\(\int \frac {-8+4 x^2-8 x^2 \log (x)+(-8+4 x^2) \log (-2+x^2)+((4 x-2 x^3) \log (x)+(4 x-2 x^3) \log (x) \log (-2+x^2)) \log ^3(\frac {1+2 \log (-2+x^2)+\log ^2(-2+x^2)}{\log ^2(x)})}{((-2 x+x^3) \log (x)+(-2 x+x^3) \log (x) \log (-2+x^2)) \log ^3(\frac {1+2 \log (-2+x^2)+\log ^2(-2+x^2)}{\log ^2(x)})} \, dx\) [520]

Optimal result

Integrand size = 142, antiderivative size = 23 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=13-2 x+\frac {1}{\log ^2\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \] Output:

1/ln((ln(x^2-2)+1)^2/ln(x)^2)^2+13-2*x
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=-2 x+\frac {1}{\log ^2\left (\frac {\left (1+\log \left (-2+x^2\right )\right )^2}{\log ^2(x)}\right )} \] Input:

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

Output:

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

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(23)=46\).

Time = 62.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.61

method	result	size
parallelrisch	\(\frac {4-8 {\ln \left (\frac {\ln \left (x^{2}-2\right )^{2}+2 \ln \left (x^{2}-2\right )+1}{\ln \left (x \right )^{2}}\right )}^{2} x}{4 {\ln \left (\frac {\ln \left (x^{2}-2\right )^{2}+2 \ln \left (x^{2}-2\right )+1}{\ln \left (x \right )^{2}}\right )}^{2}}\)	\(60\)
risch	\(-2 x -\frac {4}{{\left (\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) \operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right ) \operatorname {csgn}\left (\frac {i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}}{\ln \left (x \right )^{2}}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}}{\ln \left (x \right )^{2}}\right )}^{2}-\pi \operatorname {csgn}\left (i \ln \left (x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{2}-\pi \operatorname {csgn}\left (i \ln \left (x \right )^{2}\right )^{3}+\pi {\operatorname {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )\right )}^{2} \operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (\ln \left (x^{2}-2\right )+1\right )\right ) {\operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right )}^{2}+\pi {\operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right )}^{3}-\pi \,\operatorname {csgn}\left (i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}\right ) {\operatorname {csgn}\left (\frac {i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}}{\ln \left (x \right )^{2}}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i {\left (\ln \left (x^{2}-2\right )+1\right )}^{2}}{\ln \left (x \right )^{2}}\right )}^{3}-4 i \ln \left (\ln \left (x \right )\right )+4 i \ln \left (\ln \left (x^{2}-2\right )+1\right )\right )}^{2}}\)	\(290\)
default	\(\text {Expression too large to display}\)	\(13907\)
parts	\(\text {Expression too large to display}\)	\(13907\)

Input:

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

Output:

1/4*(4-8*ln((ln(x^2-2)^2+2*ln(x^2-2)+1)/ln(x)^2)^2*x)/ln((ln(x^2-2)^2+2*ln 
(x^2-2)+1)/ln(x)^2)^2
                                                                                    
                                                                                    
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (23) = 46\).

Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.57 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=-\frac {2 \, x \log \left (\frac {\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1}{\log \left (x\right )^{2}}\right )^{2} - 1}{\log \left (\frac {\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1}{\log \left (x\right )^{2}}\right )^{2}} \] Input:

integrate((((-2*x^3+4*x)*log(x)*log(x^2-2)+(-2*x^3+4*x)*log(x))*log((log(x 
^2-2)^2+2*log(x^2-2)+1)/log(x)^2)^3+(4*x^2-8)*log(x^2-2)-8*x^2*log(x)+4*x^ 
2-8)/((x^3-2*x)*log(x)*log(x^2-2)+(x^3-2*x)*log(x))/log((log(x^2-2)^2+2*lo 
g(x^2-2)+1)/log(x)^2)^3,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=- 2 x + \frac {1}{\log {\left (\frac {\log {\left (x^{2} - 2 \right )}^{2} + 2 \log {\left (x^{2} - 2 \right )} + 1}{\log {\left (x \right )}^{2}} \right )}^{2}} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (23) = 46\).

Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=-\frac {8 \, x \log \left (\log \left (x^{2} - 2\right ) + 1\right )^{2} - 16 \, x \log \left (\log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )\right ) + 8 \, x \log \left (\log \left (x\right )\right )^{2} - 1}{4 \, {\left (\log \left (\log \left (x^{2} - 2\right ) + 1\right )^{2} - 2 \, \log \left (\log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2}\right )}} \] Input:

integrate((((-2*x^3+4*x)*log(x)*log(x^2-2)+(-2*x^3+4*x)*log(x))*log((log(x 
^2-2)^2+2*log(x^2-2)+1)/log(x)^2)^3+(4*x^2-8)*log(x^2-2)-8*x^2*log(x)+4*x^ 
2-8)/((x^3-2*x)*log(x)*log(x^2-2)+(x^3-2*x)*log(x))/log((log(x^2-2)^2+2*lo 
g(x^2-2)+1)/log(x)^2)^3,x, algorithm="maxima")
 

Output:

-1/4*(8*x*log(log(x^2 - 2) + 1)^2 - 16*x*log(log(x^2 - 2) + 1)*log(log(x)) 
 + 8*x*log(log(x))^2 - 1)/(log(log(x^2 - 2) + 1)^2 - 2*log(log(x^2 - 2) + 
1)*log(log(x)) + log(log(x))^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (23) = 46\).

