\(\int \frac {-2 x^4-2 x^4 \log (x)+(5 x^2+x^4+(5 x^2+x^4) \log (x)) \log (5+x^2)+(2 x^2+2 x^2 \log (x)+(-5-x^2+(-5-x^2) \log (x)) \log (5+x^2)) \log (1+\log (x))+(-5+9 x^2+2 x^4+(10 x^2+2 x^4) \log (x)) \log (5+x^2) \log (\frac {x}{2 \log (5+x^2)})}{(5 x+x^3+(5 x+x^3) \log (x)) \log (5+x^2)} \, dx\) [1114]

Optimal result

Integrand size = 159, antiderivative size = 26 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (x^2-\log (1+\log (x))\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right ) \] Output:

ln(1/2*x/ln(x^2+5))*(x^2-ln(1+ln(x)))
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (x^2-\log (1+\log (x))\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right ) \] Input:

Integrate[(-2*x^4 - 2*x^4*Log[x] + (5*x^2 + x^4 + (5*x^2 + x^4)*Log[x])*Lo 
g[5 + x^2] + (2*x^2 + 2*x^2*Log[x] + (-5 - x^2 + (-5 - x^2)*Log[x])*Log[5 
+ x^2])*Log[1 + Log[x]] + (-5 + 9*x^2 + 2*x^4 + (10*x^2 + 2*x^4)*Log[x])*L 
og[5 + x^2]*Log[x/(2*Log[5 + x^2])])/((5*x + x^3 + (5*x + x^3)*Log[x])*Log 
[5 + x^2]),x]
 

Output:

(x^2 - Log[1 + Log[x]])*Log[x/(2*Log[5 + 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.

\(\displaystyle \int \frac {-2 x^4-2 x^4 \log (x)+\left (2 x^2+2 x^2 \log (x)+\left (-x^2+\left (-x^2-5\right ) \log (x)-5\right ) \log \left (x^2+5\right )\right ) \log (\log (x)+1)+\left (x^4+5 x^2+\left (x^4+5 x^2\right ) \log (x)\right ) \log \left (x^2+5\right )+\left (2 x^4+9 x^2+\left (2 x^4+10 x^2\right ) \log (x)-5\right ) \log \left (x^2+5\right ) \log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{\left (x^3+\left (x^3+5 x\right ) \log (x)+5 x\right ) \log \left (x^2+5\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-2 x^4-2 x^4 \log (x)+\left (2 x^2+2 x^2 \log (x)+\left (-x^2+\left (-x^2-5\right ) \log (x)-5\right ) \log \left (x^2+5\right )\right ) \log (\log (x)+1)+\left (x^4+5 x^2+\left (x^4+5 x^2\right ) \log (x)\right ) \log \left (x^2+5\right )+\left (2 x^4+9 x^2+\left (2 x^4+10 x^2\right ) \log (x)-5\right ) \log \left (x^2+5\right ) \log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{x \left (x^2+5\right ) (\log (x)+1) \log \left (x^2+5\right )}dx\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {\log (\log (x)+1)}{\left (i \sqrt {5}-x\right ) \log \left (x^2+5\right )}dx+\int \frac {\log (\log (x)+1)}{\left (x+i \sqrt {5}\right ) \log \left (x^2+5\right )}dx-\int \frac {\log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{x (\log (x)+1)}dx+2 \int \frac {x \log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{\log (x)+1}dx+2 \int \frac {x \log (x) \log \left (\frac {x}{2 \log \left (x^2+5\right )}\right )}{\log (x)+1}dx-\operatorname {LogIntegral}\left (x^2+5\right )+\frac {x^2}{2}+5 \log \left (\log \left (x^2+5\right )\right )+\log (x)-(\log (x)+1) \log (\log (x)+1)\)

Input:

Int[(-2*x^4 - 2*x^4*Log[x] + (5*x^2 + x^4 + (5*x^2 + x^4)*Log[x])*Log[5 + 
x^2] + (2*x^2 + 2*x^2*Log[x] + (-5 - x^2 + (-5 - x^2)*Log[x])*Log[5 + x^2] 
)*Log[1 + Log[x]] + (-5 + 9*x^2 + 2*x^4 + (10*x^2 + 2*x^4)*Log[x])*Log[5 + 
 x^2]*Log[x/(2*Log[5 + x^2])])/((5*x + x^3 + (5*x + x^3)*Log[x])*Log[5 + x 
^2]),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 0.02 (sec) , antiderivative size = 299, normalized size of antiderivative = 11.50

\[\left (-x^{2}+\ln \left (\ln \left (x \right )+1\right )\right ) \ln \left (\ln \left (x^{2}+5\right )\right )-\ln \left (x \right ) \ln \left (\ln \left (x \right )+1\right )+x^{2} \ln \left (x \right )+\frac {i \ln \left (\ln \left (x \right )+1\right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}{2}+\frac {i \pi \,x^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}+\frac {i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}-\frac {i \ln \left (\ln \left (x \right )+1\right ) \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}-\frac {i \ln \left (\ln \left (x \right )+1\right ) \pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{2}}{2}-\frac {i \pi \,x^{2} {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{3}}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x^{2}+5\right )}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}{2}+\frac {i \ln \left (\ln \left (x \right )+1\right ) \pi {\operatorname {csgn}\left (\frac {i x}{\ln \left (x^{2}+5\right )}\right )}^{3}}{2}-x^{2} \ln \left (2\right )+\ln \left (\ln \left (x \right )+1\right ) \ln \left (2\right )\]

Input:

int(((((-x^2-5)*ln(x)-x^2-5)*ln(x^2+5)+2*x^2*ln(x)+2*x^2)*ln(ln(x)+1)+((2* 
x^4+10*x^2)*ln(x)+2*x^4+9*x^2-5)*ln(x^2+5)*ln(1/2*x/ln(x^2+5))+((x^4+5*x^2 
)*ln(x)+x^4+5*x^2)*ln(x^2+5)-2*x^4*ln(x)-2*x^4)/((x^3+5*x)*ln(x)+x^3+5*x)/ 
ln(x^2+5),x)
 

Output:

(-x^2+ln(ln(x)+1))*ln(ln(x^2+5))-ln(x)*ln(ln(x)+1)+x^2*ln(x)+1/2*I*ln(ln(x 
)+1)*Pi*csgn(I*x)*csgn(I/ln(x^2+5))*csgn(I*x/ln(x^2+5))+1/2*I*Pi*x^2*csgn( 
I/ln(x^2+5))*csgn(I*x/ln(x^2+5))^2+1/2*I*Pi*x^2*csgn(I*x)*csgn(I*x/ln(x^2+ 
5))^2-1/2*I*ln(ln(x)+1)*Pi*csgn(I/ln(x^2+5))*csgn(I*x/ln(x^2+5))^2-1/2*I*l 
n(ln(x)+1)*Pi*csgn(I*x)*csgn(I*x/ln(x^2+5))^2-1/2*I*Pi*x^2*csgn(I*x/ln(x^2 
+5))^3-1/2*I*Pi*x^2*csgn(I*x)*csgn(I/ln(x^2+5))*csgn(I*x/ln(x^2+5))+1/2*I* 
ln(ln(x)+1)*Pi*csgn(I*x/ln(x^2+5))^3-x^2*ln(2)+ln(ln(x)+1)*ln(2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=x^{2} \log \left (\frac {x}{2 \, \log \left (x^{2} + 5\right )}\right ) - \log \left (\frac {x}{2 \, \log \left (x^{2} + 5\right )}\right ) \log \left (\log \left (x\right ) + 1\right ) \] Input:

integrate(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(1+l 
og(x))+((2*x^4+10*x^2)*log(x)+2*x^4+9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+ 
5))+((x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+5* 
x)*log(x)+x^3+5*x)/log(x^2+5),x, algorithm="fricas")
 

Output:

x^2*log(1/2*x/log(x^2 + 5)) - log(1/2*x/log(x^2 + 5))*log(log(x) + 1)
 

Sympy [A] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (x^{2} - \log {\left (\log {\left (x \right )} + 1 \right )}\right ) \log {\left (\frac {x}{2 \log {\left (x^{2} + 5 \right )}} \right )} \] Input:

integrate(((((-x**2-5)*ln(x)-x**2-5)*ln(x**2+5)+2*x**2*ln(x)+2*x**2)*ln(1+ 
ln(x))+((2*x**4+10*x**2)*ln(x)+2*x**4+9*x**2-5)*ln(x**2+5)*ln(1/2*x/ln(x** 
2+5))+((x**4+5*x**2)*ln(x)+x**4+5*x**2)*ln(x**2+5)-2*x**4*ln(x)-2*x**4)/(( 
x**3+5*x)*ln(x)+x**3+5*x)/ln(x**2+5),x)
 

