3.7.97 \(\int \frac {-3 x+3 x^2-3 \log (x)+(-3 x^2-3 x \log (x)) \log (\log (x))+(3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+(-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+(-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx\) [697]

3.7.97.1 Optimal result

Integrand size = 191, antiderivative size = 24 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 x}{\log (x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \]

output

3*x/ln(x)/(x+ln(3*ln(x)+3*(ln(ln(x))-1)*x))
 

3.7.97.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 x}{\log (x) (x+\log (3 (\log (x)+x (-1+\log (\log (x))))))} \]

input

Integrate[(-3*x + 3*x^2 - 3*Log[x] + (-3*x^2 - 3*x*Log[x])*Log[Log[x]] + ( 
3*x + (-3 - 3*x)*Log[x] + 3*Log[x]^2 + (-3*x + 3*x*Log[x])*Log[Log[x]])*Lo 
g[-3*x + 3*Log[x] + 3*x*Log[Log[x]]])/(-(x^3*Log[x]^2) + x^2*Log[x]^3 + x^ 
3*Log[x]^2*Log[Log[x]] + (-2*x^2*Log[x]^2 + 2*x*Log[x]^3 + 2*x^2*Log[x]^2* 
Log[Log[x]])*Log[-3*x + 3*Log[x] + 3*x*Log[Log[x]]] + (-(x*Log[x]^2) + Log 
[x]^3 + x*Log[x]^2*Log[Log[x]])*Log[-3*x + 3*Log[x] + 3*x*Log[Log[x]]]^2), 
x]
 

output

(3*x)/(Log[x]*(x + Log[3*(Log[x] + x*(-1 + Log[Log[x]]))]))
 

3.7.97.3 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.

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

Int[(-3*x + 3*x^2 - 3*Log[x] + (-3*x^2 - 3*x*Log[x])*Log[Log[x]] + (3*x + 
(-3 - 3*x)*Log[x] + 3*Log[x]^2 + (-3*x + 3*x*Log[x])*Log[Log[x]])*Log[-3*x 
 + 3*Log[x] + 3*x*Log[Log[x]]])/(-(x^3*Log[x]^2) + x^2*Log[x]^3 + x^3*Log[ 
x]^2*Log[Log[x]] + (-2*x^2*Log[x]^2 + 2*x*Log[x]^3 + 2*x^2*Log[x]^2*Log[Lo 
g[x]])*Log[-3*x + 3*Log[x] + 3*x*Log[Log[x]]] + (-(x*Log[x]^2) + Log[x]^3 
+ x*Log[x]^2*Log[Log[x]])*Log[-3*x + 3*Log[x] + 3*x*Log[Log[x]]]^2),x]
 

output

$Aborted
 

3.7.97.3.1 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] 
]
 

3.7.97.4 Maple [A] (verified)

Time = 25.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08

method	result	size
default	\(\frac {3 x}{\ln \left (x \right ) \left (\ln \left (3\right )+x +\ln \left (x \ln \left (\ln \left (x \right )\right )+\ln \left (x \right )-x \right )\right )}\)	\(26\)
risch	\(\frac {3 x}{\ln \left (x \right ) \left (\ln \left (3 x \ln \left (\ln \left (x \right )\right )+3 \ln \left (x \right )-3 x \right )+x \right )}\)	\(27\)
parallelrisch	\(\frac {3 x}{\ln \left (x \right ) \left (\ln \left (3 x \ln \left (\ln \left (x \right )\right )+3 \ln \left (x \right )-3 x \right )+x \right )}\)	\(27\)

input

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

output

3*x/ln(x)/(ln(3)+x+ln(x*ln(ln(x))+ln(x)-x))
 

3.7.97.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 \, x}{x \log \left (x\right ) + \log \left (3 \, x \log \left (\log \left (x\right )\right ) - 3 \, x + 3 \, \log \left (x\right )\right ) \log \left (x\right )} \]

input

integrate((((3*x*log(x)-3*x)*log(log(x))+3*log(x)^2+(-3*x-3)*log(x)+3*x)*l 
og(3*x*log(log(x))+3*log(x)-3*x)+(-3*x*log(x)-3*x^2)*log(log(x))-3*log(x)+ 
3*x^2-3*x)/((x*log(x)^2*log(log(x))+log(x)^3-x*log(x)^2)*log(3*x*log(log(x 
))+3*log(x)-3*x)^2+(2*x^2*log(x)^2*log(log(x))+2*x*log(x)^3-2*x^2*log(x)^2 
)*log(3*x*log(log(x))+3*log(x)-3*x)+x^3*log(x)^2*log(log(x))+x^2*log(x)^3- 
x^3*log(x)^2),x, algorithm=\
 

output

3*x/(x*log(x) + log(3*x*log(log(x)) - 3*x + 3*log(x))*log(x))
 

