\(\int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+(-60 x^3-5 x^4-10 x^7) \log ^2(x)+(-20 x+20 x^3 \log ^2(x)) \log (1-x^2 \log ^2(x))}{-16-8 x-x^2+8 x^4+2 x^5-x^8+(16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}) \log ^2(x)+(16+4 x-4 x^4+(-16 x^2-4 x^3+4 x^6) \log ^2(x)) \log (1-x^2 \log ^2(x))+(-4+4 x^2 \log ^2(x)) \log ^2(1-x^2 \log ^2(x))} \, dx\) [494]

Optimal result

Integrand size = 198, antiderivative size = 29 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 x^2}{-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )} \] Output:

5/(x^4-x+2*ln(-x^2*ln(x)^2+1)-4)*x^2
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 x^2}{-4-x+x^4+2 \log \left (1-x^2 \log ^2(x)\right )} \] Input:

Integrate[(40*x + 5*x^2 + 10*x^5 - 20*x^3*Log[x] + (-60*x^3 - 5*x^4 - 10*x 
^7)*Log[x]^2 + (-20*x + 20*x^3*Log[x]^2)*Log[1 - x^2*Log[x]^2])/(-16 - 8*x 
 - x^2 + 8*x^4 + 2*x^5 - x^8 + (16*x^2 + 8*x^3 + x^4 - 8*x^6 - 2*x^7 + x^1 
0)*Log[x]^2 + (16 + 4*x - 4*x^4 + (-16*x^2 - 4*x^3 + 4*x^6)*Log[x]^2)*Log[ 
1 - x^2*Log[x]^2] + (-4 + 4*x^2*Log[x]^2)*Log[1 - x^2*Log[x]^2]^2),x]
 

Output:

(5*x^2)/(-4 - x + x^4 + 2*Log[1 - x^2*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.

\(\displaystyle \int \frac {10 x^5-20 x^3 \log (x)+5 x^2+\left (20 x^3 \log ^2(x)-20 x\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-10 x^7-5 x^4-60 x^3\right ) \log ^2(x)+40 x}{-x^8+2 x^5+8 x^4-x^2+\left (4 x^2 \log ^2(x)-4\right ) \log ^2\left (1-x^2 \log ^2(x)\right )+\left (-4 x^4+\left (4 x^6-4 x^3-16 x^2\right ) \log ^2(x)+4 x+16\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (x^{10}-2 x^7-8 x^6+x^4+8 x^3+16 x^2\right ) \log ^2(x)-8 x-16} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {5 x \left (-2 x^4+4 \log \left (1-x^2 \log ^2(x)\right )+4 x^2 \log (x)+x^2 \log ^2(x) \left (2 x^4-4 \log \left (1-x^2 \log ^2(x)\right )+x+12\right )-x-8\right )}{\left (1-x^2 \log ^2(x)\right ) \left (-x^4-2 \log \left (1-x^2 \log ^2(x)\right )+x+4\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 5 \int -\frac {x \left (2 x^4-4 \log (x) x^2-\log ^2(x) \left (2 x^4+x-4 \log \left (1-x^2 \log ^2(x)\right )+12\right ) x^2+x-4 \log \left (1-x^2 \log ^2(x)\right )+8\right )}{\left (1-x^2 \log ^2(x)\right ) \left (-x^4+x-2 \log \left (1-x^2 \log ^2(x)\right )+4\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -5 \int \frac {x \left (2 x^4-4 \log (x) x^2-\log ^2(x) \left (2 x^4+x-4 \log \left (1-x^2 \log ^2(x)\right )+12\right ) x^2+x-4 \log \left (1-x^2 \log ^2(x)\right )+8\right )}{\left (1-x^2 \log ^2(x)\right ) \left (-x^4+x-2 \log \left (1-x^2 \log ^2(x)\right )+4\right )^2}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -5 \left (\int \frac {x^2}{\left (x^2 \log ^2(x)-1\right ) \left (x^4-x+2 \log \left (1-x^2 \log ^2(x)\right )-4\right )^2}dx-\int \frac {x^4 \log ^2(x)}{\left (x^2 \log ^2(x)-1\right ) \left (x^4-x+2 \log \left (1-x^2 \log ^2(x)\right )-4\right )^2}dx-2 \int \frac {x}{x^4-x+2 \log \left (1-x^2 \log ^2(x)\right )-4}dx+4 \int \frac {x^7 \log ^2(x)}{\left (x^2 \log ^2(x)-1\right ) \left (x^4-x+2 \log \left (1-x^2 \log ^2(x)\right )-4\right )^2}dx-4 \int \frac {x^5}{\left (x^2 \log ^2(x)-1\right ) \left (x^4-x+2 \log \left (1-x^2 \log ^2(x)\right )-4\right )^2}dx+4 \int \frac {x^3 \log (x)}{\left (x^2 \log ^2(x)-1\right ) \left (x^4-x+2 \log \left (1-x^2 \log ^2(x)\right )-4\right )^2}dx+4 \int \frac {x^3 \log ^2(x)}{\left (x^2 \log ^2(x)-1\right ) \left (x^4-x+2 \log \left (1-x^2 \log ^2(x)\right )-4\right )^2}dx\right )\)