Output:

(x**2 - log(log(x) + 1))*log(x/(2*log(x**2 + 5)))
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=-x^{2} \log \left (2\right ) + x^{2} \log \left (x\right ) + {\left (\log \left (2\right ) - \log \left (x\right )\right )} \log \left (\log \left (x\right ) + 1\right ) - {\left (x^{2} - \log \left (\log \left (x\right ) + 1\right )\right )} \log \left (\log \left (x^{2} + 5\right )\right ) \] Input:

integrate(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(1+l 
og(x))+((2*x^4+10*x^2)*log(x)+2*x^4+9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+ 
5))+((x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+5* 
x)*log(x)+x^3+5*x)/log(x^2+5),x, algorithm="maxima")
 

Output:

-x^2*log(2) + x^2*log(x) + (log(2) - log(x))*log(log(x) + 1) - (x^2 - log( 
log(x) + 1))*log(log(x^2 + 5))
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=x^{2} \log \left (x\right ) - {\left (x^{2} - \log \left (\log \left (x\right ) + 1\right )\right )} \log \left (2 \, \log \left (x^{2} + 5\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right ) + 1\right ) \] Input:

integrate(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(1+l 
og(x))+((2*x^4+10*x^2)*log(x)+2*x^4+9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+ 
5))+((x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+5* 
x)*log(x)+x^3+5*x)/log(x^2+5),x, algorithm="giac")
 

Output:

x^2*log(x) - (x^2 - log(log(x) + 1))*log(2*log(x^2 + 5)) - log(x)*log(log( 
x) + 1)
 

Mupad [B] (verification not implemented)

Time = 4.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\left (\ln \left (\ln \left (x\right )+1\right )-x^2\right )\,\left (\ln \left (\ln \left (x^2+5\right )\right )+\ln \left (2\right )-\ln \left (x\right )\right ) \] Input:

int((log(x^2 + 5)*(log(x)*(5*x^2 + x^4) + 5*x^2 + x^4) - 2*x^4*log(x) + lo 
g(log(x) + 1)*(2*x^2*log(x) - log(x^2 + 5)*(log(x)*(x^2 + 5) + x^2 + 5) + 
2*x^2) - 2*x^4 + log(x/(2*log(x^2 + 5)))*log(x^2 + 5)*(log(x)*(10*x^2 + 2* 
x^4) + 9*x^2 + 2*x^4 - 5))/(log(x^2 + 5)*(5*x + log(x)*(5*x + x^3) + x^3)) 
,x)
 

Output:

(log(log(x) + 1) - x^2)*(log(log(x^2 + 5)) + log(2) - log(x))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-2 x^4-2 x^4 \log (x)+\left (5 x^2+x^4+\left (5 x^2+x^4\right ) \log (x)\right ) \log \left (5+x^2\right )+\left (2 x^2+2 x^2 \log (x)+\left (-5-x^2+\left (-5-x^2\right ) \log (x)\right ) \log \left (5+x^2\right )\right ) \log (1+\log (x))+\left (-5+9 x^2+2 x^4+\left (10 x^2+2 x^4\right ) \log (x)\right ) \log \left (5+x^2\right ) \log \left (\frac {x}{2 \log \left (5+x^2\right )}\right )}{\left (5 x+x^3+\left (5 x+x^3\right ) \log (x)\right ) \log \left (5+x^2\right )} \, dx=\mathrm {log}\left (\frac {x}{2 \,\mathrm {log}\left (x^{2}+5\right )}\right ) \left (-\mathrm {log}\left (\mathrm {log}\left (x \right )+1\right )+x^{2}\right ) \] Input:

int(((((-x^2-5)*log(x)-x^2-5)*log(x^2+5)+2*x^2*log(x)+2*x^2)*log(1+log(x)) 
+((2*x^4+10*x^2)*log(x)+2*x^4+9*x^2-5)*log(x^2+5)*log(1/2*x/log(x^2+5))+(( 
x^4+5*x^2)*log(x)+x^4+5*x^2)*log(x^2+5)-2*x^4*log(x)-2*x^4)/((x^3+5*x)*log 
(x)+x^3+5*x)/log(x^2+5),x)
 

Output:

log(x/(2*log(x**2 + 5)))*( - log(log(x) + 1) + x**2)