\(\int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+(-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7) \log (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2})} \, dx\) [2319]

Optimal result

Integrand size = 146, antiderivative size = 31 \[ \int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+\left (-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7\right ) \log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2}\right )} \, dx=\log \left (4+\log \left (8-\frac {\left ((-3+x)^2-\frac {1}{x}\right )^2 x^2}{4 (4+x)^2}\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+\left (-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7\right ) \log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2}\right )} \, dx=\log \left (-4-\log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{4 (4+x)^2}\right )\right ) \] Input:

Integrate[(-74 + 762*x - 1320*x^2 + 754*x^3 - 132*x^4 - 12*x^5 + 4*x^6)/(- 
8176 - 6428*x - 120*x^2 - 1516*x^3 + 424*x^4 + 24*x^5 - 32*x^6 + 4*x^7 + ( 
-2044 - 1607*x - 30*x^2 - 379*x^3 + 106*x^4 + 6*x^5 - 8*x^6 + x^7)*Log[(51 
1 + 274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(64 + 32*x + 4*x^2)] 
),x]
 

Output:

Log[-4 - Log[(511 + 274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(4*( 
4 + 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.

\(\displaystyle \int \frac {4 x^6-12 x^5-132 x^4+754 x^3-1320 x^2+762 x-74}{4 x^7-32 x^6+24 x^5+424 x^4-1516 x^3-120 x^2+\left (x^7-8 x^6+6 x^5+106 x^4-379 x^3-30 x^2-1607 x-2044\right ) \log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 x^2+32 x+64}\right )-6428 x-8176} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {2 \left (-2 x^6+6 x^5+66 x^4-377 x^3+660 x^2-381 x+37\right )}{\left (-x^7+8 x^6-6 x^5-106 x^4+379 x^3+30 x^2+1607 x+2044\right ) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {-2 x^6+6 x^5+66 x^4-377 x^3+660 x^2-381 x+37}{\left (-x^7+8 x^6-6 x^5-106 x^4+379 x^3+30 x^2+1607 x+2044\right ) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle 2 \int \left (\frac {\left (x^5-16 x^4+118 x^3-582 x^2+2389 x-9830\right ) \left (-2 x^6+6 x^5+66 x^4-377 x^3+660 x^2-381 x+37\right )}{38809 \left (x^6-12 x^5+54 x^4-110 x^3+61 x^2-274 x-511\right ) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}-\frac {-2 x^6+6 x^5+66 x^4-377 x^3+660 x^2-381 x+37}{38809 (x+4) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\int \frac {1}{(x+4) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}dx-137 \int \frac {1}{\left (x^6-12 x^5+54 x^4-110 x^3+61 x^2-274 x-511\right ) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}dx+61 \int \frac {x}{\left (x^6-12 x^5+54 x^4-110 x^3+61 x^2-274 x-511\right ) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}dx-165 \int \frac {x^2}{\left (x^6-12 x^5+54 x^4-110 x^3+61 x^2-274 x-511\right ) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}dx+108 \int \frac {x^3}{\left (x^6-12 x^5+54 x^4-110 x^3+61 x^2-274 x-511\right ) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}dx-30 \int \frac {x^4}{\left (x^6-12 x^5+54 x^4-110 x^3+61 x^2-274 x-511\right ) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}dx+3 \int \frac {x^5}{\left (x^6-12 x^5+54 x^4-110 x^3+61 x^2-274 x-511\right ) \left (\log \left (\frac {-x^6+12 x^5-54 x^4+110 x^3-61 x^2+274 x+511}{4 (x+4)^2}\right )+4\right )}dx\right )\)

Input:

Int[(-74 + 762*x - 1320*x^2 + 754*x^3 - 132*x^4 - 12*x^5 + 4*x^6)/(-8176 - 
 6428*x - 120*x^2 - 1516*x^3 + 424*x^4 + 24*x^5 - 32*x^6 + 4*x^7 + (-2044 
- 1607*x - 30*x^2 - 379*x^3 + 106*x^4 + 6*x^5 - 8*x^6 + x^7)*Log[(511 + 27 
4*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6)/(64 + 32*x + 4*x^2)]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45

method	result	size
parallelrisch	\(\ln \left (\ln \left (-\frac {x^{6}-12 x^{5}+54 x^{4}-110 x^{3}+61 x^{2}-274 x -511}{4 \left (x^{2}+8 x +16\right )}\right )+4\right )\)	\(45\)
norman	\(\ln \left (\ln \left (\frac {-x^{6}+12 x^{5}-54 x^{4}+110 x^{3}-61 x^{2}+274 x +511}{4 x^{2}+32 x +64}\right )+4\right )\)	\(48\)
risch	\(\ln \left (\ln \left (\frac {-x^{6}+12 x^{5}-54 x^{4}+110 x^{3}-61 x^{2}+274 x +511}{4 x^{2}+32 x +64}\right )+4\right )\)	\(48\)
default	\(\ln \left (2 \ln \left (2\right )-\ln \left (-\frac {x^{6}-12 x^{5}+54 x^{4}-110 x^{3}+61 x^{2}-274 x -511}{x^{2}+8 x +16}\right )-4\right )\)	\(51\)

Input:

int((4*x^6-12*x^5-132*x^4+754*x^3-1320*x^2+762*x-74)/((x^7-8*x^6+6*x^5+106 
*x^4-379*x^3-30*x^2-1607*x-2044)*ln((-x^6+12*x^5-54*x^4+110*x^3-61*x^2+274 
*x+511)/(4*x^2+32*x+64))+4*x^7-32*x^6+24*x^5+424*x^4-1516*x^3-120*x^2-6428 
*x-8176),x,method=_RETURNVERBOSE)
 

Output:

ln(ln(-1/4*(x^6-12*x^5+54*x^4-110*x^3+61*x^2-274*x-511)/(x^2+8*x+16))+4)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+\left (-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7\right ) \log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2}\right )} \, dx=\log \left (\log \left (-\frac {x^{6} - 12 \, x^{5} + 54 \, x^{4} - 110 \, x^{3} + 61 \, x^{2} - 274 \, x - 511}{4 \, {\left (x^{2} + 8 \, x + 16\right )}}\right ) + 4\right ) \] Input:

integrate((4*x^6-12*x^5-132*x^4+754*x^3-1320*x^2+762*x-74)/((x^7-8*x^6+6*x 
^5+106*x^4-379*x^3-30*x^2-1607*x-2044)*log((-x^6+12*x^5-54*x^4+110*x^3-61* 
x^2+274*x+511)/(4*x^2+32*x+64))+4*x^7-32*x^6+24*x^5+424*x^4-1516*x^3-120*x 
^2-6428*x-8176),x, algorithm="fricas")
 

Output:

log(log(-1/4*(x^6 - 12*x^5 + 54*x^4 - 110*x^3 + 61*x^2 - 274*x - 511)/(x^2 
 + 8*x + 16)) + 4)
 

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+\left (-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7\right ) \log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2}\right )} \, dx=\log {\left (\log {\left (\frac {- x^{6} + 12 x^{5} - 54 x^{4} + 110 x^{3} - 61 x^{2} + 274 x + 511}{4 x^{2} + 32 x + 64} \right )} + 4 \right )} \] Input:

integrate((4*x**6-12*x**5-132*x**4+754*x**3-1320*x**2+762*x-74)/((x**7-8*x 
**6+6*x**5+106*x**4-379*x**3-30*x**2-1607*x-2044)*ln((-x**6+12*x**5-54*x** 
4+110*x**3-61*x**2+274*x+511)/(4*x**2+32*x+64))+4*x**7-32*x**6+24*x**5+424 
*x**4-1516*x**3-120*x**2-6428*x-8176),x)
 

Output:

log(log((-x**6 + 12*x**5 - 54*x**4 + 110*x**3 - 61*x**2 + 274*x + 511)/(4* 
x**2 + 32*x + 64)) + 4)
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+\left (-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7\right ) \log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2}\right )} \, dx=\log \left (-2 \, \log \left (2\right ) + \log \left (-x^{6} + 12 \, x^{5} - 54 \, x^{4} + 110 \, x^{3} - 61 \, x^{2} + 274 \, x + 511\right ) - 2 \, \log \left (x + 4\right ) + 4\right ) \] Input:

integrate((4*x^6-12*x^5-132*x^4+754*x^3-1320*x^2+762*x-74)/((x^7-8*x^6+6*x 
^5+106*x^4-379*x^3-30*x^2-1607*x-2044)*log((-x^6+12*x^5-54*x^4+110*x^3-61* 
x^2+274*x+511)/(4*x^2+32*x+64))+4*x^7-32*x^6+24*x^5+424*x^4-1516*x^3-120*x 
^2-6428*x-8176),x, algorithm="maxima")
 

