\(\int \frac {1+\log (x)+(x-x^2-x^3-4 x^4-x^5) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{(x^2-x^3-x^5) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx\) [1051]

Optimal result

Integrand size = 126, antiderivative size = 23 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x+\log \left (-x+x^2+x^4-\log (\log (\log (-x \log (x))))\right ) \] Output:

ln(x^2+x^4-x-ln(ln(ln(-x*ln(x)))))+x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x+\log \left (-x+x^2+x^4-\log (\log (\log (-x \log (x))))\right ) \] Input:

Integrate[(1 + Log[x] + (x - x^2 - x^3 - 4*x^4 - x^5)*Log[x]*Log[-(x*Log[x 
])]*Log[Log[-(x*Log[x])]] + x*Log[x]*Log[-(x*Log[x])]*Log[Log[-(x*Log[x])] 
]*Log[Log[Log[-(x*Log[x])]]])/((x^2 - x^3 - x^5)*Log[x]*Log[-(x*Log[x])]*L 
og[Log[-(x*Log[x])]] + x*Log[x]*Log[-(x*Log[x])]*Log[Log[-(x*Log[x])]]*Log 
[Log[Log[-(x*Log[x])]]]),x]
 

Output:

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

Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {7292, 7293, 2009}

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.

\(\displaystyle \int \frac {\left (-x^5-4 x^4-x^3-x^2+x\right ) \log (-x \log (x)) \log (\log (-x \log (x))) \log (x)+x \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x)))) \log (x)+\log (x)+1}{\left (-x^5-x^3+x^2\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (\log (-x \log (x)))) \log (\log (-x \log (x)))} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-x^5-4 x^4-x^3-x^2+x\right ) \log (-x \log (x)) \log (\log (-x \log (x))) \log (x)+x \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x)))) \log (x)+\log (x)+1}{x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \left (-x^4-x^2+x+\log (\log (\log (-x \log (x))))\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 x^4 \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+2 x^2 \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))-x \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))-\log (x)-1}{x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \left (x^4+x^2-x-\log (\log (\log (-x \log (x))))\right )}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \log \left (-x^4-x^2+x+\log (\log (\log (-x \log (x))))\right )+x\)

Input:

Int[(1 + Log[x] + (x - x^2 - x^3 - 4*x^4 - x^5)*Log[x]*Log[-(x*Log[x])]*Lo 
g[Log[-(x*Log[x])]] + x*Log[x]*Log[-(x*Log[x])]*Log[Log[-(x*Log[x])]]*Log[ 
Log[Log[-(x*Log[x])]]])/((x^2 - x^3 - x^5)*Log[x]*Log[-(x*Log[x])]*Log[Log 
[-(x*Log[x])]] + x*Log[x]*Log[-(x*Log[x])]*Log[Log[-(x*Log[x])]]*Log[Log[L 
og[-(x*Log[x])]]]),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [C] (warning: unable to verify)

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

Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00

\[x +\ln \left (-x^{4}-x^{2}+x +\ln \left (\ln \left (i \pi +\ln \left (x \right )+\ln \left (\ln \left (x \right )\right )-\frac {i \pi \,\operatorname {csgn}\left (i x \ln \left (x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \ln \left (x \right )\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right )}{2}+i \pi \operatorname {csgn}\left (i x \ln \left (x \right )\right )^{2} \left (\operatorname {csgn}\left (i x \ln \left (x \right )\right )-1\right )\right )\right )\right )\]

Input:

int((x*ln(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))*ln(ln(ln(-x*ln(x))))+(-x^5-4*x^ 
4-x^3-x^2+x)*ln(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))+ln(x)+1)/(x*ln(x)*ln(-x*l 
n(x))*ln(ln(-x*ln(x)))*ln(ln(ln(-x*ln(x))))+(-x^5-x^3+x^2)*ln(x)*ln(-x*ln( 
x))*ln(ln(-x*ln(x)))),x)
 

Output:

x+ln(-x^4-x^2+x+ln(ln(I*Pi+ln(x)+ln(ln(x))-1/2*I*Pi*csgn(I*x*ln(x))*(-csgn 
(I*x*ln(x))+csgn(I*x))*(-csgn(I*x*ln(x))+csgn(I*ln(x)))+I*Pi*csgn(I*x*ln(x 
))^2*(csgn(I*x*ln(x))-1))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x + \log \left (-x^{4} - x^{2} + x + \log \left (\log \left (\log \left (-x \log \left (x\right )\right )\right )\right )\right ) \] Input:

integrate((x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log( 
x))))+(-x^5-4*x^4-x^3-x^2+x)*log(x)*log(-x*log(x))*log(log(-x*log(x)))+log 
(x)+1)/(x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x)) 
))+(-x^5-x^3+x^2)*log(x)*log(-x*log(x))*log(log(-x*log(x)))),x, algorithm= 
"fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x + \log {\left (- x^{4} - x^{2} + x + \log {\left (\log {\left (\log {\left (- x \log {\left (x \right )} \right )} \right )} \right )} \right )} \] Input:

integrate((x*ln(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))*ln(ln(ln(-x*ln(x))))+(-x* 
*5-4*x**4-x**3-x**2+x)*ln(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))+ln(x)+1)/(x*ln( 
x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))*ln(ln(ln(-x*ln(x))))+(-x**5-x**3+x**2)*ln 
(x)*ln(-x*ln(x))*ln(ln(-x*ln(x)))),x)
                                                                                    
