Integrand size = 113, antiderivative size = 25 \[ \int \frac {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log \left (1-2 x+x^2\right )}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+\left (-150+610 x-626 x^2+182 x^3-16 x^4\right ) \log \left (1-2 x+x^2\right )+\left (-25+35 x-11 x^2+x^3\right ) \log ^2\left (1-2 x+x^2\right )} \, dx=\frac {4 x}{(-5+x) \left (-3+8 x-\log \left ((1-x)^2\right )\right )} \] Output:
x/(4*x-3/2-1/2*ln((1-x)^2))/(1/2*x-5/2)
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log \left (1-2 x+x^2\right )}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+\left (-150+610 x-626 x^2+182 x^3-16 x^4\right ) \log \left (1-2 x+x^2\right )+\left (-25+35 x-11 x^2+x^3\right ) \log ^2\left (1-2 x+x^2\right )} \, dx=-\frac {4 x}{(-5+x) \left (-5-8 (-1+x)+\log \left ((-1+x)^2\right )\right )} \] Input:
Integrate[(-60 + 20*x + 40*x^2 - 32*x^3 + (-20 + 20*x)*Log[1 - 2*x + x^2]) /(-225 + 1515*x - 3379*x^2 + 2777*x^3 - 752*x^4 + 64*x^5 + (-150 + 610*x - 626*x^2 + 182*x^3 - 16*x^4)*Log[1 - 2*x + x^2] + (-25 + 35*x - 11*x^2 + x ^3)*Log[1 - 2*x + x^2]^2),x]
Output:
(-4*x)/((-5 + x)*(-5 - 8*(-1 + x) + Log[(-1 + x)^2]))
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.
\(\displaystyle \int \frac {-32 x^3+40 x^2+(20 x-20) \log \left (x^2-2 x+1\right )+20 x-60}{64 x^5-752 x^4+2777 x^3-3379 x^2+\left (x^3-11 x^2+35 x-25\right ) \log ^2\left (x^2-2 x+1\right )+\left (-16 x^4+182 x^3-626 x^2+610 x-150\right ) \log \left (x^2-2 x+1\right )+1515 x-225} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {4 \left (8 x^3-10 x^2-5 x-5 (x-1) \log \left ((x-1)^2\right )+15\right )}{(1-x) (5-x)^2 \left (-8 x+\log \left ((x-1)^2\right )+3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \int \frac {8 x^3-10 x^2-5 x+5 (1-x) \log \left ((x-1)^2\right )+15}{(1-x) (5-x)^2 \left (-8 x+\log \left ((x-1)^2\right )+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 4 \int \left (-\frac {2 x (4 x-5)}{(x-5) (x-1) \left (8 x-\log \left ((x-1)^2\right )-3\right )^2}-\frac {5}{(x-5)^2 \left (8 x-\log \left ((x-1)^2\right )-3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (-8 \int \frac {1}{\left (8 x-\log \left ((x-1)^2\right )-3\right )^2}dx-\frac {75}{2} \int \frac {1}{(x-5) \left (8 x-\log \left ((x-1)^2\right )-3\right )^2}dx-\frac {1}{2} \int \frac {1}{(x-1) \left (8 x-\log \left ((x-1)^2\right )-3\right )^2}dx-5 \int \frac {1}{(x-5)^2 \left (8 x-\log \left ((x-1)^2\right )-3\right )}dx\right )\) |
Input:
Int[(-60 + 20*x + 40*x^2 - 32*x^3 + (-20 + 20*x)*Log[1 - 2*x + x^2])/(-225 + 1515*x - 3379*x^2 + 2777*x^3 - 752*x^4 + 64*x^5 + (-150 + 610*x - 626*x ^2 + 182*x^3 - 16*x^4)*Log[1 - 2*x + x^2] + (-25 + 35*x - 11*x^2 + x^3)*Lo g[1 - 2*x + x^2]^2),x]
Output:
$Aborted
Time = 0.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08
method | result | size |
risch | \(\frac {4 x}{\left (-5+x \right ) \left (8 x -\ln \left (x^{2}-2 x +1\right )-3\right )}\) | \(27\) |
norman | \(\frac {4 x}{8 x^{2}-\ln \left (x^{2}-2 x +1\right ) x -43 x +5 \ln \left (x^{2}-2 x +1\right )+15}\) | \(39\) |
parallelrisch | \(\frac {4 x}{8 x^{2}-\ln \left (x^{2}-2 x +1\right ) x -43 x +5 \ln \left (x^{2}-2 x +1\right )+15}\) | \(39\) |
Input:
int(((20*x-20)*ln(x^2-2*x+1)-32*x^3+40*x^2+20*x-60)/((x^3-11*x^2+35*x-25)* ln(x^2-2*x+1)^2+(-16*x^4+182*x^3-626*x^2+610*x-150)*ln(x^2-2*x+1)+64*x^5-7 52*x^4+2777*x^3-3379*x^2+1515*x-225),x,method=_RETURNVERBOSE)
Output:
4*x/(-5+x)/(8*x-ln(x^2-2*x+1)-3)
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log \left (1-2 x+x^2\right )}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+\left (-150+610 x-626 x^2+182 x^3-16 x^4\right ) \log \left (1-2 x+x^2\right )+\left (-25+35 x-11 x^2+x^3\right ) \log ^2\left (1-2 x+x^2\right )} \, dx=\frac {4 \, x}{8 \, x^{2} - {\left (x - 5\right )} \log \left (x^{2} - 2 \, x + 1\right ) - 43 \, x + 15} \] Input:
integrate(((20*x-20)*log(x^2-2*x+1)-32*x^3+40*x^2+20*x-60)/((x^3-11*x^2+35 *x-25)*log(x^2-2*x+1)^2+(-16*x^4+182*x^3-626*x^2+610*x-150)*log(x^2-2*x+1) +64*x^5-752*x^4+2777*x^3-3379*x^2+1515*x-225),x, algorithm="fricas")
Output:
4*x/(8*x^2 - (x - 5)*log(x^2 - 2*x + 1) - 43*x + 15)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log \left (1-2 x+x^2\right )}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+\left (-150+610 x-626 x^2+182 x^3-16 x^4\right ) \log \left (1-2 x+x^2\right )+\left (-25+35 x-11 x^2+x^3\right ) \log ^2\left (1-2 x+x^2\right )} \, dx=- \frac {4 x}{- 8 x^{2} + 43 x + \left (x - 5\right ) \log {\left (x^{2} - 2 x + 1 \right )} - 15} \] Input:
integrate(((20*x-20)*ln(x**2-2*x+1)-32*x**3+40*x**2+20*x-60)/((x**3-11*x** 2+35*x-25)*ln(x**2-2*x+1)**2+(-16*x**4+182*x**3-626*x**2+610*x-150)*ln(x** 2-2*x+1)+64*x**5-752*x**4+2777*x**3-3379*x**2+1515*x-225),x)
Output:
-4*x/(-8*x**2 + 43*x + (x - 5)*log(x**2 - 2*x + 1) - 15)
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log \left (1-2 x+x^2\right )}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+\left (-150+610 x-626 x^2+182 x^3-16 x^4\right ) \log \left (1-2 x+x^2\right )+\left (-25+35 x-11 x^2+x^3\right ) \log ^2\left (1-2 x+x^2\right )} \, dx=\frac {4 \, x}{8 \, x^{2} - 2 \, {\left (x - 5\right )} \log \left (x - 1\right ) - 43 \, x + 15} \] Input:
integrate(((20*x-20)*log(x^2-2*x+1)-32*x^3+40*x^2+20*x-60)/((x^3-11*x^2+35 *x-25)*log(x^2-2*x+1)^2+(-16*x^4+182*x^3-626*x^2+610*x-150)*log(x^2-2*x+1) +64*x^5-752*x^4+2777*x^3-3379*x^2+1515*x-225),x, algorithm="maxima")
Output:
4*x/(8*x^2 - 2*(x - 5)*log(x - 1) - 43*x + 15)
Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log \left (1-2 x+x^2\right )}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+\left (-150+610 x-626 x^2+182 x^3-16 x^4\right ) \log \left (1-2 x+x^2\right )+\left (-25+35 x-11 x^2+x^3\right ) \log ^2\left (1-2 x+x^2\right )} \, dx=\frac {4 \, x}{8 \, x^{2} - x \log \left (x^{2} - 2 \, x + 1\right ) - 43 \, x + 5 \, \log \left (x^{2} - 2 \, x + 1\right ) + 15} \] Input:
integrate(((20*x-20)*log(x^2-2*x+1)-32*x^3+40*x^2+20*x-60)/((x^3-11*x^2+35 *x-25)*log(x^2-2*x+1)^2+(-16*x^4+182*x^3-626*x^2+610*x-150)*log(x^2-2*x+1) +64*x^5-752*x^4+2777*x^3-3379*x^2+1515*x-225),x, algorithm="giac")
Output:
4*x/(8*x^2 - x*log(x^2 - 2*x + 1) - 43*x + 5*log(x^2 - 2*x + 1) + 15)
Time = 3.88 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log \left (1-2 x+x^2\right )}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+\left (-150+610 x-626 x^2+182 x^3-16 x^4\right ) \log \left (1-2 x+x^2\right )+\left (-25+35 x-11 x^2+x^3\right ) \log ^2\left (1-2 x+x^2\right )} \, dx=-\frac {4\,x}{\left (x-5\right )\,\left (\ln \left (x^2-2\,x+1\right )-8\,x+3\right )} \] Input:
int((20*x + log(x^2 - 2*x + 1)*(20*x - 20) + 40*x^2 - 32*x^3 - 60)/(1515*x + log(x^2 - 2*x + 1)^2*(35*x - 11*x^2 + x^3 - 25) - log(x^2 - 2*x + 1)*(6 26*x^2 - 610*x - 182*x^3 + 16*x^4 + 150) - 3379*x^2 + 2777*x^3 - 752*x^4 + 64*x^5 - 225),x)
Output:
-(4*x)/((x - 5)*(log(x^2 - 2*x + 1) - 8*x + 3))
Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log \left (1-2 x+x^2\right )}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+\left (-150+610 x-626 x^2+182 x^3-16 x^4\right ) \log \left (1-2 x+x^2\right )+\left (-25+35 x-11 x^2+x^3\right ) \log ^2\left (1-2 x+x^2\right )} \, dx=-\frac {4 x}{\mathrm {log}\left (x^{2}-2 x +1\right ) x -5 \,\mathrm {log}\left (x^{2}-2 x +1\right )-8 x^{2}+43 x -15} \] Input:
int(((20*x-20)*log(x^2-2*x+1)-32*x^3+40*x^2+20*x-60)/((x^3-11*x^2+35*x-25) *log(x^2-2*x+1)^2+(-16*x^4+182*x^3-626*x^2+610*x-150)*log(x^2-2*x+1)+64*x^ 5-752*x^4+2777*x^3-3379*x^2+1515*x-225),x)
Output:
( - 4*x)/(log(x**2 - 2*x + 1)*x - 5*log(x**2 - 2*x + 1) - 8*x**2 + 43*x - 15)