Output:

log(-2*log(2) + log(-x^6 + 12*x^5 - 54*x^4 + 110*x^3 - 61*x^2 + 274*x + 51 
1) - 2*log(x + 4) + 4)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+\left (-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7\right ) \log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2}\right )} \, dx=\log \left (\log \left (-\frac {x^{6} - 12 \, x^{5} + 54 \, x^{4} - 110 \, x^{3} + 61 \, x^{2} - 274 \, x - 511}{4 \, {\left (x^{2} + 8 \, x + 16\right )}}\right ) + 4\right ) \] Input:

integrate((4*x^6-12*x^5-132*x^4+754*x^3-1320*x^2+762*x-74)/((x^7-8*x^6+6*x 
^5+106*x^4-379*x^3-30*x^2-1607*x-2044)*log((-x^6+12*x^5-54*x^4+110*x^3-61* 
x^2+274*x+511)/(4*x^2+32*x+64))+4*x^7-32*x^6+24*x^5+424*x^4-1516*x^3-120*x 
^2-6428*x-8176),x, algorithm="giac")
 

Output:

log(log(-1/4*(x^6 - 12*x^5 + 54*x^4 - 110*x^3 + 61*x^2 - 274*x - 511)/(x^2 
 + 8*x + 16)) + 4)
 

Mupad [B] (verification not implemented)

Time = 2.83 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+\left (-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7\right ) \log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2}\right )} \, dx=\ln \left (\ln \left (\frac {-x^6+12\,x^5-54\,x^4+110\,x^3-61\,x^2+274\,x+511}{4\,x^2+32\,x+64}\right )+4\right ) \] Input:

int((1320*x^2 - 762*x - 754*x^3 + 132*x^4 + 12*x^5 - 4*x^6 + 74)/(6428*x + 
 log((274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6 + 511)/(32*x + 4*x^ 
2 + 64))*(1607*x + 30*x^2 + 379*x^3 - 106*x^4 - 6*x^5 + 8*x^6 - x^7 + 2044 
) + 120*x^2 + 1516*x^3 - 424*x^4 - 24*x^5 + 32*x^6 - 4*x^7 + 8176),x)
 

Output:

log(log((274*x - 61*x^2 + 110*x^3 - 54*x^4 + 12*x^5 - x^6 + 511)/(32*x + 4 
*x^2 + 64)) + 4)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {-74+762 x-1320 x^2+754 x^3-132 x^4-12 x^5+4 x^6}{-8176-6428 x-120 x^2-1516 x^3+424 x^4+24 x^5-32 x^6+4 x^7+\left (-2044-1607 x-30 x^2-379 x^3+106 x^4+6 x^5-8 x^6+x^7\right ) \log \left (\frac {511+274 x-61 x^2+110 x^3-54 x^4+12 x^5-x^6}{64+32 x+4 x^2}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\frac {-x^{6}+12 x^{5}-54 x^{4}+110 x^{3}-61 x^{2}+274 x +511}{4 x^{2}+32 x +64}\right )+4\right ) \] Input:

int((4*x^6-12*x^5-132*x^4+754*x^3-1320*x^2+762*x-74)/((x^7-8*x^6+6*x^5+106 
*x^4-379*x^3-30*x^2-1607*x-2044)*log((-x^6+12*x^5-54*x^4+110*x^3-61*x^2+27 
4*x+511)/(4*x^2+32*x+64))+4*x^7-32*x^6+24*x^5+424*x^4-1516*x^3-120*x^2-642 
8*x-8176),x)
 

Output:

log(log(( - x**6 + 12*x**5 - 54*x**4 + 110*x**3 - 61*x**2 + 274*x + 511)/( 
4*x**2 + 32*x + 64)) + 4)