\(\int \frac {-2 x \log (x)+(3 x+x \log ^2(x)) \log (3+\log ^2(x))+(6+2 \log ^2(x)) \log ^2(3+\log ^2(x))+((-3 x-x \log ^2(x)) \log (3+\log ^2(x))+(-15+3 \log (5)+3 \log (\frac {3}{4 x^2})+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log ^2(x)) \log ^2(3+\log ^2(x))) \log (\frac {-x+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log (3+\log ^2(x))}{\log (3+\log ^2(x))})}{((-3 x-x \log ^2(x)) \log (3+\log ^2(x))+(-15+3 \log (5)+3 \log (\frac {3}{4 x^2})+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log ^2(x)) \log ^2(3+\log ^2(x))) \log ^2(\frac {-x+(-5+\log (5)+\log (\frac {3}{4 x^2})) \log (3+\log ^2(x))}{\log (3+\log ^2(x))})} \, dx\) [2536]

Optimal result

Integrand size = 244, antiderivative size = 29 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \] Output:

x/ln(ln(5)+ln(3/4/x^2)-5-x/ln(ln(x)^2+3))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \] Input:

Integrate[(-2*x*Log[x] + (3*x + x*Log[x]^2)*Log[3 + Log[x]^2] + (6 + 2*Log 
[x]^2)*Log[3 + Log[x]^2]^2 + ((-3*x - x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 
 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)* 
Log[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[ 
x]^2])/Log[3 + Log[x]^2]])/(((-3*x - x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 
+ 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*L 
og[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x 
]^2])/Log[3 + Log[x]^2]]^2),x]
 

Output:

x/Log[-5 + Log[15/(4*x^2)] - x/Log[3 + Log[x]^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.

\(\displaystyle \int \frac {\left (\left (\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log ^2(x)+3 \log \left (\frac {3}{4 x^2}\right )-15+3 \log (5)\right ) \log ^2\left (\log ^2(x)+3\right )+\left (-3 x-x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )\right ) \log \left (\frac {\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log \left (\log ^2(x)+3\right )-x}{\log \left (\log ^2(x)+3\right )}\right )+\left (2 \log ^2(x)+6\right ) \log ^2\left (\log ^2(x)+3\right )+\left (3 x+x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )-2 x \log (x)}{\left (\left (\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log ^2(x)+3 \log \left (\frac {3}{4 x^2}\right )-15+3 \log (5)\right ) \log ^2\left (\log ^2(x)+3\right )+\left (-3 x-x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (\frac {\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log \left (\log ^2(x)+3\right )-x}{\log \left (\log ^2(x)+3\right )}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-\left (\left (\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log ^2(x)+3 \log \left (\frac {3}{4 x^2}\right )-15+3 \log (5)\right ) \log ^2\left (\log ^2(x)+3\right )+\left (-3 x-x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )\right ) \log \left (\frac {\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log \left (\log ^2(x)+3\right )-x}{\log \left (\log ^2(x)+3\right )}\right )-\left (\left (2 \log ^2(x)+6\right ) \log ^2\left (\log ^2(x)+3\right )\right )-\left (3 x+x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )+2 x \log (x)}{\left (\log ^2(x)+3\right ) \log \left (\log ^2(x)+3\right ) \left (-\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )+x+5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (\frac {\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log \left (\log ^2(x)+3\right )-x}{\log \left (\log ^2(x)+3\right )}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \log ^2\left (\log ^2(x)+3\right ) \log ^2(x)+x \log \left (\log ^2(x)+3\right ) \log ^2(x)+6 \log ^2\left (\log ^2(x)+3\right )+3 x \log \left (\log ^2(x)+3\right )-2 x \log (x)}{\left (\log ^2(x)+3\right ) \log \left (\log ^2(x)+3\right ) \left (\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )-x-5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (\log ^2(x)+3\right )}-5\right )}+\frac {1}{\log \left (\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (\log ^2(x)+3\right )}-5\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -3 \int \frac {x}{\left (\log ^2(x)+3\right ) \left (x-\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )+5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx-\int \frac {x \log ^2(x)}{\left (\log ^2(x)+3\right ) \left (x-\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )+5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx-2 \int \frac {x \log (x)}{\left (\log ^2(x)+3\right ) \log \left (\log ^2(x)+3\right ) \left (-x+\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )-5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx+6 \int \frac {\log \left (\log ^2(x)+3\right )}{\left (\log ^2(x)+3\right ) \left (-x+\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )-5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx+2 \int \frac {\log ^2(x) \log \left (\log ^2(x)+3\right )}{\left (\log ^2(x)+3\right ) \left (-x+\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )-5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx+\int \frac {1}{\log \left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx\)

Input:

Int[(-2*x*Log[x] + (3*x + x*Log[x]^2)*Log[3 + Log[x]^2] + (6 + 2*Log[x]^2) 
*Log[3 + Log[x]^2]^2 + ((-3*x - x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*L 
og[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*Log[3 
+ Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2]) 
/Log[3 + Log[x]^2]])/(((-3*x - x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Lo 
g[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*Log[3 + 
 Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/ 
Log[3 + Log[x]^2]]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 0.04 (sec) , antiderivative size = 928, normalized size of antiderivative = 32.00

\[\text {Expression too large to display}\]

