3.5.83 \(\int \frac {-16-8 x^2 \log (x)+(-8 x^2 \log (x)-32 \log (x) \log (\log (x))) \log (\frac {1}{4} (x^2+4 \log (\log (x))))}{(9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))) \log ^3(\frac {1}{4} (x^2+4 \log (\log (x))))} \, dx\) [483]

3.5.83.1 Optimal result

Integrand size = 76, antiderivative size = 21 \begin {dmath*} \int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx=\frac {2}{9 x^4 \log ^2\left (\frac {x^2}{4}+\log (\log (x))\right )} \end {dmath*}

output

2/9/x^4/ln(ln(ln(x))+1/4*x^2)^2
 

3.5.83.2 Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx=\frac {2}{9 x^4 \log ^2\left (\frac {x^2}{4}+\log (\log (x))\right )} \end {dmath*}

input

Integrate[(-16 - 8*x^2*Log[x] + (-8*x^2*Log[x] - 32*Log[x]*Log[Log[x]])*Lo 
g[(x^2 + 4*Log[Log[x]])/4])/((9*x^7*Log[x] + 36*x^5*Log[x]*Log[Log[x]])*Lo 
g[(x^2 + 4*Log[Log[x]])/4]^3),x]
 

output

2/(9*x^4*Log[x^2/4 + Log[Log[x]]]^2)
 

3.5.83.3 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.

\(\displaystyle \int \frac {-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )-16}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )-16}{9 x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int -\frac {8 \left (\log (x) x^2+\left (\log (x) x^2+4 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )+2\right )}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {8}{9} \int \frac {\log (x) x^2+\left (\log (x) x^2+4 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )+2}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {8}{9} \int \left (\frac {\log (x) x^2+2}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}+\frac {1}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {8}{9} \left (2 \int \frac {1}{x^5 \log (x) \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}dx+\int \frac {1}{x^5 \log ^2\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}dx+\int \frac {1}{x^3 \left (x^2+4 \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}dx\right )\)

input

Int[(-16 - 8*x^2*Log[x] + (-8*x^2*Log[x] - 32*Log[x]*Log[Log[x]])*Log[(x^2 
 + 4*Log[Log[x]])/4])/((9*x^7*Log[x] + 36*x^5*Log[x]*Log[Log[x]])*Log[(x^2 
 + 4*Log[Log[x]])/4]^3),x]
 

output

$Aborted
 

3.5.83.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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] 
]
 

3.5.83.4 Maple [A] (verified)

Time = 37.67 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
risch	\(\frac {2}{9 x^{4} \ln \left (\ln \left (\ln \left (x \right )\right )+\frac {x^{2}}{4}\right )^{2}}\)	\(18\)
parallelrisch	\(\frac {2}{9 x^{4} \ln \left (\ln \left (\ln \left (x \right )\right )+\frac {x^{2}}{4}\right )^{2}}\)	\(18\)
default	\(-\frac {2 \ln \left (x \right ) \left (-x^{6} \ln \left (x \right )^{2}+12 \ln \left (2\right ) x^{4} \ln \left (x \right )-6 x^{4} \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right ) \ln \left (x \right )+48 \ln \left (2\right ) x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (x \right )+2 x^{4} \ln \left (2\right )-4 x^{4} \ln \left (x \right )-24 x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right ) \ln \left (x \right )-x^{4} \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right )+8 \ln \left (2\right ) x^{2} \ln \left (\ln \left (x \right )\right )-4 x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right )+16 x^{2} \ln \left (2\right )-8 \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right ) x^{2}+64 \ln \left (2\right ) \ln \left (\ln \left (x \right )\right )-4 x^{2}-32 \ln \left (\ln \left (x \right )\right ) \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right )\right )}{9 \left (x^{2} \ln \left (x \right )+2\right )^{3} x^{4} {\left (2 \ln \left (2\right )-\ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right )\right )}^{2}}+\frac {\frac {8 \ln \left (2\right ) \ln \left (x \right )^{2} x^{4}}{3}-\frac {4 \ln \left (x \right )^{2} x^{4} \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right )}{3}+\frac {32 \ln \left (2\right ) \ln \left (x \right )^{2} x^{2} \ln \left (\ln \left (x \right )\right )}{3}+\frac {4 \ln \left (2\right ) x^{4} \ln \left (x \right )}{9}+\frac {4 x^{4} \ln \left (x \right )^{2}}{9}-\frac {16 \ln \left (x \right )^{2} x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right )}{3}-\frac {2 x^{4} \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right ) \ln \left (x \right )}{9}+\frac {16 \ln \left (2\right ) x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (x \right )}{9}-\frac {8 x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right ) \ln \left (x \right )}{9}+\frac {32 x^{2} \ln \left (2\right ) \ln \left (x \right )}{9}-\frac {16 x^{2} \ln \left (x \right ) \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right )}{9}+\frac {128 \ln \left (2\right ) \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )}{9}+\frac {16 x^{2} \ln \left (x \right )}{9}-\frac {64 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right ) \ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right )}{9}+\frac {16}{9}}{\left (x^{2} \ln \left (x \right )+2\right )^{3} x^{4} {\left (2 \ln \left (2\right )-\ln \left (x^{2}+4 \ln \left (\ln \left (x \right )\right )\right )\right )}^{2}}\)	\(440\)

