\(\int \frac {(392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6) \log (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2})}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx\) [2335]

Optimal result

Integrand size = 100, antiderivative size = 30 \[ \int \frac {\left (392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx=\log ^2\left (4 x \left ((7-x)^2-x+\frac {2}{2 x-x^3}\right )\right ) \] Output:

ln(4*((-x+7)^2-x+2/(-x^3+2*x))*x)^2
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {\left (392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx=\log ^2\left (\frac {4 \left (-2-98 x+30 x^2+47 x^3-15 x^4+x^5\right )}{-2+x^2}\right ) \] Input:

Integrate[((392 - 232*x - 368*x^2 + 240*x^3 + 74*x^4 - 60*x^5 + 6*x^6)*Log 
[(-8 - 392*x + 120*x^2 + 188*x^3 - 60*x^4 + 4*x^5)/(-2 + x^2)])/(4 + 196*x 
 - 62*x^2 - 192*x^3 + 60*x^4 + 45*x^5 - 15*x^6 + x^7),x]
 

Output:

Log[(4*(-2 - 98*x + 30*x^2 + 47*x^3 - 15*x^4 + x^5))/(-2 + 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.

\(\displaystyle \int \frac {\left (6 x^6-60 x^5+74 x^4+240 x^3-368 x^2-232 x+392\right ) \log \left (\frac {4 x^5-60 x^4+188 x^3+120 x^2-392 x-8}{x^2-2}\right )}{x^7-15 x^6+45 x^5+60 x^4-192 x^3-62 x^2+196 x+4} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {x \left (x^2-15 x+49\right ) \left (6 x^6-60 x^5+74 x^4+240 x^3-368 x^2-232 x+392\right ) \log \left (\frac {4 x^5-60 x^4+188 x^3+120 x^2-392 x-8}{x^2-2}\right )}{2 \left (x^5-15 x^4+47 x^3+30 x^2-98 x-2\right )}-\frac {\left (6 x^6-60 x^5+74 x^4+240 x^3-368 x^2-232 x+392\right ) \log \left (\frac {4 x^5-60 x^4+188 x^3+120 x^2-392 x-8}{x^2-2}\right )}{2 \left (x^2-2\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \left (3 x^6-30 x^5+37 x^4+120 x^3-184 x^2-116 x+196\right ) \log \left (\frac {4 \left (x^5-15 x^4+47 x^3+30 x^2-98 x-2\right )}{x^2-2}\right )}{\left (2-x^2\right ) \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {\left (3 x^6-30 x^5+37 x^4+120 x^3-184 x^2-116 x+196\right ) \log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right )}{\left (2-x^2\right ) \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}dx\)