Input:

int(((((ln(3/4/x^2)+ln(5)-5)*ln(x)^2+3*ln(3/4/x^2)+3*ln(5)-15)*ln(ln(x)^2+ 
3)^2+(-x*ln(x)^2-3*x)*ln(ln(x)^2+3))*ln(((ln(3/4/x^2)+ln(5)-5)*ln(ln(x)^2+ 
3)-x)/ln(ln(x)^2+3))+(2*ln(x)^2+6)*ln(ln(x)^2+3)^2+(x*ln(x)^2+3*x)*ln(ln(x 
)^2+3)-2*x*ln(x))/(((ln(3/4/x^2)+ln(5)-5)*ln(x)^2+3*ln(3/4/x^2)+3*ln(5)-15 
)*ln(ln(x)^2+3)^2+(-x*ln(x)^2-3*x)*ln(ln(x)^2+3))/ln(((ln(3/4/x^2)+ln(5)-5 
)*ln(ln(x)^2+3)-x)/ln(ln(x)^2+3))^2,x)
 

Output:

2*I*x/(Pi*csgn(I*(ln(ln(x)^2+3)*ln(3/4)-2*ln(ln(x)^2+3)*ln(x)+1/2*I*ln(ln( 
x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-I*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2 
)^2+1/2*I*ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+ln(ln(x)^2+3)*ln(5)-5*ln(ln(x)^2+ 
3)-x))*csgn(I/ln(ln(x)^2+3))*csgn(I/ln(ln(x)^2+3)*(ln(ln(x)^2+3)*ln(3/4)-2 
*ln(ln(x)^2+3)*ln(x)+1/2*I*ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-I*ln(l 
n(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2+1/2*I*ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+ 
ln(ln(x)^2+3)*ln(5)-5*ln(ln(x)^2+3)-x))-Pi*csgn(I*(ln(ln(x)^2+3)*ln(3/4)-2 
*ln(ln(x)^2+3)*ln(x)+1/2*I*ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-I*ln(l 
n(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2+1/2*I*ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+ 
ln(ln(x)^2+3)*ln(5)-5*ln(ln(x)^2+3)-x))*csgn(I/ln(ln(x)^2+3)*(ln(ln(x)^2+3 
)*ln(3/4)-2*ln(ln(x)^2+3)*ln(x)+1/2*I*ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I* 
x^2)-I*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2+1/2*I*ln(ln(x)^2+3)*Pi*csg 
n(I*x^2)^3+ln(ln(x)^2+3)*ln(5)-5*ln(ln(x)^2+3)-x))^2-Pi*csgn(I/ln(ln(x)^2+ 
3))*csgn(I/ln(ln(x)^2+3)*(ln(ln(x)^2+3)*ln(3/4)-2*ln(ln(x)^2+3)*ln(x)+1/2* 
I*ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-I*ln(ln(x)^2+3)*Pi*csgn(I*x)*cs 
gn(I*x^2)^2+1/2*I*ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+ln(ln(x)^2+3)*ln(5)-5*ln( 
ln(x)^2+3)-x))^2+Pi*csgn(I/ln(ln(x)^2+3)*(ln(ln(x)^2+3)*ln(3/4)-2*ln(ln(x) 
^2+3)*ln(x)+1/2*I*ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-I*ln(ln(x)^2+3) 
*Pi*csgn(I*x)*csgn(I*x^2)^2+1/2*I*ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+ln(ln(x)^ 
2+3)*ln(5)-5*ln(ln(x)^2+3)-x))^3-2*I*ln(ln(ln(x)^2+3))+2*I*ln(ln(ln(x)^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (27) = 54\).

Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (\frac {{\left (\log \left (5\right ) + \log \left (\frac {3}{4 \, x^{2}}\right ) - 5\right )} \log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right ) - x}{\log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right )}\right )} \] Input:

integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)* 
log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))*log(((log(3/4/x^2)+lo 
g(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+3) 
^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)* 
log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*l 
og(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x) 
^2+3))^2,x, algorithm="fricas")
 

Output:

x/log(((log(5) + log(3/4/x^2) - 5)*log(1/4*log(3/4)^2 - 1/2*log(3/4)*log(3 
/4/x^2) + 1/4*log(3/4/x^2)^2 + 3) - x)/log(1/4*log(3/4)^2 - 1/2*log(3/4)*l 
og(3/4/x^2) + 1/4*log(3/4/x^2)^2 + 3))
 

Sympy [A] (verification not implemented)

