Integrand size = 86, antiderivative size = 26 \[ \int \frac {-4 x+16 x^4+4 x^5-4 x^6+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{8-8 x+2 x^2+\left (16-16 x+4 x^2\right ) \log (x)+\left (8-8 x+2 x^2\right ) \log ^2(x)} \, dx=\frac {x (2+x) \left (1-x^4\right )}{2 (-2+x) (1+\log (x))} \] Output:
1/2*(-x^4+1)*(2+x)/(1+ln(x))/(-2+x)*x
Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-4 x+16 x^4+4 x^5-4 x^6+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{8-8 x+2 x^2+\left (16-16 x+4 x^2\right ) \log (x)+\left (8-8 x+2 x^2\right ) \log ^2(x)} \, dx=-\frac {x \left (-2-x+2 x^4+x^5\right )}{2 (-2+x) (1+\log (x))} \] Input:
Integrate[(-4*x + 16*x^4 + 4*x^5 - 4*x^6 + (-4 - 4*x + x^2 + 20*x^4 + 4*x^ 5 - 5*x^6)*Log[x])/(8 - 8*x + 2*x^2 + (16 - 16*x + 4*x^2)*Log[x] + (8 - 8* x + 2*x^2)*Log[x]^2),x]
Output:
-1/2*(x*(-2 - x + 2*x^4 + x^5))/((-2 + x)*(1 + Log[x]))
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.
\(\displaystyle \int \frac {-4 x^6+4 x^5+16 x^4+\left (-5 x^6+4 x^5+20 x^4+x^2-4 x-4\right ) \log (x)-4 x}{2 x^2+\left (2 x^2-8 x+8\right ) \log ^2(x)+\left (4 x^2-16 x+16\right ) \log (x)-8 x+8} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-4 x^6+4 x^5+16 x^4+\left (-5 x^6+4 x^5+20 x^4+x^2-4 x-4\right ) \log (x)-4 x}{2 (2-x)^2 (\log (x)+1)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int -\frac {4 x^6-4 x^5-16 x^4+4 x+\left (5 x^6-4 x^5-20 x^4-x^2+4 x+4\right ) \log (x)}{(2-x)^2 (\log (x)+1)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {4 x^6-4 x^5-16 x^4+4 x+\left (5 x^6-4 x^5-20 x^4-x^2+4 x+4\right ) \log (x)}{(2-x)^2 (\log (x)+1)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{2} \int \left (\frac {-x^5-2 x^4+x+2}{(x-2) (\log (x)+1)^2}+\frac {5 x^6-4 x^5-20 x^4-x^2+4 x+4}{(x-2)^2 (\log (x)+1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {-x^5-2 x^4+x+2}{(x-2) (\log (x)+1)^2}dx-\int \frac {5 x^6-4 x^5-20 x^4-x^2+4 x+4}{(x-2)^2 (\log (x)+1)}dx\right )\) |
Input:
Int[(-4*x + 16*x^4 + 4*x^5 - 4*x^6 + (-4 - 4*x + x^2 + 20*x^4 + 4*x^5 - 5* x^6)*Log[x])/(8 - 8*x + 2*x^2 + (16 - 16*x + 4*x^2)*Log[x] + (8 - 8*x + 2* x^2)*Log[x]^2),x]
Output:
$Aborted
Time = 0.47 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08
method | result | size |
risch | \(-\frac {x \left (x^{5}+2 x^{4}-x -2\right )}{2 \left (-2+x \right ) \left (\ln \left (x \right )+1\right )}\) | \(28\) |
norman | \(\frac {x +\frac {1}{2} x^{2}-x^{5}-\frac {1}{2} x^{6}}{\left (\ln \left (x \right )+1\right ) \left (-2+x \right )}\) | \(30\) |
default | \(-\frac {x^{6}+2 x^{5}-x^{2}-2 x}{2 \left (-2+x \right ) \left (\ln \left (x \right )+1\right )}\) | \(31\) |
parallelrisch | \(\frac {-x^{6}-2 x^{5}+x^{2}+2 x}{2 \left (\ln \left (x \right )+1\right ) \left (-2+x \right )}\) | \(31\) |
Input:
int(((-5*x^6+4*x^5+20*x^4+x^2-4*x-4)*ln(x)-4*x^6+4*x^5+16*x^4-4*x)/((2*x^2 -8*x+8)*ln(x)^2+(4*x^2-16*x+16)*ln(x)+2*x^2-8*x+8),x,method=_RETURNVERBOSE )
Output:
-1/2*x*(x^5+2*x^4-x-2)/(-2+x)/(ln(x)+1)
Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-4 x+16 x^4+4 x^5-4 x^6+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{8-8 x+2 x^2+\left (16-16 x+4 x^2\right ) \log (x)+\left (8-8 x+2 x^2\right ) \log ^2(x)} \, dx=-\frac {x^{6} + 2 \, x^{5} - x^{2} - 2 \, x}{2 \, {\left ({\left (x - 2\right )} \log \left (x\right ) + x - 2\right )}} \] Input:
integrate(((-5*x^6+4*x^5+20*x^4+x^2-4*x-4)*log(x)-4*x^6+4*x^5+16*x^4-4*x)/ ((2*x^2-8*x+8)*log(x)^2+(4*x^2-16*x+16)*log(x)+2*x^2-8*x+8),x, algorithm=" fricas")
Output:
-1/2*(x^6 + 2*x^5 - x^2 - 2*x)/((x - 2)*log(x) + x - 2)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-4 x+16 x^4+4 x^5-4 x^6+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{8-8 x+2 x^2+\left (16-16 x+4 x^2\right ) \log (x)+\left (8-8 x+2 x^2\right ) \log ^2(x)} \, dx=\frac {- x^{6} - 2 x^{5} + x^{2} + 2 x}{2 x + \left (2 x - 4\right ) \log {\left (x \right )} - 4} \] Input:
integrate(((-5*x**6+4*x**5+20*x**4+x**2-4*x-4)*ln(x)-4*x**6+4*x**5+16*x**4 -4*x)/((2*x**2-8*x+8)*ln(x)**2+(4*x**2-16*x+16)*ln(x)+2*x**2-8*x+8),x)
Output:
(-x**6 - 2*x**5 + x**2 + 2*x)/(2*x + (2*x - 4)*log(x) - 4)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-4 x+16 x^4+4 x^5-4 x^6+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{8-8 x+2 x^2+\left (16-16 x+4 x^2\right ) \log (x)+\left (8-8 x+2 x^2\right ) \log ^2(x)} \, dx=-\frac {x^{6} + 2 \, x^{5} - x^{2} - 2 \, x}{2 \, {\left ({\left (x - 2\right )} \log \left (x\right ) + x - 2\right )}} \] Input:
integrate(((-5*x^6+4*x^5+20*x^4+x^2-4*x-4)*log(x)-4*x^6+4*x^5+16*x^4-4*x)/ ((2*x^2-8*x+8)*log(x)^2+(4*x^2-16*x+16)*log(x)+2*x^2-8*x+8),x, algorithm=" maxima")
Output:
-1/2*(x^6 + 2*x^5 - x^2 - 2*x)/((x - 2)*log(x) + x - 2)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-4 x+16 x^4+4 x^5-4 x^6+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{8-8 x+2 x^2+\left (16-16 x+4 x^2\right ) \log (x)+\left (8-8 x+2 x^2\right ) \log ^2(x)} \, dx=-\frac {x^{6} + 2 \, x^{5} - x^{2} - 2 \, x}{2 \, {\left (x \log \left (x\right ) + x - 2 \, \log \left (x\right ) - 2\right )}} \] Input:
integrate(((-5*x^6+4*x^5+20*x^4+x^2-4*x-4)*log(x)-4*x^6+4*x^5+16*x^4-4*x)/ ((2*x^2-8*x+8)*log(x)^2+(4*x^2-16*x+16)*log(x)+2*x^2-8*x+8),x, algorithm=" giac")
Output:
-1/2*(x^6 + 2*x^5 - x^2 - 2*x)/(x*log(x) + x - 2*log(x) - 2)
Time = 0.70 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.08 \[ \int \frac {-4 x+16 x^4+4 x^5-4 x^6+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{8-8 x+2 x^2+\left (16-16 x+4 x^2\right ) \log (x)+\left (8-8 x+2 x^2\right ) \log ^2(x)} \, dx=\frac {60\,x}{x^2-4\,x+4}-\frac {31\,x}{2}-\frac {\frac {2\,x^2\,\left (-x^5+x^4+4\,x^3-1\right )}{{\left (x-2\right )}^2}-\frac {x\,\ln \left (x\right )\,\left (5\,x^6-4\,x^5-20\,x^4-x^2+4\,x+4\right )}{2\,{\left (x-2\right )}^2}}{\ln \left (x\right )+1}-16\,x^2-12\,x^3-8\,x^4-\frac {5\,x^5}{2} \] Input:
int(-(4*x + log(x)*(4*x - x^2 - 20*x^4 - 4*x^5 + 5*x^6 + 4) - 16*x^4 - 4*x ^5 + 4*x^6)/(log(x)^2*(2*x^2 - 8*x + 8) - 8*x + log(x)*(4*x^2 - 16*x + 16) + 2*x^2 + 8),x)
Output:
(60*x)/(x^2 - 4*x + 4) - (31*x)/2 - ((2*x^2*(4*x^3 + x^4 - x^5 - 1))/(x - 2)^2 - (x*log(x)*(4*x - x^2 - 20*x^4 - 4*x^5 + 5*x^6 + 4))/(2*(x - 2)^2))/ (log(x) + 1) - 16*x^2 - 12*x^3 - 8*x^4 - (5*x^5)/2
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-4 x+16 x^4+4 x^5-4 x^6+\left (-4-4 x+x^2+20 x^4+4 x^5-5 x^6\right ) \log (x)}{8-8 x+2 x^2+\left (16-16 x+4 x^2\right ) \log (x)+\left (8-8 x+2 x^2\right ) \log ^2(x)} \, dx=\frac {x \left (-x^{5}-2 x^{4}+x +2\right )}{2 \,\mathrm {log}\left (x \right ) x -4 \,\mathrm {log}\left (x \right )+2 x -4} \] Input:
int(((-5*x^6+4*x^5+20*x^4+x^2-4*x-4)*log(x)-4*x^6+4*x^5+16*x^4-4*x)/((2*x^ 2-8*x+8)*log(x)^2+(4*x^2-16*x+16)*log(x)+2*x^2-8*x+8),x)
Output:
(x*( - x**5 - 2*x**4 + x + 2))/(2*(log(x)*x - 2*log(x) + x - 2))