                                                                                    
 

Output:

x + log(-x**4 - x**2 + x + log(log(log(-x*log(x)))))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x + \log \left (-x^{4} - x^{2} + x + \log \left (\log \left (\log \left (x\right ) + \log \left (-\log \left (x\right )\right )\right )\right )\right ) \] Input:

integrate((x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log( 
x))))+(-x^5-4*x^4-x^3-x^2+x)*log(x)*log(-x*log(x))*log(log(-x*log(x)))+log 
(x)+1)/(x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x)) 
))+(-x^5-x^3+x^2)*log(x)*log(-x*log(x))*log(log(-x*log(x)))),x, algorithm= 
"maxima")
 

Output:

x + log(-x^4 - x^2 + x + log(log(log(x) + log(-log(x)))))
 

Giac [A] (verification not implemented)

Time = 3.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=x + \log \left (-x^{4} - x^{2} + x + \log \left (\log \left (\log \left (x\right ) + \log \left (-\log \left (x\right )\right )\right )\right )\right ) \] Input:

integrate((x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log( 
x))))+(-x^5-4*x^4-x^3-x^2+x)*log(x)*log(-x*log(x))*log(log(-x*log(x)))+log 
(x)+1)/(x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x)) 
))+(-x^5-x^3+x^2)*log(x)*log(-x*log(x))*log(log(-x*log(x)))),x, algorithm= 
"giac")
 

Output:

x + log(-x^4 - x^2 + x + log(log(log(x) + log(-log(x)))))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=-\int \frac {\ln \left (x\right )-\ln \left (-x\,\ln \left (x\right )\right )\,\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\,\ln \left (x\right )\,\left (x^5+4\,x^4+x^3+x^2-x\right )+x\,\ln \left (\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\right )\,\ln \left (-x\,\ln \left (x\right )\right )\,\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\,\ln \left (x\right )+1}{\ln \left (-x\,\ln \left (x\right )\right )\,\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\,\ln \left (x\right )\,\left (x^5+x^3-x^2\right )-x\,\ln \left (\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\right )\,\ln \left (-x\,\ln \left (x\right )\right )\,\ln \left (\ln \left (-x\,\ln \left (x\right )\right )\right )\,\ln \left (x\right )} \,d x \] Input:

int(-(log(x) - log(-x*log(x))*log(log(-x*log(x)))*log(x)*(x^2 - x + x^3 + 
4*x^4 + x^5) + x*log(log(log(-x*log(x))))*log(-x*log(x))*log(log(-x*log(x) 
))*log(x) + 1)/(log(-x*log(x))*log(log(-x*log(x)))*log(x)*(x^3 - x^2 + x^5 
) - x*log(log(log(-x*log(x))))*log(-x*log(x))*log(log(-x*log(x)))*log(x)), 
x)
 

Output:

-int((log(x) - log(-x*log(x))*log(log(-x*log(x)))*log(x)*(x^2 - x + x^3 + 
4*x^4 + x^5) + x*log(log(log(-x*log(x))))*log(-x*log(x))*log(log(-x*log(x) 
))*log(x) + 1)/(log(-x*log(x))*log(log(-x*log(x)))*log(x)*(x^3 - x^2 + x^5 
) - x*log(log(log(-x*log(x))))*log(-x*log(x))*log(log(-x*log(x)))*log(x)), 
 x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1+\log (x)+\left (x-x^2-x^3-4 x^4-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))}{\left (x^2-x^3-x^5\right ) \log (x) \log (-x \log (x)) \log (\log (-x \log (x)))+x \log (x) \log (-x \log (x)) \log (\log (-x \log (x))) \log (\log (\log (-x \log (x))))} \, dx=\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (-\mathrm {log}\left (x \right ) x \right )\right )\right )-x^{4}-x^{2}+x \right )+x \] Input:

int((x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x))))+ 
(-x^5-4*x^4-x^3-x^2+x)*log(x)*log(-x*log(x))*log(log(-x*log(x)))+log(x)+1) 
/(x*log(x)*log(-x*log(x))*log(log(-x*log(x)))*log(log(log(-x*log(x))))+(-x 
^5-x^3+x^2)*log(x)*log(-x*log(x))*log(log(-x*log(x)))),x)
 

Output:

log(log(log(log( - log(x)*x))) - x**4 - x**2 + x) + x