Time = 40.27 (sec) , antiderivative size = 393, normalized size of antiderivative = 17.09 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=-2 \, x + \frac {x^{2} \log \left (x^{2} - 2\right ) - 2 \, x^{2} \log \left (x\right ) + x^{2} - 2 \, \log \left (x^{2} - 2\right ) - 2}{x^{2} \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} - 2 \, x^{2} \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )^{2}\right ) + 4 \, x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right ) + x^{2} \log \left (x^{2} - 2\right ) \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} \log \left (x\right ) + x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} - 2 \, x^{2} \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )^{2}\right ) + x^{2} \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} + 4 \, \log \left (x^{2} - 2\right ) \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )^{2}\right ) - 2 \, \log \left (x^{2} - 2\right ) \log \left (\log \left (x\right )^{2}\right )^{2} - 2 \, \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right )^{2} + 4 \, \log \left (\log \left (x^{2} - 2\right )^{2} + 2 \, \log \left (x^{2} - 2\right ) + 1\right ) \log \left (\log \left (x\right )^{2}\right ) - 2 \, \log \left (\log \left (x\right )^{2}\right )^{2}} \] Input:

integrate((((-2*x^3+4*x)*log(x)*log(x^2-2)+(-2*x^3+4*x)*log(x))*log((log(x 
^2-2)^2+2*log(x^2-2)+1)/log(x)^2)^3+(4*x^2-8)*log(x^2-2)-8*x^2*log(x)+4*x^ 
2-8)/((x^3-2*x)*log(x)*log(x^2-2)+(x^3-2*x)*log(x))/log((log(x^2-2)^2+2*lo 
g(x^2-2)+1)/log(x)^2)^3,x, algorithm="giac")
 

Output:

-2*x + (x^2*log(x^2 - 2) - 2*x^2*log(x) + x^2 - 2*log(x^2 - 2) - 2)/(x^2*l 
og(x^2 - 2)*log(log(x^2 - 2)^2 + 2*log(x^2 - 2) + 1)^2 - 2*x^2*log(x^2 - 2 
)*log(log(x^2 - 2)^2 + 2*log(x^2 - 2) + 1)*log(log(x)^2) + 4*x^2*log(log(x 
^2 - 2)^2 + 2*log(x^2 - 2) + 1)*log(x)*log(log(x)^2) + x^2*log(x^2 - 2)*lo 
g(log(x)^2)^2 - 2*x^2*log(x)*log(log(x)^2)^2 - 2*x^2*log(log(x^2 - 2)^2 + 
2*log(x^2 - 2) + 1)^2*log(x) + x^2*log(log(x^2 - 2)^2 + 2*log(x^2 - 2) + 1 
)^2 - 2*x^2*log(log(x^2 - 2)^2 + 2*log(x^2 - 2) + 1)*log(log(x)^2) + x^2*l 
og(log(x)^2)^2 - 2*log(x^2 - 2)*log(log(x^2 - 2)^2 + 2*log(x^2 - 2) + 1)^2 
 + 4*log(x^2 - 2)*log(log(x^2 - 2)^2 + 2*log(x^2 - 2) + 1)*log(log(x)^2) - 
 2*log(x^2 - 2)*log(log(x)^2)^2 - 2*log(log(x^2 - 2)^2 + 2*log(x^2 - 2) + 
1)^2 + 4*log(log(x^2 - 2)^2 + 2*log(x^2 - 2) + 1)*log(log(x)^2) - 2*log(lo 
g(x)^2)^2)
 

Mupad [B] (verification not implemented)

Time = 3.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=\frac {1}{{\ln \left (\frac {{\ln \left (x^2-2\right )}^2+2\,\ln \left (x^2-2\right )+1}{{\ln \left (x\right )}^2}\right )}^2}-2\,x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {-8+4 x^2-8 x^2 \log (x)+\left (-8+4 x^2\right ) \log \left (-2+x^2\right )+\left (\left (4 x-2 x^3\right ) \log (x)+\left (4 x-2 x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )}{\left (\left (-2 x+x^3\right ) \log (x)+\left (-2 x+x^3\right ) \log (x) \log \left (-2+x^2\right )\right ) \log ^3\left (\frac {1+2 \log \left (-2+x^2\right )+\log ^2\left (-2+x^2\right )}{\log ^2(x)}\right )} \, dx=\frac {-2 {\mathrm {log}\left (\frac {\mathrm {log}\left (x^{2}-2\right )^{2}+2 \,\mathrm {log}\left (x^{2}-2\right )+1}{\mathrm {log}\left (x \right )^{2}}\right )}^{2} x +1}{{\mathrm {log}\left (\frac {\mathrm {log}\left (x^{2}-2\right )^{2}+2 \,\mathrm {log}\left (x^{2}-2\right )+1}{\mathrm {log}\left (x \right )^{2}}\right )}^{2}} \] Input:

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

Output:

( - 2*log((log(x**2 - 2)**2 + 2*log(x**2 - 2) + 1)/log(x)**2)**2*x + 1)/lo 
g((log(x**2 - 2)**2 + 2*log(x**2 - 2) + 1)/log(x)**2)**2