\(\Big \downarrow \) 3008

\(\displaystyle 2 \int \left (\frac {\left (5 x^4-60 x^3+141 x^2+60 x-98\right ) \log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}-\frac {2 x \log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right )}{x^2-2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {1}{2} \log ^2\left (\sqrt {2}-x\right )-\log \left (\frac {x+\sqrt {2}}{2 \sqrt {2}}\right ) \log \left (\sqrt {2}-x\right )-\log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right ) \log \left (\sqrt {2}-x\right )-\frac {1}{2} \log ^2\left (x+\sqrt {2}\right )-\log \left (\frac {\sqrt {2}-x}{2 \sqrt {2}}\right ) \log \left (x+\sqrt {2}\right )-\log \left (x+\sqrt {2}\right ) \log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {2}-x}{2 \sqrt {2}}\right )-\operatorname {PolyLog}\left (2,\frac {x+\sqrt {2}}{2 \sqrt {2}}\right )-98 \log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right ) \int \frac {1}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+60 \log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right ) \int \frac {x}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+141 \log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right ) \int \frac {x^2}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-60 \log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right ) \int \frac {x^3}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-98 \int \frac {\log \left (\sqrt {2}-x\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+60 \int \frac {x \log \left (\sqrt {2}-x\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+141 \int \frac {x^2 \log \left (\sqrt {2}-x\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-60 \int \frac {x^3 \log \left (\sqrt {2}-x\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+5 \int \frac {x^4 \log \left (\sqrt {2}-x\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-98 \int \frac {\log \left (x+\sqrt {2}\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+60 \int \frac {x \log \left (x+\sqrt {2}\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+141 \int \frac {x^2 \log \left (x+\sqrt {2}\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-60 \int \frac {x^3 \log \left (x+\sqrt {2}\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+5 \int \frac {x^4 \log \left (x+\sqrt {2}\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+5 \int \frac {x^4 \log \left (\frac {4 \left (-x^5+15 x^4-47 x^3-30 x^2+98 x+2\right )}{2-x^2}\right )}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+98 \int \frac {\int \frac {1}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{\sqrt {2}-x}dx-98 \int \frac {\int \frac {1}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x+\sqrt {2}}dx-9604 \int \frac {\int \frac {1}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+5880 \int \frac {x \int \frac {1}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+13818 \int \frac {x^2 \int \frac {1}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-5880 \int \frac {x^3 \int \frac {1}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+490 \int \frac {x^4 \int \frac {1}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-60 \int \frac {\int \frac {x}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{\sqrt {2}-x}dx+60 \int \frac {\int \frac {x}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x+\sqrt {2}}dx+5880 \int \frac {\int \frac {x}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-3600 \int \frac {x \int \frac {x}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-8460 \int \frac {x^2 \int \frac {x}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+3600 \int \frac {x^3 \int \frac {x}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-300 \int \frac {x^4 \int \frac {x}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-141 \int \frac {\int \frac {x^2}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{\sqrt {2}-x}dx+141 \int \frac {\int \frac {x^2}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x+\sqrt {2}}dx+13818 \int \frac {\int \frac {x^2}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-8460 \int \frac {x \int \frac {x^2}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-19881 \int \frac {x^2 \int \frac {x^2}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+8460 \int \frac {x^3 \int \frac {x^2}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-705 \int \frac {x^4 \int \frac {x^2}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+60 \int \frac {\int \frac {x^3}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{\sqrt {2}-x}dx-60 \int \frac {\int \frac {x^3}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x+\sqrt {2}}dx-5880 \int \frac {\int \frac {x^3}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+3600 \int \frac {x \int \frac {x^3}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+8460 \int \frac {x^2 \int \frac {x^3}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx-3600 \int \frac {x^3 \int \frac {x^3}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx+300 \int \frac {x^4 \int \frac {x^3}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx}{x^5-15 x^4+47 x^3+30 x^2-98 x-2}dx\right )\)

Input:

Int[((392 - 232*x - 368*x^2 + 240*x^3 + 74*x^4 - 60*x^5 + 6*x^6)*Log[(-8 - 
 392*x + 120*x^2 + 188*x^3 - 60*x^4 + 4*x^5)/(-2 + x^2)])/(4 + 196*x - 62* 
x^2 - 192*x^3 + 60*x^4 + 45*x^5 - 15*x^6 + x^7),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23

method	result	size
norman	\(\ln \left (\frac {4 x^{5}-60 x^{4}+188 x^{3}+120 x^{2}-392 x -8}{x^{2}-2}\right )^{2}\)	\(37\)
default	\(\text {Expression too large to display}\)	\(1543\)
parts	\(\text {Expression too large to display}\)	\(1664\)
risch	\(\text {Expression too large to display}\)	\(30279\)

Input:

int((6*x^6-60*x^5+74*x^4+240*x^3-368*x^2-232*x+392)*ln((4*x^5-60*x^4+188*x 
^3+120*x^2-392*x-8)/(x^2-2))/(x^7-15*x^6+45*x^5+60*x^4-192*x^3-62*x^2+196* 
x+4),x,method=_RETURNVERBOSE)
 

Output:

ln((4*x^5-60*x^4+188*x^3+120*x^2-392*x-8)/(x^2-2))^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {\left (392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx=\log \left (\frac {4 \, {\left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right )}}{x^{2} - 2}\right )^{2} \] Input:

integrate((6*x^6-60*x^5+74*x^4+240*x^3-368*x^2-232*x+392)*log((4*x^5-60*x^ 
4+188*x^3+120*x^2-392*x-8)/(x^2-2))/(x^7-15*x^6+45*x^5+60*x^4-192*x^3-62*x 
^2+196*x+4),x, algorithm="fricas")
 

Output:

log(4*(x^5 - 15*x^4 + 47*x^3 + 30*x^2 - 98*x - 2)/(x^2 - 2))^2
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {\left (392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx=\log {\left (\frac {4 x^{5} - 60 x^{4} + 188 x^{3} + 120 x^{2} - 392 x - 8}{x^{2} - 2} \right )}^{2} \] Input:

integrate((6*x**6-60*x**5+74*x**4+240*x**3-368*x**2-232*x+392)*ln((4*x**5- 
60*x**4+188*x**3+120*x**2-392*x-8)/(x**2-2))/(x**7-15*x**6+45*x**5+60*x**4 
-192*x**3-62*x**2+196*x+4),x)
 

Output:

log((4*x**5 - 60*x**4 + 188*x**3 + 120*x**2 - 392*x - 8)/(x**2 - 2))**2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (26) = 52\).

