Integrand size = 67, antiderivative size = 22 \[ \int \frac {-10+4 x+\left (-2+2 x-x^2\right ) \log (x)+(2-x) \log \left (-\frac {x}{-2+x}\right )}{-2 x+x^2+\left (-4 x+2 x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {-4-x+\log \left (\frac {x}{2-x}\right )}{1+\log (x)} \] Output:
(-4-x+ln(x/(2-x)))/(1+ln(x))
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-10+4 x+\left (-2+2 x-x^2\right ) \log (x)+(2-x) \log \left (-\frac {x}{-2+x}\right )}{-2 x+x^2+\left (-4 x+2 x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log ^2(x)} \, dx=-\frac {4+x-\log \left (-\frac {x}{-2+x}\right )}{1+\log (x)} \] Input:
Integrate[(-10 + 4*x + (-2 + 2*x - x^2)*Log[x] + (2 - x)*Log[-(x/(-2 + x)) ])/(-2*x + x^2 + (-4*x + 2*x^2)*Log[x] + (-2*x + x^2)*Log[x]^2),x]
Output:
-((4 + x - Log[-(x/(-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.
\(\displaystyle \int \frac {\left (-x^2+2 x-2\right ) \log (x)+4 x+(2-x) \log \left (-\frac {x}{x-2}\right )-10}{x^2+\left (x^2-2 x\right ) \log ^2(x)+\left (2 x^2-4 x\right ) \log (x)-2 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\left (-x^2+2 x-2\right ) \log (x)-4 x-(2-x) \log \left (-\frac {x}{x-2}\right )+10}{(2-x) x (\log (x)+1)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {\left (x^2-2 x+2\right ) \log (x)}{(x-2) x (\log (x)+1)^2}-\frac {\log \left (-\frac {x}{x-2}\right )}{x (\log (x)+1)^2}+\frac {4}{(x-2) (\log (x)+1)^2}-\frac {10}{(x-2) x (\log (x)+1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {-x^2+2 x-2}{(x-2) x (\log (x)+1)^2}dx-\int \frac {x^2-2 x+2}{(x-2) x (\log (x)+1)}dx+4 \int \frac {1}{(x-2) (\log (x)+1)^2}dx-10 \int \frac {1}{(x-2) x (\log (x)+1)^2}dx-\int \frac {\log \left (-\frac {x}{x-2}\right )}{x (\log (x)+1)^2}dx\) |
Input:
Int[(-10 + 4*x + (-2 + 2*x - x^2)*Log[x] + (2 - x)*Log[-(x/(-2 + x))])/(-2 *x + x^2 + (-4*x + 2*x^2)*Log[x] + (-2*x + x^2)*Log[x]^2),x]
Output:
$Aborted
Time = 1.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {-4-x +\ln \left (-\frac {x}{-2+x}\right )}{\ln \left (x \right )+1}\) | \(22\) |
risch | \(-\frac {\ln \left (-2+x \right )}{\ln \left (x \right )+1}-\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i x}{-2+x}\right )-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{-2+x}\right )^{2}+2 i \pi \operatorname {csgn}\left (\frac {i x}{-2+x}\right )^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i x}{-2+x}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i x}{-2+x}\right )^{3}-2 i \pi +2 x +10}{2 \left (\ln \left (x \right )+1\right )}\) | \(137\) |
Input:
int(((-x^2+2*x-2)*ln(x)+(2-x)*ln(-x/(-2+x))+4*x-10)/((x^2-2*x)*ln(x)^2+(2* x^2-4*x)*ln(x)+x^2-2*x),x,method=_RETURNVERBOSE)
Output:
(-4-x+ln(-x/(-2+x)))/(ln(x)+1)
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-10+4 x+\left (-2+2 x-x^2\right ) \log (x)+(2-x) \log \left (-\frac {x}{-2+x}\right )}{-2 x+x^2+\left (-4 x+2 x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log ^2(x)} \, dx=-\frac {x - \log \left (-\frac {x}{x - 2}\right ) + 4}{\log \left (x\right ) + 1} \] Input:
