\(\int \frac {48 x+48 x^2+(-24 x^2-24 x \log (x)) \log (x+\log (x)) \log (16 \log ^2(x+\log (x)))+(x+\log (x)) \log (x+\log (x)) \log ^3(16 \log ^2(x+\log (x)))}{(x+\log (x)) \log (x+\log (x)) \log ^3(16 \log ^2(x+\log (x)))} \, dx\) [1682]

Optimal result

Integrand size = 84, antiderivative size = 19 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x-\frac {12 x^2}{\log ^2\left (16 \log ^2(x+\log (x))\right )} \] Output:

x-12/ln(16*ln(x+ln(x))^2)^2*x^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x-\frac {12 x^2}{\log ^2\left (16 \log ^2(x+\log (x))\right )} \] Input:

Integrate[(48*x + 48*x^2 + (-24*x^2 - 24*x*Log[x])*Log[x + Log[x]]*Log[16* 
Log[x + Log[x]]^2] + (x + Log[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2 
]^3)/((x + Log[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2]^3),x]
 

Output:

x - (12*x^2)/Log[16*Log[x + Log[x]]^2]^2
 

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.

\(\displaystyle \int \frac {48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+48 x+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (-\frac {24 x}{\log ^2\left (16 \log ^2(x+\log (x))\right )}+\frac {48 (x+1) x}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 48 \int \frac {x^2}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}dx-24 \int \frac {x}{\log ^2\left (16 \log ^2(x+\log (x))\right )}dx+48 \int \frac {x}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}dx+x\)

Input:

Int[(48*x + 48*x^2 + (-24*x^2 - 24*x*Log[x])*Log[x + Log[x]]*Log[16*Log[x 
+ Log[x]]^2] + (x + Log[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2]^3)/( 
(x + Log[x])*Log[x + Log[x]]*Log[16*Log[x + Log[x]]^2]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 6.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89

method	result	size
parallelrisch	\(\frac {2 \ln \left (16 \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2} x -24 x^{2}}{2 \ln \left (16 \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}}\)	\(36\)
risch	\(x +\frac {48 x^{2}}{\left (\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3}+8 i \ln \left (2\right )+4 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )^{2}}\)	\(89\)
default	\(x +\frac {24 i \ln \left (x +\ln \left (x \right )\right ) \left (x +\ln \left (x \right )\right ) x^{2}}{\left (1+x \right ) \left (\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3}+8 i \ln \left (2\right )+4 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )}-\frac {24 i \left (2 i x^{3}+2 i x^{2}+\pi \ln \left (x \right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right ) x^{2} \ln \left (x +\ln \left (x \right )\right )-2 \pi \ln \left (x \right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2} x^{2} \ln \left (x +\ln \left (x \right )\right )+\pi \ln \left (x \right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3} x^{2} \ln \left (x +\ln \left (x \right )\right )+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right ) x^{3} \ln \left (x +\ln \left (x \right )\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2} x^{3} \ln \left (x +\ln \left (x \right )\right )+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3} x^{3} \ln \left (x +\ln \left (x \right )\right )+8 i \ln \left (2\right ) \ln \left (x \right ) x^{2} \ln \left (x +\ln \left (x \right )\right )+8 i \ln \left (2\right ) x^{3} \ln \left (x +\ln \left (x \right )\right )+4 i \ln \left (x \right ) x^{2} \ln \left (x +\ln \left (x \right )\right ) \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )+4 i x^{3} \ln \left (x +\ln \left (x \right )\right ) \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )}{\left (\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \ln \left (x +\ln \left (x \right )\right )^{2}\right )^{3}+8 i \ln \left (2\right )+4 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )\right )^{2} \left (1+x \right )}\)	\(452\)