Input:

Int[(40*x + 5*x^2 + 10*x^5 - 20*x^3*Log[x] + (-60*x^3 - 5*x^4 - 10*x^7)*Lo 
g[x]^2 + (-20*x + 20*x^3*Log[x]^2)*Log[1 - x^2*Log[x]^2])/(-16 - 8*x - x^2 
 + 8*x^4 + 2*x^5 - x^8 + (16*x^2 + 8*x^3 + x^4 - 8*x^6 - 2*x^7 + x^10)*Log 
[x]^2 + (16 + 4*x - 4*x^4 + (-16*x^2 - 4*x^3 + 4*x^6)*Log[x]^2)*Log[1 - x^ 
2*Log[x]^2] + (-4 + 4*x^2*Log[x]^2)*Log[1 - x^2*Log[x]^2]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 141.44 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {5 x^{2}}{x^{4}-x +2 \ln \left (-x^{2} \ln \left (x \right )^{2}+1\right )-4}\)	\(30\)
risch	\(\frac {5 x^{2}}{x^{4}-x +2 \ln \left (-x^{2} \ln \left (x \right )^{2}+1\right )-4}\)	\(30\)
parallelrisch	\(\frac {5 x^{2}}{x^{4}-x +2 \ln \left (-x^{2} \ln \left (x \right )^{2}+1\right )-4}\)	\(30\)

Input:

int(((20*x^3*ln(x)^2-20*x)*ln(-x^2*ln(x)^2+1)+(-10*x^7-5*x^4-60*x^3)*ln(x) 
^2-20*x^3*ln(x)+10*x^5+5*x^2+40*x)/((4*x^2*ln(x)^2-4)*ln(-x^2*ln(x)^2+1)^2 
+((4*x^6-4*x^3-16*x^2)*ln(x)^2-4*x^4+4*x+16)*ln(-x^2*ln(x)^2+1)+(x^10-2*x^ 
7-8*x^6+x^4+8*x^3+16*x^2)*ln(x)^2-x^8+2*x^5+8*x^4-x^2-8*x-16),x,method=_RE 
TURNVERBOSE)
 

Output:

5/(x^4-x+2*ln(-x^2*ln(x)^2+1)-4)*x^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 \, x^{2}}{x^{4} - x + 2 \, \log \left (-x^{2} \log \left (x\right )^{2} + 1\right ) - 4} \] Input:

integrate(((20*x^3*log(x)^2-20*x)*log(-x^2*log(x)^2+1)+(-10*x^7-5*x^4-60*x 
^3)*log(x)^2-20*x^3*log(x)+10*x^5+5*x^2+40*x)/((4*x^2*log(x)^2-4)*log(-x^2 
*log(x)^2+1)^2+((4*x^6-4*x^3-16*x^2)*log(x)^2-4*x^4+4*x+16)*log(-x^2*log(x 
)^2+1)+(x^10-2*x^7-8*x^6+x^4+8*x^3+16*x^2)*log(x)^2-x^8+2*x^5+8*x^4-x^2-8* 
x-16),x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 x^{2}}{x^{4} - x + 2 \log {\left (- x^{2} \log {\left (x \right )}^{2} + 1 \right )} - 4} \] Input:

integrate(((20*x**3*ln(x)**2-20*x)*ln(-x**2*ln(x)**2+1)+(-10*x**7-5*x**4-6 
0*x**3)*ln(x)**2-20*x**3*ln(x)+10*x**5+5*x**2+40*x)/((4*x**2*ln(x)**2-4)*l 
n(-x**2*ln(x)**2+1)**2+((4*x**6-4*x**3-16*x**2)*ln(x)**2-4*x**4+4*x+16)*ln 
(-x**2*ln(x)**2+1)+(x**10-2*x**7-8*x**6+x**4+8*x**3+16*x**2)*ln(x)**2-x**8 
+2*x**5+8*x**4-x**2-8*x-16),x)
 

Output:

5*x**2/(x**4 - x + 2*log(-x**2*log(x)**2 + 1) - 4)
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 \, x^{2}}{x^{4} - x + 2 \, \log \left (x \log \left (x\right ) + 1\right ) + 2 \, \log \left (-x \log \left (x\right ) + 1\right ) - 4} \] Input:

integrate(((20*x^3*log(x)^2-20*x)*log(-x^2*log(x)^2+1)+(-10*x^7-5*x^4-60*x 
^3)*log(x)^2-20*x^3*log(x)+10*x^5+5*x^2+40*x)/((4*x^2*log(x)^2-4)*log(-x^2 
*log(x)^2+1)^2+((4*x^6-4*x^3-16*x^2)*log(x)^2-4*x^4+4*x+16)*log(-x^2*log(x 
)^2+1)+(x^10-2*x^7-8*x^6+x^4+8*x^3+16*x^2)*log(x)^2-x^8+2*x^5+8*x^4-x^2-8* 
x-16),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 \, x^{2}}{x^{4} - x + 2 \, \log \left (-x^{2} \log \left (x\right )^{2} + 1\right ) - 4} \] Input:

integrate(((20*x^3*log(x)^2-20*x)*log(-x^2*log(x)^2+1)+(-10*x^7-5*x^4-60*x 
^3)*log(x)^2-20*x^3*log(x)+10*x^5+5*x^2+40*x)/((4*x^2*log(x)^2-4)*log(-x^2 
*log(x)^2+1)^2+((4*x^6-4*x^3-16*x^2)*log(x)^2-4*x^4+4*x+16)*log(-x^2*log(x 
)^2+1)+(x^10-2*x^7-8*x^6+x^4+8*x^3+16*x^2)*log(x)^2-x^8+2*x^5+8*x^4-x^2-8* 
x-16),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\int \frac {40\,x-20\,x^3\,\ln \left (x\right )-\ln \left (1-x^2\,{\ln \left (x\right )}^2\right )\,\left (20\,x-20\,x^3\,{\ln \left (x\right )}^2\right )-{\ln \left (x\right )}^2\,\left (10\,x^7+5\,x^4+60\,x^3\right )+5\,x^2+10\,x^5}{{\ln \left (x\right )}^2\,\left (x^{10}-2\,x^7-8\,x^6+x^4+8\,x^3+16\,x^2\right )-8\,x+\ln \left (1-x^2\,{\ln \left (x\right )}^2\right )\,\left (4\,x-{\ln \left (x\right )}^2\,\left (-4\,x^6+4\,x^3+16\,x^2\right )-4\,x^4+16\right )-x^2+8\,x^4+2\,x^5-x^8+{\ln \left (1-x^2\,{\ln \left (x\right )}^2\right )}^2\,\left (4\,x^2\,{\ln \left (x\right )}^2-4\right )-16} \,d x \] Input:

int((40*x - 20*x^3*log(x) - log(1 - x^2*log(x)^2)*(20*x - 20*x^3*log(x)^2) 
 - log(x)^2*(60*x^3 + 5*x^4 + 10*x^7) + 5*x^2 + 10*x^5)/(log(x)^2*(16*x^2 
+ 8*x^3 + x^4 - 8*x^6 - 2*x^7 + x^10) - 8*x + log(1 - x^2*log(x)^2)*(4*x - 
 log(x)^2*(16*x^2 + 4*x^3 - 4*x^6) - 4*x^4 + 16) - x^2 + 8*x^4 + 2*x^5 - x 
^8 + log(1 - x^2*log(x)^2)^2*(4*x^2*log(x)^2 - 4) - 16),x)
 

Output:

int((40*x - 20*x^3*log(x) - log(1 - x^2*log(x)^2)*(20*x - 20*x^3*log(x)^2) 
 - log(x)^2*(60*x^3 + 5*x^4 + 10*x^7) + 5*x^2 + 10*x^5)/(log(x)^2*(16*x^2 
+ 8*x^3 + x^4 - 8*x^6 - 2*x^7 + x^10) - 8*x + log(1 - x^2*log(x)^2)*(4*x - 
 log(x)^2*(16*x^2 + 4*x^3 - 4*x^6) - 4*x^4 + 16) - x^2 + 8*x^4 + 2*x^5 - x 
^8 + log(1 - x^2*log(x)^2)^2*(4*x^2*log(x)^2 - 4) - 16), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {40 x+5 x^2+10 x^5-20 x^3 \log (x)+\left (-60 x^3-5 x^4-10 x^7\right ) \log ^2(x)+\left (-20 x+20 x^3 \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )}{-16-8 x-x^2+8 x^4+2 x^5-x^8+\left (16 x^2+8 x^3+x^4-8 x^6-2 x^7+x^{10}\right ) \log ^2(x)+\left (16+4 x-4 x^4+\left (-16 x^2-4 x^3+4 x^6\right ) \log ^2(x)\right ) \log \left (1-x^2 \log ^2(x)\right )+\left (-4+4 x^2 \log ^2(x)\right ) \log ^2\left (1-x^2 \log ^2(x)\right )} \, dx=\frac {5 x^{2}}{2 \,\mathrm {log}\left (-\mathrm {log}\left (x \right )^{2} x^{2}+1\right )+x^{4}-x -4} \] Input:

int(((20*x^3*log(x)^2-20*x)*log(-x^2*log(x)^2+1)+(-10*x^7-5*x^4-60*x^3)*lo 
g(x)^2-20*x^3*log(x)+10*x^5+5*x^2+40*x)/((4*x^2*log(x)^2-4)*log(-x^2*log(x 
)^2+1)^2+((4*x^6-4*x^3-16*x^2)*log(x)^2-4*x^4+4*x+16)*log(-x^2*log(x)^2+1) 
+(x^10-2*x^7-8*x^6+x^4+8*x^3+16*x^2)*log(x)^2-x^8+2*x^5+8*x^4-x^2-8*x-16), 
x)
 

Output:

(5*x**2)/(2*log( - log(x)**2*x**2 + 1) + x**4 - x - 4)