\(\int \frac {8-2 x+6 x^2+(4+x-x^2) \log (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}) \log (\log (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}))}{(-4 x^2-x^3+x^4) \log (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2})} \, dx\) [2379]

Optimal result

Integrand size = 177, antiderivative size = 23 \[ \int \frac {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{\left (-4 x^2-x^3+x^4\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )} \, dx=\frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x} \] Output:

ln(ln(1/128*(-x^2+x+4)^4/x^2))/x
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{\left (-4 x^2-x^3+x^4\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )} \, dx=\frac {\log \left (\log \left (\frac {\left (4+x-x^2\right )^4}{128 x^2}\right )\right )}{x} \] Input:

Integrate[(8 - 2*x + 6*x^2 + (4 + x - x^2)*Log[(256 + 256*x - 160*x^2 - 17 
6*x^3 + 49*x^4 + 44*x^5 - 10*x^6 - 4*x^7 + x^8)/(128*x^2)]*Log[Log[(256 + 
256*x - 160*x^2 - 176*x^3 + 49*x^4 + 44*x^5 - 10*x^6 - 4*x^7 + x^8)/(128*x 
^2)]])/((-4*x^2 - x^3 + x^4)*Log[(256 + 256*x - 160*x^2 - 176*x^3 + 49*x^4 
 + 44*x^5 - 10*x^6 - 4*x^7 + x^8)/(128*x^2)]),x]
 

Output:

Log[Log[(4 + x - x^2)^4/(128*x^2)]]/x
 

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.

\(\displaystyle \int \frac {6 x^2+\left (-x^2+x+4\right ) \log \left (\frac {x^8-4 x^7-10 x^6+44 x^5+49 x^4-176 x^3-160 x^2+256 x+256}{128 x^2}\right ) \log \left (\log \left (\frac {x^8-4 x^7-10 x^6+44 x^5+49 x^4-176 x^3-160 x^2+256 x+256}{128 x^2}\right )\right )-2 x+8}{\left (x^4-x^3-4 x^2\right ) \log \left (\frac {x^8-4 x^7-10 x^6+44 x^5+49 x^4-176 x^3-160 x^2+256 x+256}{128 x^2}\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {6 x^2+\left (-x^2+x+4\right ) \log \left (\frac {x^8-4 x^7-10 x^6+44 x^5+49 x^4-176 x^3-160 x^2+256 x+256}{128 x^2}\right ) \log \left (\log \left (\frac {x^8-4 x^7-10 x^6+44 x^5+49 x^4-176 x^3-160 x^2+256 x+256}{128 x^2}\right )\right )-2 x+8}{x^2 \left (x^2-x-4\right ) \log \left (\frac {x^8-4 x^7-10 x^6+44 x^5+49 x^4-176 x^3-160 x^2+256 x+256}{128 x^2}\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-6 x^2-\left (-x^2+x+4\right ) \log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right ) \log \left (\log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )\right )+2 x-8}{x^2 \left (-x^2+x+4\right ) \log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {2 \left (3 x^2-x+4\right )}{x^2 \left (x^2-x-4\right ) \log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )}-\frac {\log \left (\log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {18 \int \frac {1}{\left (-2 x+\sqrt {17}+1\right ) \log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )}dx}{\sqrt {17}}-2 \int \frac {1}{x^2 \log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )}dx+\int \frac {1}{x \log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )}dx-\frac {1}{17} \left (17+\sqrt {17}\right ) \int \frac {1}{\left (2 x-\sqrt {17}-1\right ) \log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )}dx-\frac {1}{17} \left (17-\sqrt {17}\right ) \int \frac {1}{\left (2 x+\sqrt {17}-1\right ) \log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )}dx-\frac {18 \int \frac {1}{\left (2 x+\sqrt {17}-1\right ) \log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )}dx}{\sqrt {17}}-\int \frac {\log \left (\log \left (\frac {\left (-x^2+x+4\right )^4}{128 x^2}\right )\right )}{x^2}dx\)

Input:

Int[(8 - 2*x + 6*x^2 + (4 + x - x^2)*Log[(256 + 256*x - 160*x^2 - 176*x^3 
+ 49*x^4 + 44*x^5 - 10*x^6 - 4*x^7 + x^8)/(128*x^2)]*Log[Log[(256 + 256*x 
- 160*x^2 - 176*x^3 + 49*x^4 + 44*x^5 - 10*x^6 - 4*x^7 + x^8)/(128*x^2)]]) 
/((-4*x^2 - x^3 + x^4)*Log[(256 + 256*x - 160*x^2 - 176*x^3 + 49*x^4 + 44* 
x^5 - 10*x^6 - 4*x^7 + x^8)/(128*x^2)]),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(21)=42\).

Time = 0.98 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.17

method	result	size
parallelrisch	\(\frac {\ln \left (\ln \left (\frac {x^{8}-4 x^{7}-10 x^{6}+44 x^{5}+49 x^{4}-176 x^{3}-160 x^{2}+256 x +256}{128 x^{2}}\right )\right )}{x}\)	\(50\)

Input:

int(((-x^2+x+4)*ln(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^3-160*x^2+2 
56*x+256)/x^2)*ln(ln(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^3-160*x^2 
+256*x+256)/x^2))+6*x^2-2*x+8)/(x^4-x^3-4*x^2)/ln(1/128*(x^8-4*x^7-10*x^6+ 
44*x^5+49*x^4-176*x^3-160*x^2+256*x+256)/x^2),x,method=_RETURNVERBOSE)
 

Output:

ln(ln(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^3-160*x^2+256*x+256)/x^2 
))/x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (21) = 42\).

Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{\left (-4 x^2-x^3+x^4\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )} \, dx=\frac {\log \left (\log \left (\frac {x^{8} - 4 \, x^{7} - 10 \, x^{6} + 44 \, x^{5} + 49 \, x^{4} - 176 \, x^{3} - 160 \, x^{2} + 256 \, x + 256}{128 \, x^{2}}\right )\right )}{x} \] Input:

integrate(((-x^2+x+4)*log(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^3-16 
0*x^2+256*x+256)/x^2)*log(log(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^ 
3-160*x^2+256*x+256)/x^2))+6*x^2-2*x+8)/(x^4-x^3-4*x^2)/log(1/128*(x^8-4*x 
^7-10*x^6+44*x^5+49*x^4-176*x^3-160*x^2+256*x+256)/x^2),x, algorithm="fric 
as")
 

Output:

log(log(1/128*(x^8 - 4*x^7 - 10*x^6 + 44*x^5 + 49*x^4 - 176*x^3 - 160*x^2 
+ 256*x + 256)/x^2))/x
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (17) = 34\).

Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int \frac {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{\left (-4 x^2-x^3+x^4\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )} \, dx=\frac {\log {\left (\log {\left (\frac {\frac {x^{8}}{128} - \frac {x^{7}}{32} - \frac {5 x^{6}}{64} + \frac {11 x^{5}}{32} + \frac {49 x^{4}}{128} - \frac {11 x^{3}}{8} - \frac {5 x^{2}}{4} + 2 x + 2}{x^{2}} \right )} \right )}}{x} \] Input:

integrate(((-x**2+x+4)*ln(1/128*(x**8-4*x**7-10*x**6+44*x**5+49*x**4-176*x 
**3-160*x**2+256*x+256)/x**2)*ln(ln(1/128*(x**8-4*x**7-10*x**6+44*x**5+49* 
x**4-176*x**3-160*x**2+256*x+256)/x**2))+6*x**2-2*x+8)/(x**4-x**3-4*x**2)/ 
ln(1/128*(x**8-4*x**7-10*x**6+44*x**5+49*x**4-176*x**3-160*x**2+256*x+256) 
/x**2),x)
 