Input:

int(((x+ln(x))*ln(x+ln(x))*ln(16*ln(x+ln(x))^2)^3+(-24*x*ln(x)-24*x^2)*ln( 
x+ln(x))*ln(16*ln(x+ln(x))^2)+48*x^2+48*x)/(x+ln(x))/ln(x+ln(x))/ln(16*ln( 
x+ln(x))^2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/2*(2*ln(16*ln(x+ln(x))^2)^2*x-24*x^2)/ln(16*ln(x+ln(x))^2)^2
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=\frac {x \log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2} - 12 \, x^{2}}{\log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2}} \] Input:

integrate(((x+log(x))*log(x+log(x))*log(16*log(x+log(x))^2)^3+(-24*x*log(x 
)-24*x^2)*log(x+log(x))*log(16*log(x+log(x))^2)+48*x^2+48*x)/(x+log(x))/lo 
g(x+log(x))/log(16*log(x+log(x))^2)^3,x, algorithm="fricas")
 

Output:

(x*log(16*log(x + log(x))^2)^2 - 12*x^2)/log(16*log(x + log(x))^2)^2
 

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=- \frac {12 x^{2}}{\log {\left (16 \log {\left (x + \log {\left (x \right )} \right )}^{2} \right )}^{2}} + x \] Input:

integrate(((x+ln(x))*ln(x+ln(x))*ln(16*ln(x+ln(x))**2)**3+(-24*x*ln(x)-24* 
x**2)*ln(x+ln(x))*ln(16*ln(x+ln(x))**2)+48*x**2+48*x)/(x+ln(x))/ln(x+ln(x) 
)/ln(16*ln(x+ln(x))**2)**3,x)
 

Output:

-12*x**2/log(16*log(x + log(x))**2)**2 + x
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).

Time = 0.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.26 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=\frac {4 \, x \log \left (2\right )^{2} + 4 \, x \log \left (2\right ) \log \left (\log \left (x + \log \left (x\right )\right )\right ) + x \log \left (\log \left (x + \log \left (x\right )\right )\right )^{2} - 3 \, x^{2}}{4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (\log \left (x + \log \left (x\right )\right )\right ) + \log \left (\log \left (x + \log \left (x\right )\right )\right )^{2}} \] Input:

integrate(((x+log(x))*log(x+log(x))*log(16*log(x+log(x))^2)^3+(-24*x*log(x 
)-24*x^2)*log(x+log(x))*log(16*log(x+log(x))^2)+48*x^2+48*x)/(x+log(x))/lo 
g(x+log(x))/log(16*log(x+log(x))^2)^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).

Time = 18.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=x - \frac {12 \, {\left (x^{3} + x^{2}\right )}}{x \log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2} + \log \left (16 \, \log \left (x + \log \left (x\right )\right )^{2}\right )^{2}} \] Input:

integrate(((x+log(x))*log(x+log(x))*log(16*log(x+log(x))^2)^3+(-24*x*log(x 
)-24*x^2)*log(x+log(x))*log(16*log(x+log(x))^2)+48*x^2+48*x)/(x+log(x))/lo 
g(x+log(x))/log(16*log(x+log(x))^2)^3,x, algorithm="giac")
 

Output:

x - 12*(x^3 + x^2)/(x*log(16*log(x + log(x))^2)^2 + log(16*log(x + log(x)) 
^2)^2)
 

Mupad [B] (verification not implemented)

Time = 2.70 (sec) , antiderivative size = 739, normalized size of antiderivative = 38.89 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx =\text {Too large to display} \] Input:

int((48*x + 48*x^2 - log(x + log(x))*log(16*log(x + log(x))^2)*(24*x*log(x 
) + 24*x^2) + log(x + log(x))*log(16*log(x + log(x))^2)^3*(x + log(x)))/(l 
og(x + log(x))*log(16*log(x + log(x))^2)^3*(x + log(x))),x)
 