Time = 70.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log {\left (\frac {- x + \left (- 2 \log {\left (x \right )} - 5 + \log {\left (\frac {3}{4} \right )} + \log {\left (5 \right )}\right ) \log {\left (\log {\left (x \right )}^{2} + 3 \right )}}{\log {\left (\log {\left (x \right )}^{2} + 3 \right )}} \right )}} \] Input:

integrate(((((ln(3/4/x**2)+ln(5)-5)*ln(x)**2+3*ln(3/4/x**2)+3*ln(5)-15)*ln 
(ln(x)**2+3)**2+(-x*ln(x)**2-3*x)*ln(ln(x)**2+3))*ln(((ln(3/4/x**2)+ln(5)- 
5)*ln(ln(x)**2+3)-x)/ln(ln(x)**2+3))+(2*ln(x)**2+6)*ln(ln(x)**2+3)**2+(x*l 
n(x)**2+3*x)*ln(ln(x)**2+3)-2*x*ln(x))/(((ln(3/4/x**2)+ln(5)-5)*ln(x)**2+3 
*ln(3/4/x**2)+3*ln(5)-15)*ln(ln(x)**2+3)**2+(-x*ln(x)**2-3*x)*ln(ln(x)**2+ 
3))/ln(((ln(3/4/x**2)+ln(5)-5)*ln(ln(x)**2+3)-x)/ln(ln(x)**2+3))**2,x)
 

Output:

x/log((-x + (-2*log(x) - 5 + log(3/4) + log(5))*log(log(x)**2 + 3))/log(lo 
g(x)**2 + 3))
 

Maxima [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left ({\left (\log \left (5\right ) + \log \left (3\right ) - 2 \, \log \left (2\right ) - 5\right )} \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (\log \left (x\right )^{2} + 3\right ) \log \left (x\right ) - x\right ) - \log \left (\log \left (\log \left (x\right )^{2} + 3\right )\right )} \] Input:

integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)* 
log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))*log(((log(3/4/x^2)+lo 
g(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+3) 
^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)* 
log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*l 
og(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x) 
^2+3))^2,x, algorithm="maxima")
 

Output:

x/(log((log(5) + log(3) - 2*log(2) - 5)*log(log(x)^2 + 3) - 2*log(log(x)^2 
 + 3)*log(x) - x) - log(log(log(x)^2 + 3)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (27) = 54\).

Time = 2.61 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (\log \left (5\right ) \log \left (\log \left (x\right )^{2} + 3\right ) + \log \left (3\right ) \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (2\right ) \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (\log \left (x\right )^{2} + 3\right ) \log \left (x\right ) - x - 5 \, \log \left (\log \left (x\right )^{2} + 3\right )\right ) - \log \left (\log \left (\log \left (x\right )^{2} + 3\right )\right )} \] Input:

integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)* 
log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))*log(((log(3/4/x^2)+lo 
g(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+3) 
^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)* 
log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*l 
og(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x) 
^2+3))^2,x, algorithm="giac")
 

Output:

x/(log(log(5)*log(log(x)^2 + 3) + log(3)*log(log(x)^2 + 3) - 2*log(2)*log( 
log(x)^2 + 3) - 2*log(log(x)^2 + 3)*log(x) - x - 5*log(log(x)^2 + 3)) - lo 
g(log(log(x)^2 + 3)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\int \frac {\ln \left (-\frac {x-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \left (x\right )}^2+3\right )}\right )\,\left ({\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \left (x\right )}^2+3\,\ln \left (5\right )+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )\right )-2\,x\,\ln \left (x\right )+\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )+{\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (2\,{\ln \left (x\right )}^2+6\right )}{{\ln \left (-\frac {x-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \left (x\right )}^2+3\right )}\right )}^2\,\left ({\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \left (x\right )}^2+3\,\ln \left (5\right )+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )\right )} \,d x \] Input:

int((log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x) 
^2 + 3))*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2)) + log(x)^2*(log 
(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2)) - 
2*x*log(x) + log(log(x)^2 + 3)*(3*x + x*log(x)^2) + log(log(x)^2 + 3)^2*(2 
*log(x)^2 + 6))/(log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5) 
)/log(log(x)^2 + 3))^2*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2)) + 
 log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x 
*log(x)^2))),x)
 

Output:

int((log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x) 
^2 + 3))*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2)) + log(x)^2*(log 
(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2)) - 
2*x*log(x) + log(log(x)^2 + 3)*(3*x + x*log(x)^2) + log(log(x)^2 + 3)^2*(2 
*log(x)^2 + 6))/(log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5) 
)/log(log(x)^2 + 3))^2*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2)) + 
 log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x 
*log(x)^2))), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\mathrm {log}\left (\frac {\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}+3\right ) \mathrm {log}\left (\frac {3}{4 x^{2}}\right )+\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}+3\right ) \mathrm {log}\left (5\right )-5 \,\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}+3\right )-x}{\mathrm {log}\left (\mathrm {log}\left (x \right )^{2}+3\right )}\right )} \] Input:

int(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(lo 
g(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))*log(((log(3/4/x^2)+log(5)-5 
)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+3)^2+(x* 
log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)*log(x) 
^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log 
(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3)) 
^2,x)
 

Output:

x/log((log(log(x)**2 + 3)*log(3/(4*x**2)) + log(log(x)**2 + 3)*log(5) - 5* 
log(log(x)**2 + 3) - x)/log(log(x)**2 + 3))