integrate(((-x^2+2*x-2)*log(x)+(2-x)*log(-x/(-2+x))+4*x-10)/((x^2-2*x)*log (x)^2+(2*x^2-4*x)*log(x)+x^2-2*x),x, algorithm="fricas")
Output:
-(x - log(-x/(x - 2)) + 4)/(log(x) + 1)
Exception generated. \[ \int \frac {-10+4 x+\left (-2+2 x-x^2\right ) \log (x)+(2-x) \log \left (-\frac {x}{-2+x}\right )}{-2 x+x^2+\left (-4 x+2 x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((-x**2+2*x-2)*ln(x)+(2-x)*ln(-x/(-2+x))+4*x-10)/((x**2-2*x)*ln( x)**2+(2*x**2-4*x)*ln(x)+x**2-2*x),x)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {-10+4 x+\left (-2+2 x-x^2\right ) \log (x)+(2-x) \log \left (-\frac {x}{-2+x}\right )}{-2 x+x^2+\left (-4 x+2 x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log ^2(x)} \, dx=-\frac {x + \log \left (-x + 2\right ) + 5}{\log \left (x\right ) + 1} \] Input:
integrate(((-x^2+2*x-2)*log(x)+(2-x)*log(-x/(-2+x))+4*x-10)/((x^2-2*x)*log (x)^2+(2*x^2-4*x)*log(x)+x^2-2*x),x, algorithm="maxima")
Output:
-(x + log(-x + 2) + 5)/(log(x) + 1)
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {-10+4 x+\left (-2+2 x-x^2\right ) \log (x)+(2-x) \log \left (-\frac {x}{-2+x}\right )}{-2 x+x^2+\left (-4 x+2 x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log ^2(x)} \, dx=-\frac {-i \, \pi + x + 5}{\log \left (x\right ) + 1} - \frac {\log \left (x - 2\right )}{\log \left (x\right ) + 1} \] Input:
integrate(((-x^2+2*x-2)*log(x)+(2-x)*log(-x/(-2+x))+4*x-10)/((x^2-2*x)*log (x)^2+(2*x^2-4*x)*log(x)+x^2-2*x),x, algorithm="giac")
Output:
-(-I*pi + x + 5)/(log(x) + 1) - log(x - 2)/(log(x) + 1)
Time = 0.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-10+4 x+\left (-2+2 x-x^2\right ) \log (x)+(2-x) \log \left (-\frac {x}{-2+x}\right )}{-2 x+x^2+\left (-4 x+2 x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log ^2(x)} \, dx=-\frac {x-\ln \left (-\frac {x}{x-2}\right )+4}{\ln \left (x\right )+1} \] Input:
int((log(x)*(x^2 - 2*x + 2) - 4*x + log(-x/(x - 2))*(x - 2) + 10)/(2*x + l og(x)^2*(2*x - x^2) + log(x)*(4*x - 2*x^2) - x^2),x)
Output:
-(x - log(-x/(x - 2)) + 4)/(log(x) + 1)
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \frac {-10+4 x+\left (-2+2 x-x^2\right ) \log (x)+(2-x) \log \left (-\frac {x}{-2+x}\right )}{-2 x+x^2+\left (-4 x+2 x^2\right ) \log (x)+\left (-2 x+x^2\right ) \log ^2(x)} \, dx=\frac {-\mathrm {log}\left (x -2\right ) \mathrm {log}\left (x \right )-\mathrm {log}\left (x -2\right )-\mathrm {log}\left (-\frac {x}{x -2}\right ) \mathrm {log}\left (x \right )+\mathrm {log}\left (x \right )^{2}+5 \,\mathrm {log}\left (x \right )-x}{\mathrm {log}\left (x \right )+1} \] Input:
int(((-x^2+2*x-2)*log(x)+(2-x)*log(-x/(-2+x))+4*x-10)/((x^2-2*x)*log(x)^2+ (2*x^2-4*x)*log(x)+x^2-2*x),x)
Output:
( - log(x - 2)*log(x) - log(x - 2) - log(( - x)/(x - 2))*log(x) + log(x)** 2 + 5*log(x) - x)/(log(x) + 1)