Output:

x + log(x + log(x))*((6*x + 6*x^2)/(x + 1) - (3*x + 3*x^2)/(x + 1) + (3*x 
+ 12*x^2 + 6*x^3)/(x + 1) - (6*x + 15*x^2 + 9*x^3)/(x + 1) + log(x)*((6*x 
+ 3*x^2)/(x + 1) - (6*x + 6*x^2)/(x + 1))) - (12*x^2 - (6*x^2*log(x + log( 
x))*log(16*log(x + log(x))^2)*(x + log(x)))/(x + 1))/log(16*log(x + log(x) 
)^2)^2 - ((6*x^2*log(x + log(x))*(x + log(x)))/(x + 1) - (3*x^2*log(x + lo 
g(x))*log(16*log(x + log(x))^2)*(x + log(x))*(2*x + log(x + log(x)) + 2*x^ 
2*log(x + log(x)) + 2*log(x + log(x))*log(x) + x^2 + 4*x*log(x + log(x)) + 
 x*log(x + log(x))*log(x) + 1))/(x + 1)^3)/log(16*log(x + log(x))^2) - log 
(x + log(x))^2*((84*x + 312*x^2 + 432*x^3 + 264*x^4 + 60*x^5)/(3*x + 3*x^2 
 + x^3 + 1) - ((213*x)/2 + (639*x^2)/2 + (831*x^3)/2 + 243*x^4 + 54*x^5 + 
21/2)/(3*x + 3*x^2 + x^3 + 1) - (18*x + 6*x^2 + 50/3)/(3*x + 3*x^2 + x^3 + 
 1) - log(x)^2*((9*x + 3*x^2 + 3)/(3*x + 3*x^2 + x^3 + 1) - 3) + log(x)*(( 
3*(16*x + 13*x^2 + (11*x^3)/3 + 22/3))/(3*x + 3*x^2 + x^3 + 1) - (18*x + 6 
*x^2 + 14)/(3*x + 3*x^2 + x^3 + 1) + (36*x + 108*x^2 + 108*x^3 + 36*x^4)/( 
3*x + 3*x^2 + x^3 + 1) - (120*x + 192*x^2 + 119*x^3 + 27*x^4 + 26)/(3*x + 
3*x^2 + x^3 + 1) + 18) + (18*(8*x + (13*x^2)/2 + (11*x^3)/6 + 11/3))/(3*x 
+ 3*x^2 + x^3 + 1) + (3*((209*x)/6 + (155*x^2)/6 + (121*x^3)/18 + 109/6))/ 
(3*x + 3*x^2 + x^3 + 1) - (36*x + 108*x^2 + 108*x^3 + 36*x^4)/(3*x + 3*x^2 
 + x^3 + 1) + (108*x + 324*x^2 + 324*x^3 + 108*x^4)/(3*x + 3*x^2 + x^3 + 1 
) - (613*x + 730*x^2 + (1181*x^3)/3 + 81*x^4 + 613/3)/(3*x + 3*x^2 + x^...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {48 x+48 x^2+\left (-24 x^2-24 x \log (x)\right ) \log (x+\log (x)) \log \left (16 \log ^2(x+\log (x))\right )+(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )}{(x+\log (x)) \log (x+\log (x)) \log ^3\left (16 \log ^2(x+\log (x))\right )} \, dx=\frac {x \left (\mathrm {log}\left (16 \mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )^{2}\right )^{2}-12 x \right )}{\mathrm {log}\left (16 \mathrm {log}\left (\mathrm {log}\left (x \right )+x \right )^{2}\right )^{2}} \] Input:

int(((x+log(x))*log(x+log(x))*log(16*log(x+log(x))^2)^3+(-24*x*log(x)-24*x 
^2)*log(x+log(x))*log(16*log(x+log(x))^2)+48*x^2+48*x)/(x+log(x))/log(x+lo 
g(x))/log(16*log(x+log(x))^2)^3,x)
 

Output:

(x*(log(16*log(log(x) + x)**2)**2 - 12*x))/log(16*log(log(x) + x)**2)**2