\(\int \frac {4+8 x+(8 x+4 x^2) \log (x)+(-2+(-2-2 x) \log (x)) \log (\frac {1}{1+(2+2 x) \log (x)+(1+2 x+x^2) \log ^2(x)})}{5 x^4+(5 x^4+5 x^5) \log (x)+(-10 x^3+(-10 x^3-10 x^4) \log (x)) \log (\frac {1}{1+(2+2 x) \log (x)+(1+2 x+x^2) \log ^2(x)})+(5 x^2+(5 x^2+5 x^3) \log (x)) \log ^2(\frac {1}{1+(2+2 x) \log (x)+(1+2 x+x^2) \log ^2(x)})} \, dx\) [2100]

Optimal result

Integrand size = 173, antiderivative size = 31 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=5-\frac {2}{5 x \left (x-\log \left (\frac {x^2}{(x+x (1+x) \log (x))^2}\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 6.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=\frac {2}{5 x \left (-x+\log \left (\frac {1}{(1+\log (x)+x \log (x))^2}\right )\right )} \] Input:

Integrate[(4 + 8*x + (8*x + 4*x^2)*Log[x] + (-2 + (-2 - 2*x)*Log[x])*Log[( 
1 + (2 + 2*x)*Log[x] + (1 + 2*x + x^2)*Log[x]^2)^(-1)])/(5*x^4 + (5*x^4 + 
5*x^5)*Log[x] + (-10*x^3 + (-10*x^3 - 10*x^4)*Log[x])*Log[(1 + (2 + 2*x)*L 
og[x] + (1 + 2*x + x^2)*Log[x]^2)^(-1)] + (5*x^2 + (5*x^2 + 5*x^3)*Log[x]) 
*Log[(1 + (2 + 2*x)*Log[x] + (1 + 2*x + x^2)*Log[x]^2)^(-1)]^2),x]
 

Output:

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

\(\displaystyle \int \frac {((-2 x-2) \log (x)-2) \log \left (\frac {1}{\left (x^2+2 x+1\right ) \log ^2(x)+(2 x+2) \log (x)+1}\right )+\left (4 x^2+8 x\right ) \log (x)+8 x+4}{5 x^4+\left (5 x^5+5 x^4\right ) \log (x)+\left (5 x^2+\left (5 x^3+5 x^2\right ) \log (x)\right ) \log ^2\left (\frac {1}{\left (x^2+2 x+1\right ) \log ^2(x)+(2 x+2) \log (x)+1}\right )+\left (\left (-10 x^4-10 x^3\right ) \log (x)-10 x^3\right ) \log \left (\frac {1}{\left (x^2+2 x+1\right ) \log ^2(x)+(2 x+2) \log (x)+1}\right )} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(4 + 8*x + (8*x + 4*x^2)*Log[x] + (-2 + (-2 - 2*x)*Log[x])*Log[(1 + (2 
 + 2*x)*Log[x] + (1 + 2*x + x^2)*Log[x]^2)^(-1)])/(5*x^4 + (5*x^4 + 5*x^5) 
*Log[x] + (-10*x^3 + (-10*x^3 - 10*x^4)*Log[x])*Log[(1 + (2 + 2*x)*Log[x] 
+ (1 + 2*x + x^2)*Log[x]^2)^(-1)] + (5*x^2 + (5*x^2 + 5*x^3)*Log[x])*Log[( 
1 + (2 + 2*x)*Log[x] + (1 + 2*x + x^2)*Log[x]^2)^(-1)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.96 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45

method	result	size
parallelrisch	\(-\frac {2}{5 x \left (-\ln \left (\frac {1}{x^{2} \ln \left (x \right )^{2}+2 x \ln \left (x \right )^{2}+\ln \left (x \right )^{2}+2 x \ln \left (x \right )+2 \ln \left (x \right )+1}\right )+x \right )}\)	\(45\)
default	\(-\frac {4 i}{5 \left (\pi \operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )\right )^{2} \operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )\right ) \operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \left (1+\ln \left (x \right ) \left (1+x \right )\right )^{2}\right )^{3}+2 i x +4 i \ln \left (1+\ln \left (x \right ) \left (1+x \right )\right )\right ) x}\)	\(105\)
risch	\(-\frac {4 i}{5 x \left (\pi \operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )\right )^{2} \operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )\right ) \operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )^{2}\right )^{3}+2 i x +4 i \ln \left (x \ln \left (x \right )+\ln \left (x \right )+1\right )\right )}\)	\(105\)

Input:

int((((-2-2*x)*ln(x)-2)*ln(1/((x^2+2*x+1)*ln(x)^2+(2+2*x)*ln(x)+1))+(4*x^2 
+8*x)*ln(x)+8*x+4)/(((5*x^3+5*x^2)*ln(x)+5*x^2)*ln(1/((x^2+2*x+1)*ln(x)^2+ 
(2+2*x)*ln(x)+1))^2+((-10*x^4-10*x^3)*ln(x)-10*x^3)*ln(1/((x^2+2*x+1)*ln(x 
)^2+(2+2*x)*ln(x)+1))+(5*x^5+5*x^4)*ln(x)+5*x^4),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=-\frac {2}{5 \, {\left (x^{2} - x \log \left (\frac {1}{{\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 2 \, {\left (x + 1\right )} \log \left (x\right ) + 1}\right )\right )}} \] Input:

integrate((((-2-2*x)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+ 
1))+(4*x^2+8*x)*log(x)+8*x+4)/(((5*x^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2* 
x+1)*log(x)^2+(2+2*x)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(1/ 
((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x, al 
gorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=\frac {2}{- 5 x^{2} + 5 x \log {\left (\frac {1}{\left (2 x + 2\right ) \log {\left (x \right )} + \left (x^{2} + 2 x + 1\right ) \log {\left (x \right )}^{2} + 1} \right )}} \] Input:

integrate((((-2-2*x)*ln(x)-2)*ln(1/((x**2+2*x+1)*ln(x)**2+(2+2*x)*ln(x)+1) 
)+(4*x**2+8*x)*ln(x)+8*x+4)/(((5*x**3+5*x**2)*ln(x)+5*x**2)*ln(1/((x**2+2* 
x+1)*ln(x)**2+(2+2*x)*ln(x)+1))**2+((-10*x**4-10*x**3)*ln(x)-10*x**3)*ln(1 
/((x**2+2*x+1)*ln(x)**2+(2+2*x)*ln(x)+1))+(5*x**5+5*x**4)*ln(x)+5*x**4),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=-\frac {2}{5 \, {\left (x^{2} + 2 \, x \log \left ({\left (x + 1\right )} \log \left (x\right ) + 1\right )\right )}} \] Input:

integrate((((-2-2*x)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+ 
1))+(4*x^2+8*x)*log(x)+8*x+4)/(((5*x^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2* 
x+1)*log(x)^2+(2+2*x)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(1/ 
((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x, al 
gorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 1.95 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=-\frac {2}{5 \, {\left (x^{2} + x \log \left (x^{2} \log \left (x\right )^{2} + 2 \, x \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )\right )}} \] Input:

integrate((((-2-2*x)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+ 
1))+(4*x^2+8*x)*log(x)+8*x+4)/(((5*x^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2* 
x+1)*log(x)^2+(2+2*x)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(1/ 
((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x, al 
gorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=\int \frac {8\,x-\ln \left (\frac {1}{\left (x^2+2\,x+1\right )\,{\ln \left (x\right )}^2+\left (2\,x+2\right )\,\ln \left (x\right )+1}\right )\,\left (\ln \left (x\right )\,\left (2\,x+2\right )+2\right )+\ln \left (x\right )\,\left (4\,x^2+8\,x\right )+4}{\ln \left (x\right )\,\left (5\,x^5+5\,x^4\right )-\ln \left (\frac {1}{\left (x^2+2\,x+1\right )\,{\ln \left (x\right )}^2+\left (2\,x+2\right )\,\ln \left (x\right )+1}\right )\,\left (\ln \left (x\right )\,\left (10\,x^4+10\,x^3\right )+10\,x^3\right )+{\ln \left (\frac {1}{\left (x^2+2\,x+1\right )\,{\ln \left (x\right )}^2+\left (2\,x+2\right )\,\ln \left (x\right )+1}\right )}^2\,\left (\ln \left (x\right )\,\left (5\,x^3+5\,x^2\right )+5\,x^2\right )+5\,x^4} \,d x \] Input:

int((8*x - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x 
)*(2*x + 2) + 2) + log(x)*(8*x + 4*x^2) + 4)/(log(x)*(5*x^4 + 5*x^5) - log 
(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(10*x^3 + 10 
*x^4) + 10*x^3) + log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1)) 
^2*(log(x)*(5*x^2 + 5*x^3) + 5*x^2) + 5*x^4),x)
 

Output:

int((8*x - log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x 
)*(2*x + 2) + 2) + log(x)*(8*x + 4*x^2) + 4)/(log(x)*(5*x^4 + 5*x^5) - log 
(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1))*(log(x)*(10*x^3 + 10 
*x^4) + 10*x^3) + log(1/(log(x)*(2*x + 2) + log(x)^2*(2*x + x^2 + 1) + 1)) 
^2*(log(x)*(5*x^2 + 5*x^3) + 5*x^2) + 5*x^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {4+8 x+\left (8 x+4 x^2\right ) \log (x)+(-2+(-2-2 x) \log (x)) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )}{5 x^4+\left (5 x^4+5 x^5\right ) \log (x)+\left (-10 x^3+\left (-10 x^3-10 x^4\right ) \log (x)\right ) \log \left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )+\left (5 x^2+\left (5 x^2+5 x^3\right ) \log (x)\right ) \log ^2\left (\frac {1}{1+(2+2 x) \log (x)+\left (1+2 x+x^2\right ) \log ^2(x)}\right )} \, dx=-\frac {2}{5 x \left (\mathrm {log}\left (\mathrm {log}\left (x \right )^{2} x^{2}+2 \mathrm {log}\left (x \right )^{2} x +\mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x \right ) x +2 \,\mathrm {log}\left (x \right )+1\right )+x \right )} \] Input:

int((((-2-2*x)*log(x)-2)*log(1/((x^2+2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(4 
*x^2+8*x)*log(x)+8*x+4)/(((5*x^3+5*x^2)*log(x)+5*x^2)*log(1/((x^2+2*x+1)*l 
og(x)^2+(2+2*x)*log(x)+1))^2+((-10*x^4-10*x^3)*log(x)-10*x^3)*log(1/((x^2+ 
2*x+1)*log(x)^2+(2+2*x)*log(x)+1))+(5*x^5+5*x^4)*log(x)+5*x^4),x)
 

Output:

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