3.7.97.6 Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 x}{x \log {\left (x \right )} + \log {\left (x \right )} \log {\left (3 x \log {\left (\log {\left (x \right )} \right )} - 3 x + 3 \log {\left (x \right )} \right )}} \]

input

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

output

3*x/(x*log(x) + log(x)*log(3*x*log(log(x)) - 3*x + 3*log(x)))
 

3.7.97.7 Maxima [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 \, x}{{\left (x + \log \left (3\right )\right )} \log \left (x\right ) + \log \left (x \log \left (\log \left (x\right )\right ) - x + \log \left (x\right )\right ) \log \left (x\right )} \]

input

integrate((((3*x*log(x)-3*x)*log(log(x))+3*log(x)^2+(-3*x-3)*log(x)+3*x)*l 
og(3*x*log(log(x))+3*log(x)-3*x)+(-3*x*log(x)-3*x^2)*log(log(x))-3*log(x)+ 
3*x^2-3*x)/((x*log(x)^2*log(log(x))+log(x)^3-x*log(x)^2)*log(3*x*log(log(x 
))+3*log(x)-3*x)^2+(2*x^2*log(x)^2*log(log(x))+2*x*log(x)^3-2*x^2*log(x)^2 
)*log(3*x*log(log(x))+3*log(x)-3*x)+x^3*log(x)^2*log(log(x))+x^2*log(x)^3- 
x^3*log(x)^2),x, algorithm=\
 

output

3*x/((x + log(3))*log(x) + log(x*log(log(x)) - x + log(x))*log(x))
 

3.7.97.8 Giac [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-3 x+3 x^2-3 \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log (\log (x))+\left (3 x+(-3-3 x) \log (x)+3 \log ^2(x)+(-3 x+3 x \log (x)) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))}{-x^3 \log ^2(x)+x^2 \log ^3(x)+x^3 \log ^2(x) \log (\log (x))+\left (-2 x^2 \log ^2(x)+2 x \log ^3(x)+2 x^2 \log ^2(x) \log (\log (x))\right ) \log (-3 x+3 \log (x)+3 x \log (\log (x)))+\left (-x \log ^2(x)+\log ^3(x)+x \log ^2(x) \log (\log (x))\right ) \log ^2(-3 x+3 \log (x)+3 x \log (\log (x)))} \, dx=\frac {3 \, x}{x \log \left (x\right ) + \log \left (3 \, x \log \left (\log \left (x\right )\right ) - 3 \, x + 3 \, \log \left (x\right )\right ) \log \left (x\right )} \]

input

integrate((((3*x*log(x)-3*x)*log(log(x))+3*log(x)^2+(-3*x-3)*log(x)+3*x)*l 
og(3*x*log(log(x))+3*log(x)-3*x)+(-3*x*log(x)-3*x^2)*log(log(x))-3*log(x)+ 
3*x^2-3*x)/((x*log(x)^2*log(log(x))+log(x)^3-x*log(x)^2)*log(3*x*log(log(x 
))+3*log(x)-3*x)^2+(2*x^2*log(x)^2*log(log(x))+2*x*log(x)^3-2*x^2*log(x)^2 
)*log(3*x*log(log(x))+3*log(x)-3*x)+x^3*log(x)^2*log(log(x))+x^2*log(x)^3- 
x^3*log(x)^2),x, algorithm=\
 

output

3*x/(x*log(x) + log(3*x*log(log(x)) - 3*x + 3*log(x))*log(x))
 

3.7.97.9 Mupad [F(-1)]

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

input

int(-(3*x + 3*log(x) + log(log(x))*(3*x*log(x) + 3*x^2) - log(3*log(x) - 3 
*x + 3*x*log(log(x)))*(3*x + 3*log(x)^2 - log(x)*(3*x + 3) - log(log(x))*( 
3*x - 3*x*log(x))) - 3*x^2)/(log(3*log(x) - 3*x + 3*x*log(log(x)))*(2*x*lo 
g(x)^3 - 2*x^2*log(x)^2 + 2*x^2*log(log(x))*log(x)^2) + x^2*log(x)^3 - x^3 
*log(x)^2 + log(3*log(x) - 3*x + 3*x*log(log(x)))^2*(log(x)^3 - x*log(x)^2 
 + x*log(log(x))*log(x)^2) + x^3*log(log(x))*log(x)^2),x)
 

output

int(-(3*x + 3*log(x) + log(log(x))*(3*x*log(x) + 3*x^2) - log(3*log(x) - 3 
*x + 3*x*log(log(x)))*(3*x + 3*log(x)^2 - log(x)*(3*x + 3) - log(log(x))*( 
3*x - 3*x*log(x))) - 3*x^2)/(log(3*log(x) - 3*x + 3*x*log(log(x)))*(2*x*lo 
g(x)^3 - 2*x^2*log(x)^2 + 2*x^2*log(log(x))*log(x)^2) + x^2*log(x)^3 - x^3 
*log(x)^2 + log(3*log(x) - 3*x + 3*x*log(log(x)))^2*(log(x)^3 - x*log(x)^2 
 + x*log(log(x))*log(x)^2) + x^3*log(log(x))*log(x)^2), x)