\frac {2\,x}{x+\ln \left (16\,\ln \left (\frac {x}{\ln \left (x\right )}\right )-x\right )} \] Input:

int((log(x)*(2*x - 32) - log(16*log(x/log(x)) - x)*(2*x*log(x) - 32*log(x/ 
log(x))*log(x)) + 32)/(x^3*log(x) + log(16*log(x/log(x)) - x)^2*(x*log(x) 
- 16*log(x/log(x))*log(x)) + log(16*log(x/log(x)) - x)*(2*x^2*log(x) - 32* 
x*log(x/log(x))*log(x)) - 16*x^2*log(x/log(x))*log(x)),x)
 

Output:

-(2*x)/(x + log(16*log(x/log(x)) - x))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-32+(32-2 x) \log (x)+\left (2 x \log (x)-32 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )}{-x^3 \log (x)+16 x^2 \log (x) \log \left (\frac {x}{\log (x)}\right )+\left (-2 x^2 \log (x)+32 x \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log \left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )+\left (-x \log (x)+16 \log (x) \log \left (\frac {x}{\log (x)}\right )\right ) \log ^2\left (-x+16 \log \left (\frac {x}{\log (x)}\right )\right )} \, dx=\frac {2 \,\mathrm {log}\left (16 \,\mathrm {log}\left (\frac {x}{\mathrm {log}\left (x \right )}\right )-x \right )}{\mathrm {log}\left (16 \,\mathrm {log}\left (\frac {x}{\mathrm {log}\left (x \right )}\right )-x \right )+x} \] Input:

int(((-32*log(x)*log(x/log(x))+2*x*log(x))*log(16*log(x/log(x))-x)+(-2*x+3 
2)*log(x)-32)/((16*log(x)*log(x/log(x))-x*log(x))*log(16*log(x/log(x))-x)^ 
2+(32*x*log(x)*log(x/log(x))-2*x^2*log(x))*log(16*log(x/log(x))-x)+16*x^2* 
log(x)*log(x/log(x))-x^3*log(x)),x)
 

Output:

(2*log(16*log(x/log(x)) - x))/(log(16*log(x/log(x)) - x) + x)