\(\int \frac {8 x^2+8 \log (x)+(4 x^2+4 \log (x)) \log (x^2)+e^{\log ^2(x^2+\log (x))} (-4 x^2+(-4-4 x^2) \log (x)-4 \log ^2(x)+(-8-16 x^2) \log (x) \log (x^2+\log (x)))}{e^{2 \log ^2(x^2+\log (x))} (x^4 \log ^2(x)+x^2 \log ^3(x))+e^{\log ^2(x^2+\log (x))} (-2 x^4 \log (x)-2 x^2 \log ^2(x)) \log (x^2)+(x^4+x^2 \log (x)) \log ^2(x^2)} \, dx\) [806]

Optimal result

Integrand size = 161, antiderivative size = 28 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \] Output:

4/x/(exp(ln(ln(x)+x^2)^2)*ln(x)-ln(x^2))
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )} \] Input:

Integrate[(8*x^2 + 8*Log[x] + (4*x^2 + 4*Log[x])*Log[x^2] + E^Log[x^2 + Lo 
g[x]]^2*(-4*x^2 + (-4 - 4*x^2)*Log[x] - 4*Log[x]^2 + (-8 - 16*x^2)*Log[x]* 
Log[x^2 + Log[x]]))/(E^(2*Log[x^2 + Log[x]]^2)*(x^4*Log[x]^2 + x^2*Log[x]^ 
3) + E^Log[x^2 + Log[x]]^2*(-2*x^4*Log[x] - 2*x^2*Log[x]^2)*Log[x^2] + (x^ 
4 + x^2*Log[x])*Log[x^2]^2),x]
 

Output:

4/(x*(E^Log[x^2 + Log[x]]^2*Log[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.

\(\displaystyle \int \frac {8 x^2+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4 x^2-4\right ) \log (x)+\left (-16 x^2-8\right ) \log (x) \log \left (x^2+\log (x)\right )-4 \log ^2(x)\right )+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+8 \log (x)}{\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {8 x^2+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4 x^2-4\right ) \log (x)+\left (-16 x^2-8\right ) \log (x) \log \left (x^2+\log (x)\right )-4 \log ^2(x)\right )+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+8 \log (x)}{x^2 \left (x^2+\log (x)\right ) \left (e^{\log ^2\left (x^2+\log (x)\right )} \log (x)-\log \left (x^2\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(8*x^2 + 8*Log[x] + (4*x^2 + 4*Log[x])*Log[x^2] + E^Log[x^2 + Log[x]]^ 
2*(-4*x^2 + (-4 - 4*x^2)*Log[x] - 4*Log[x]^2 + (-8 - 16*x^2)*Log[x]*Log[x^ 
2 + Log[x]]))/(E^(2*Log[x^2 + Log[x]]^2)*(x^4*Log[x]^2 + x^2*Log[x]^3) + E 
^Log[x^2 + Log[x]]^2*(-2*x^4*Log[x] - 2*x^2*Log[x]^2)*Log[x^2] + (x^4 + x^ 
2*Log[x])*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 = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.64

\[-\frac {8 i}{x \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 i \ln \left (x \right ) {\mathrm e}^{\ln \left (\ln \left (x \right )+x^{2}\right )^{2}}+4 i \ln \left (x \right )\right )}\]

Input:

int((((-16*x^2-8)*ln(x)*ln(ln(x)+x^2)-4*ln(x)^2+(-4*x^2-4)*ln(x)-4*x^2)*ex 
p(ln(ln(x)+x^2)^2)+(4*ln(x)+4*x^2)*ln(x^2)+8*ln(x)+8*x^2)/((x^2*ln(x)^3+x^ 
4*ln(x)^2)*exp(ln(ln(x)+x^2)^2)^2+(-2*x^2*ln(x)^2-2*x^4*ln(x))*ln(x^2)*exp 
(ln(ln(x)+x^2)^2)+(x^2*ln(x)+x^4)*ln(x^2)^2),x)
 

Output:

-8*I/x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I* 
x^2)^3-2*I*ln(x)*exp(ln(ln(x)+x^2)^2)+4*I*ln(x))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x e^{\left (\log \left (x^{2} + \log \left (x\right )\right )^{2}\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )} \] Input:

integrate((((-16*x^2-8)*log(x)*log(log(x)+x^2)-4*log(x)^2+(-4*x^2-4)*log(x 
)-4*x^2)*exp(log(log(x)+x^2)^2)+(4*log(x)+4*x^2)*log(x^2)+8*log(x)+8*x^2)/ 
((x^2*log(x)^3+x^4*log(x)^2)*exp(log(log(x)+x^2)^2)^2+(-2*x^2*log(x)^2-2*x 
^4*log(x))*log(x^2)*exp(log(log(x)+x^2)^2)+(x^2*log(x)+x^4)*log(x^2)^2),x, 
 algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x e^{\log {\left (x^{2} + \log {\left (x \right )} \right )}^{2}} \log {\left (x \right )} - 2 x \log {\left (x \right )}} \] Input:

integrate((((-16*x**2-8)*ln(x)*ln(ln(x)+x**2)-4*ln(x)**2+(-4*x**2-4)*ln(x) 
-4*x**2)*exp(ln(ln(x)+x**2)**2)+(4*ln(x)+4*x**2)*ln(x**2)+8*ln(x)+8*x**2)/ 
((x**2*ln(x)**3+x**4*ln(x)**2)*exp(ln(ln(x)+x**2)**2)**2+(-2*x**2*ln(x)**2 
-2*x**4*ln(x))*ln(x**2)*exp(ln(ln(x)+x**2)**2)+(x**2*ln(x)+x**4)*ln(x**2)* 
*2),x)
 

Output:

4/(x*exp(log(x**2 + log(x))**2)*log(x) - 2*x*log(x))
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x e^{\left (\log \left (x^{2} + \log \left (x\right )\right )^{2}\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )} \] Input:

integrate((((-16*x^2-8)*log(x)*log(log(x)+x^2)-4*log(x)^2+(-4*x^2-4)*log(x 
)-4*x^2)*exp(log(log(x)+x^2)^2)+(4*log(x)+4*x^2)*log(x^2)+8*log(x)+8*x^2)/ 
((x^2*log(x)^3+x^4*log(x)^2)*exp(log(log(x)+x^2)^2)^2+(-2*x^2*log(x)^2-2*x 
^4*log(x))*log(x^2)*exp(log(log(x)+x^2)^2)+(x^2*log(x)+x^4)*log(x^2)^2),x, 
 algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x e^{\left (\log \left (x^{2} + \log \left (x\right )\right )^{2}\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )} \] Input:

integrate((((-16*x^2-8)*log(x)*log(log(x)+x^2)-4*log(x)^2+(-4*x^2-4)*log(x 
)-4*x^2)*exp(log(log(x)+x^2)^2)+(4*log(x)+4*x^2)*log(x^2)+8*log(x)+8*x^2)/ 
((x^2*log(x)^3+x^4*log(x)^2)*exp(log(log(x)+x^2)^2)^2+(-2*x^2*log(x)^2-2*x 
^4*log(x))*log(x^2)*exp(log(log(x)+x^2)^2)+(x^2*log(x)+x^4)*log(x^2)^2),x, 
 algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.18 (sec) , antiderivative size = 310, normalized size of antiderivative = 11.07 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4\,\left (2\,x^2\,\ln \left (x\right )+2\,{\ln \left (x\right )}^2\right )\,{\left (x\,\ln \left (x\right )+x^3\right )}^2-4\,\ln \left (x^2\right )\,{\left (x\,\ln \left (x\right )+x^3\right )}^2\,\left (\ln \left (x\right )+2\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )+x^2+4\,x^2\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )\right )}{x^2\,\left (\ln \left (x\right )+x^2\right )\,\left (\ln \left (x^2\right )-{\mathrm {e}}^{{\ln \left (\ln \left (x\right )+x^2\right )}^2}\,\ln \left (x\right )\right )\,\left (x^5\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+x\,{\ln \left (x\right )}^2\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+4\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2+8\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^3+8\,x^5\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2+4\,x\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^3+2\,x\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+2\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+4\,x^5\,\ln \left (\ln \left (x\right )+x^2\right )\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+4\,x^3\,\ln \left (\ln \left (x\right )+x^2\right )\,{\ln \left (x\right )}^2\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )} \] Input:

int((8*log(x) - exp(log(log(x) + x^2)^2)*(4*log(x)^2 + 4*x^2 + log(x)*(4*x 
^2 + 4) + log(log(x) + x^2)*log(x)*(16*x^2 + 8)) + log(x^2)*(4*log(x) + 4* 
x^2) + 8*x^2)/(log(x^2)^2*(x^2*log(x) + x^4) + exp(2*log(log(x) + x^2)^2)* 
(x^2*log(x)^3 + x^4*log(x)^2) - log(x^2)*exp(log(log(x) + x^2)^2)*(2*x^4*l 
og(x) + 2*x^2*log(x)^2)),x)
 

Output:

(4*(2*x^2*log(x) + 2*log(x)^2)*(x*log(x) + x^3)^2 - 4*log(x^2)*(x*log(x) + 
 x^3)^2*(log(x) + 2*log(log(x) + x^2)*log(x) + x^2 + 4*x^2*log(log(x) + x^ 
2)*log(x)))/(x^2*(log(x) + x^2)*(log(x^2) - exp(log(log(x) + x^2)^2)*log(x 
))*(x^5*(log(x^2) - 2*log(x)) + x*log(x)^2*(log(x^2) - 2*log(x)) + 2*x^3*l 
og(x)*(log(x^2) - 2*log(x)) + 4*x^3*log(log(x) + x^2)*log(x)^2 + 8*x^3*log 
(log(x) + x^2)*log(x)^3 + 8*x^5*log(log(x) + x^2)*log(x)^2 + 4*x*log(log(x 
) + x^2)*log(x)^3 + 2*x*log(log(x) + x^2)*log(x)^2*(log(x^2) - 2*log(x)) + 
 2*x^3*log(log(x) + x^2)*log(x)*(log(x^2) - 2*log(x)) + 4*x^5*log(log(x) + 
 x^2)*log(x)*(log(x^2) - 2*log(x)) + 4*x^3*log(log(x) + x^2)*log(x)^2*(log 
(x^2) - 2*log(x))))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {8 x^2+8 \log (x)+\left (4 x^2+4 \log (x)\right ) \log \left (x^2\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-4 x^2+\left (-4-4 x^2\right ) \log (x)-4 \log ^2(x)+\left (-8-16 x^2\right ) \log (x) \log \left (x^2+\log (x)\right )\right )}{e^{2 \log ^2\left (x^2+\log (x)\right )} \left (x^4 \log ^2(x)+x^2 \log ^3(x)\right )+e^{\log ^2\left (x^2+\log (x)\right )} \left (-2 x^4 \log (x)-2 x^2 \log ^2(x)\right ) \log \left (x^2\right )+\left (x^4+x^2 \log (x)\right ) \log ^2\left (x^2\right )} \, dx=\frac {4}{x \left (e^{\mathrm {log}\left (\mathrm {log}\left (x \right )+x^{2}\right )^{2}} \mathrm {log}\left (x \right )-\mathrm {log}\left (x^{2}\right )\right )} \] Input:

int((((-16*x^2-8)*log(x)*log(log(x)+x^2)-4*log(x)^2+(-4*x^2-4)*log(x)-4*x^ 
2)*exp(log(log(x)+x^2)^2)+(4*log(x)+4*x^2)*log(x^2)+8*log(x)+8*x^2)/((x^2* 
log(x)^3+x^4*log(x)^2)*exp(log(log(x)+x^2)^2)^2+(-2*x^2*log(x)^2-2*x^4*log 
(x))*log(x^2)*exp(log(log(x)+x^2)^2)+(x^2*log(x)+x^4)*log(x^2)^2),x)
 

Output:

4/(x*(e**(log(log(x) + x**2)**2)*log(x) - log(x**2)))