Output:

log(log((x**8/128 - x**7/32 - 5*x**6/64 + 11*x**5/32 + 49*x**4/128 - 11*x* 
*3/8 - 5*x**2/4 + 2*x + 2)/x**2))/x
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{\left (-4 x^2-x^3+x^4\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )} \, dx=\frac {\log \left (-7 \, \log \left (2\right ) + 4 \, \log \left (x^{2} - x - 4\right ) - 2 \, \log \left (x\right )\right )}{x} \] Input:

integrate(((-x^2+x+4)*log(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^3-16 
0*x^2+256*x+256)/x^2)*log(log(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^ 
3-160*x^2+256*x+256)/x^2))+6*x^2-2*x+8)/(x^4-x^3-4*x^2)/log(1/128*(x^8-4*x 
^7-10*x^6+44*x^5+49*x^4-176*x^3-160*x^2+256*x+256)/x^2),x, algorithm="maxi 
ma")
 

Output:

log(-7*log(2) + 4*log(x^2 - x - 4) - 2*log(x))/x
 

Giac [F(-2)]

Exception generated. \[ \int \frac {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{\left (-4 x^2-x^3+x^4\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((-x^2+x+4)*log(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^3-16 
0*x^2+256*x+256)/x^2)*log(log(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^ 
3-160*x^2+256*x+256)/x^2))+6*x^2-2*x+8)/(x^4-x^3-4*x^2)/log(1/128*(x^8-4*x 
^7-10*x^6+44*x^5+49*x^4-176*x^3-160*x^2+256*x+256)/x^2),x, algorithm="giac 
")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:simplify: Polynomials do not have t 
he same dimension Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 3.34 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{\left (-4 x^2-x^3+x^4\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )} \, dx=\frac {\ln \left (\ln \left (\frac {x^8-4\,x^7-10\,x^6+44\,x^5+49\,x^4-176\,x^3-160\,x^2+256\,x+256}{128\,x^2}\right )\right )}{x} \] Input:

int(-(6*x^2 - 2*x + log((2*x - (5*x^2)/4 - (11*x^3)/8 + (49*x^4)/128 + (11 
*x^5)/32 - (5*x^6)/64 - x^7/32 + x^8/128 + 2)/x^2)*log(log((2*x - (5*x^2)/ 
4 - (11*x^3)/8 + (49*x^4)/128 + (11*x^5)/32 - (5*x^6)/64 - x^7/32 + x^8/12 
8 + 2)/x^2))*(x - x^2 + 4) + 8)/(log((2*x - (5*x^2)/4 - (11*x^3)/8 + (49*x 
^4)/128 + (11*x^5)/32 - (5*x^6)/64 - x^7/32 + x^8/128 + 2)/x^2)*(4*x^2 + x 
^3 - x^4)),x)
 

Output:

log(log((256*x - 160*x^2 - 176*x^3 + 49*x^4 + 44*x^5 - 10*x^6 - 4*x^7 + x^ 
8 + 256)/(128*x^2)))/x
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {8-2 x+6 x^2+\left (4+x-x^2\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right ) \log \left (\log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )\right )}{\left (-4 x^2-x^3+x^4\right ) \log \left (\frac {256+256 x-160 x^2-176 x^3+49 x^4+44 x^5-10 x^6-4 x^7+x^8}{128 x^2}\right )} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (\frac {x^{8}-4 x^{7}-10 x^{6}+44 x^{5}+49 x^{4}-176 x^{3}-160 x^{2}+256 x +256}{128 x^{2}}\right )\right )}{x} \] Input:

int(((-x^2+x+4)*log(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^3-160*x^2+ 
256*x+256)/x^2)*log(log(1/128*(x^8-4*x^7-10*x^6+44*x^5+49*x^4-176*x^3-160* 
x^2+256*x+256)/x^2))+6*x^2-2*x+8)/(x^4-x^3-4*x^2)/log(1/128*(x^8-4*x^7-10* 
x^6+44*x^5+49*x^4-176*x^3-160*x^2+256*x+256)/x^2),x)
 

Output:

log(log((x**8 - 4*x**7 - 10*x**6 + 44*x**5 + 49*x**4 - 176*x**3 - 160*x**2 
 + 256*x + 256)/(128*x**2)))/x