Time = 0.04 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.63 \[ \int \frac {\left (392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx=-\log \left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right )^{2} + 2 \, \log \left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right ) \log \left (x^{2} - 2\right ) - \log \left (x^{2} - 2\right )^{2} + 2 \, {\left (\log \left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right ) - \log \left (x^{2} - 2\right )\right )} \log \left (\frac {4 \, {\left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right )}}{x^{2} - 2}\right ) \] Input:

integrate((6*x^6-60*x^5+74*x^4+240*x^3-368*x^2-232*x+392)*log((4*x^5-60*x^ 
4+188*x^3+120*x^2-392*x-8)/(x^2-2))/(x^7-15*x^6+45*x^5+60*x^4-192*x^3-62*x 
^2+196*x+4),x, algorithm="maxima")
 

Output:

-log(x^5 - 15*x^4 + 47*x^3 + 30*x^2 - 98*x - 2)^2 + 2*log(x^5 - 15*x^4 + 4 
7*x^3 + 30*x^2 - 98*x - 2)*log(x^2 - 2) - log(x^2 - 2)^2 + 2*(log(x^5 - 15 
*x^4 + 47*x^3 + 30*x^2 - 98*x - 2) - log(x^2 - 2))*log(4*(x^5 - 15*x^4 + 4 
7*x^3 + 30*x^2 - 98*x - 2)/(x^2 - 2))
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {\left (392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx=\log \left (\frac {4 \, {\left (x^{5} - 15 \, x^{4} + 47 \, x^{3} + 30 \, x^{2} - 98 \, x - 2\right )}}{x^{2} - 2}\right )^{2} \] Input:

integrate((6*x^6-60*x^5+74*x^4+240*x^3-368*x^2-232*x+392)*log((4*x^5-60*x^ 
4+188*x^3+120*x^2-392*x-8)/(x^2-2))/(x^7-15*x^6+45*x^5+60*x^4-192*x^3-62*x 
^2+196*x+4),x, algorithm="giac")
 

Output:

log(4*(x^5 - 15*x^4 + 47*x^3 + 30*x^2 - 98*x - 2)/(x^2 - 2))^2
 

Mupad [B] (verification not implemented)

Time = 3.77 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx={\ln \left (196\,x-\frac {8}{x^2-2}-60\,x^2+4\,x^3\right )}^2 \] Input:

int((log(-(392*x - 120*x^2 - 188*x^3 + 60*x^4 - 4*x^5 + 8)/(x^2 - 2))*(240 
*x^3 - 368*x^2 - 232*x + 74*x^4 - 60*x^5 + 6*x^6 + 392))/(196*x - 62*x^2 - 
 192*x^3 + 60*x^4 + 45*x^5 - 15*x^6 + x^7 + 4),x)
 

Output:

log(196*x - 8/(x^2 - 2) - 60*x^2 + 4*x^3)^2
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {\left (392-232 x-368 x^2+240 x^3+74 x^4-60 x^5+6 x^6\right ) \log \left (\frac {-8-392 x+120 x^2+188 x^3-60 x^4+4 x^5}{-2+x^2}\right )}{4+196 x-62 x^2-192 x^3+60 x^4+45 x^5-15 x^6+x^7} \, dx=\mathrm {log}\left (\frac {4 x^{5}-60 x^{4}+188 x^{3}+120 x^{2}-392 x -8}{x^{2}-2}\right )^{2} \] Input:

int((6*x^6-60*x^5+74*x^4+240*x^3-368*x^2-232*x+392)*log((4*x^5-60*x^4+188* 
x^3+120*x^2-392*x-8)/(x^2-2))/(x^7-15*x^6+45*x^5+60*x^4-192*x^3-62*x^2+196 
*x+4),x)
 

Output:

log((4*x**5 - 60*x**4 + 188*x**3 + 120*x**2 - 392*x - 8)/(x**2 - 2))**2