input

int(((-32*ln(x)*ln(ln(x))-8*x^2*ln(x))*ln(ln(ln(x))+1/4*x^2)-8*x^2*ln(x)-1 
6)/(36*x^5*ln(x)*ln(ln(x))+9*x^7*ln(x))/ln(ln(ln(x))+1/4*x^2)^3,x,method=_ 
RETURNVERBOSE)
 

output

2/9/x^4/ln(ln(ln(x))+1/4*x^2)^2
 

3.5.83.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \begin {dmath*} \int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx=\frac {2}{9 \, x^{4} \log \left (\frac {1}{4} \, x^{2} + \log \left (\log \left (x\right )\right )\right )^{2}} \end {dmath*}

input

integrate(((-32*log(x)*log(log(x))-8*x^2*log(x))*log(log(log(x))+1/4*x^2)- 
8*x^2*log(x)-16)/(36*x^5*log(x)*log(log(x))+9*x^7*log(x))/log(log(log(x))+ 
1/4*x^2)^3,x, algorithm=\
 

output

2/9/(x^4*log(1/4*x^2 + log(log(x)))^2)
 

3.5.83.6 Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \begin {dmath*} \int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx=\frac {2}{9 x^{4} \log {\left (\frac {x^{2}}{4} + \log {\left (\log {\left (x \right )} \right )} \right )}^{2}} \end {dmath*}

input

integrate(((-32*ln(x)*ln(ln(x))-8*x**2*ln(x))*ln(ln(ln(x))+1/4*x**2)-8*x** 
2*ln(x)-16)/(36*x**5*ln(x)*ln(ln(x))+9*x**7*ln(x))/ln(ln(ln(x))+1/4*x**2)* 
*3,x)
 

output

2/(9*x**4*log(x**2/4 + log(log(x)))**2)
 

3.5.83.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (17) = 34\).

Time = 0.33 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24 \begin {dmath*} \int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx=\frac {2}{9 \, {\left (4 \, x^{4} \log \left (2\right )^{2} - 4 \, x^{4} \log \left (2\right ) \log \left (x^{2} + 4 \, \log \left (\log \left (x\right )\right )\right ) + x^{4} \log \left (x^{2} + 4 \, \log \left (\log \left (x\right )\right )\right )^{2}\right )}} \end {dmath*}

input

integrate(((-32*log(x)*log(log(x))-8*x^2*log(x))*log(log(log(x))+1/4*x^2)- 
8*x^2*log(x)-16)/(36*x^5*log(x)*log(log(x))+9*x^7*log(x))/log(log(log(x))+ 
1/4*x^2)^3,x, algorithm=\
 

output

2/9/(4*x^4*log(2)^2 - 4*x^4*log(2)*log(x^2 + 4*log(log(x))) + x^4*log(x^2 
+ 4*log(log(x)))^2)
 

3.5.83.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (17) = 34\).

Time = 0.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.95 \begin {dmath*} \int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx=\frac {2 \, {\left (x^{2} \log \left (x\right ) + 2\right )}}{9 \, {\left (4 \, x^{6} \log \left (2\right )^{2} \log \left (x\right ) - 4 \, x^{6} \log \left (2\right ) \log \left (x^{2} + 4 \, \log \left (\log \left (x\right )\right )\right ) \log \left (x\right ) + x^{6} \log \left (x^{2} + 4 \, \log \left (\log \left (x\right )\right )\right )^{2} \log \left (x\right ) + 8 \, x^{4} \log \left (2\right )^{2} - 8 \, x^{4} \log \left (2\right ) \log \left (x^{2} + 4 \, \log \left (\log \left (x\right )\right )\right ) + 2 \, x^{4} \log \left (x^{2} + 4 \, \log \left (\log \left (x\right )\right )\right )^{2}\right )}} \end {dmath*}

input

integrate(((-32*log(x)*log(log(x))-8*x^2*log(x))*log(log(log(x))+1/4*x^2)- 
8*x^2*log(x)-16)/(36*x^5*log(x)*log(log(x))+9*x^7*log(x))/log(log(log(x))+ 
1/4*x^2)^3,x, algorithm=\
 

output

2/9*(x^2*log(x) + 2)/(4*x^6*log(2)^2*log(x) - 4*x^6*log(2)*log(x^2 + 4*log 
(log(x)))*log(x) + x^6*log(x^2 + 4*log(log(x)))^2*log(x) + 8*x^4*log(2)^2 
- 8*x^4*log(2)*log(x^2 + 4*log(log(x))) + 2*x^4*log(x^2 + 4*log(log(x)))^2 
)
 

3.5.83.9 Mupad [B] (verification not implemented)

Time = 15.96 (sec) , antiderivative size = 519, normalized size of antiderivative = 24.71 \begin {dmath*} \int \frac {-16-8 x^2 \log (x)+\left (-8 x^2 \log (x)-32 \log (x) \log (\log (x))\right ) \log \left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )}{\left (9 x^7 \log (x)+36 x^5 \log (x) \log (\log (x))\right ) \log ^3\left (\frac {1}{4} \left (x^2+4 \log (\log (x))\right )\right )} \, dx =\text {Too large to display} \end {dmath*}

input

int(-(8*x^2*log(x) + log(log(log(x)) + x^2/4)*(8*x^2*log(x) + 32*log(log(x 
))*log(x)) + 16)/(log(log(log(x)) + x^2/4)^3*(9*x^7*log(x) + 36*x^5*log(lo 
g(x))*log(x))),x)
 

output

4/(9*x^4) - ((2*log(x)*(4*log(log(x)) + x^2))/(9*x^4*(x^2*log(x) + 2)) - ( 
2*log(log(log(x)) + x^2/4)*log(x)*(4*log(log(x)) + x^2)*(4*log(log(x)) - 2 
*x^4*log(x)^2 - 16*log(log(x))*log(x) + x^2 - 12*x^2*log(log(x))*log(x)^2 
+ 4))/(9*x^4*(x^2*log(x) + 2)^3))/log(log(log(x)) + x^2/4) + (2/(9*x^4) + 
(2*log(log(log(x)) + x^2/4)*log(x)*(4*log(log(x)) + x^2))/(9*x^4*(x^2*log( 
x) + 2)))/log(log(log(x)) + x^2/4)^2 - (log(log(x))*(log(x)*(x^2*((16*(x^2 
 - 20))/(9*x^6) + 416/(9*x^6)) - 64/(9*x^4)) - (40*log(x)^3)/9 - (32*(x^2 
- 4))/(9*x^6) + (32*(x^2 - 20))/(9*x^6) - (32*log(x)^2)/(9*x^2) + 512/(9*x 
^6)))/(12*x^2*log(x) + 6*x^4*log(x)^2 + x^6*log(x)^3 + 8) - (log(log(x))^2 
*(log(x)*(x^2*((32*(x^2 - 20))/(9*x^8) + 896/(9*x^8)) - 256/(9*x^6)) - (64 
*(x^2 - 4))/(9*x^8) + (64*(x^2 - 20))/(9*x^8) - (32*log(x)^3)/(3*x^2) - (1 
28*log(x)^2)/(9*x^4) + 1024/(9*x^8)))/(12*x^2*log(x) + 6*x^4*log(x)^2 + x^ 
6*log(x)^3 + 8) + (8*(x^2 - 4))/(3*x^3*(4*x - x^3)*(x^2*log(x) + 2)) - (4* 
(16*x - 8*x^3 + x^5))/(9*x^4*(4*x - x^3)*(12*x^2*log(x) + 6*x^4*log(x)^2 + 
 x^6*log(x)^3 + 8)) + (2*(x^4 - 24*x^2 + 80))/(9*x^3*(4*x - x^3)*(4*x^2*lo 
g(x) + x^4*log(x)^2 + 4))