Integrand size = 70, antiderivative size = 25 \[ \int \frac {-2 x+2 x^2+e^{2+2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )} \left (2 x-2 \log (2)+\left (-2 x+2 x^2\right ) \log \left (\frac {2-2 x}{x}\right )\right )}{-x+x^2} \, dx=-5+e^{2-2 (-x+\log (2)) \log \left (-2+\frac {2}{x}\right )}+2 x \] Output:
2*x-5+exp(1-ln(2/x-2)*(ln(2)-x))^2
Time = 2.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-2 x+2 x^2+e^{2+2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )} \left (2 x-2 \log (2)+\left (-2 x+2 x^2\right ) \log \left (\frac {2-2 x}{x}\right )\right )}{-x+x^2} \, dx=e^2 \left (-2+\frac {2}{x}\right )^{2 x-\log (4)}+2 x \] Input:
Integrate[(-2*x + 2*x^2 + E^(2 + 2*(x - Log[2])*Log[(2 - 2*x)/x])*(2*x - 2 *Log[2] + (-2*x + 2*x^2)*Log[(2 - 2*x)/x]))/(-x + x^2),x]
Output:
E^2*(-2 + 2/x)^(2*x - Log[4]) + 2*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.
\(\displaystyle \int \frac {2 x^2+e^{2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )+2} \left (\left (2 x^2-2 x\right ) \log \left (\frac {2-2 x}{x}\right )+2 x-2 \log (2)\right )-2 x}{x^2-x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2 x^2+e^{2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )+2} \left (\left (2 x^2-2 x\right ) \log \left (\frac {2-2 x}{x}\right )+2 x-2 \log (2)\right )-2 x}{(x-1) x}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 e^2 \left (x^2 \log \left (\frac {2}{x}-2\right )+x-x \log \left (\frac {2}{x}-2\right )-\log (2)\right ) \left (\frac {2}{x}-2\right )^{2 x-\log (4)}}{(x-1) x}+2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 e^2 \log \left (\frac {2}{x}-2\right ) \int \left (\frac {2}{x}-2\right )^{2 x-\log (4)}dx+2 e^2 (1-\log (2)) \int \frac {\left (\frac {2}{x}-2\right )^{2 x-\log (4)}}{x-1}dx+2 e^2 \log (2) \int \frac {\left (\frac {2}{x}-2\right )^{2 x-\log (4)}}{x}dx-2 e^2 \int \frac {\int \left (\frac {2}{x}-2\right )^{2 x-\log (4)}dx}{x-1}dx+2 e^2 \int \frac {\int \left (\frac {2}{x}-2\right )^{2 x-\log (4)}dx}{x}dx+2 x\) |
Input:
Int[(-2*x + 2*x^2 + E^(2 + 2*(x - Log[2])*Log[(2 - 2*x)/x])*(2*x - 2*Log[2 ] + (-2*x + 2*x^2)*Log[(2 - 2*x)/x]))/(-x + x^2),x]
Output:
$Aborted
Time = 1.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
risch | \(2 x +\left (\frac {2-2 x}{x}\right )^{-2 \ln \left (2\right )+2 x} {\mathrm e}^{2}\) | \(26\) |
default | \(2 x +{\mathrm e}^{\left (-2 \ln \left (2\right )+2 x \right ) \ln \left (\frac {2-2 x}{x}\right )+2}\) | \(27\) |
norman | \(2 x +{\mathrm e}^{\left (-2 \ln \left (2\right )+2 x \right ) \ln \left (\frac {2-2 x}{x}\right )+2}\) | \(27\) |
parallelrisch | \(4+{\mathrm e}^{\left (-2 \ln \left (2\right )+2 x \right ) \ln \left (-\frac {2 \left (-1+x \right )}{x}\right )+2}+2 x\) | \(27\) |
parts | \(2 x +{\mathrm e}^{\left (-2 \ln \left (2\right )+2 x \right ) \ln \left (\frac {2-2 x}{x}\right )+2}\) | \(27\) |
Input:
int((((2*x^2-2*x)*ln((2-2*x)/x)-2*ln(2)+2*x)*exp((x-ln(2))*ln((2-2*x)/x)+1 )^2+2*x^2-2*x)/(x^2-x),x,method=_RETURNVERBOSE)
Output:
2*x+(((2-2*x)/x)^(x-ln(2)))^2*exp(2)
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-2 x+2 x^2+e^{2+2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )} \left (2 x-2 \log (2)+\left (-2 x+2 x^2\right ) \log \left (\frac {2-2 x}{x}\right )\right )}{-x+x^2} \, dx=2 \, x + e^{\left (2 \, {\left (x - \log \left (2\right )\right )} \log \left (-\frac {2 \, {\left (x - 1\right )}}{x}\right ) + 2\right )} \] Input:
integrate((((2*x^2-2*x)*log((2-2*x)/x)-2*log(2)+2*x)*exp((x-log(2))*log((2 -2*x)/x)+1)^2+2*x^2-2*x)/(x^2-x),x, algorithm="fricas")
Output:
2*x + e^(2*(x - log(2))*log(-2*(x - 1)/x) + 2)
Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-2 x+2 x^2+e^{2+2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )} \left (2 x-2 \log (2)+\left (-2 x+2 x^2\right ) \log \left (\frac {2-2 x}{x}\right )\right )}{-x+x^2} \, dx=2 x + e^{2 \left (x - \log {\left (2 \right )}\right ) \log {\left (\frac {2 - 2 x}{x} \right )} + 2} \] Input:
integrate((((2*x**2-2*x)*ln((2-2*x)/x)-2*ln(2)+2*x)*exp((x-ln(2))*ln((2-2* x)/x)+1)**2+2*x**2-2*x)/(x**2-x),x)
Output:
2*x + exp(2*(x - log(2))*log((2 - 2*x)/x) + 2)
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {-2 x+2 x^2+e^{2+2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )} \left (2 x-2 \log (2)+\left (-2 x+2 x^2\right ) \log \left (\frac {2-2 x}{x}\right )\right )}{-x+x^2} \, dx=\left (-1\right )^{2 \, \log \left (2\right )} e^{\left (2 i \, \pi x + 2 \, x \log \left (2\right ) - 2 \, \log \left (2\right )^{2} + 2 \, x \log \left (x - 1\right ) - 2 \, \log \left (2\right ) \log \left (x - 1\right ) - 2 \, x \log \left (x\right ) + 2 \, \log \left (2\right ) \log \left (x\right ) + 2\right )} + 2 \, x \] Input:
integrate((((2*x^2-2*x)*log((2-2*x)/x)-2*log(2)+2*x)*exp((x-log(2))*log((2 -2*x)/x)+1)^2+2*x^2-2*x)/(x^2-x),x, algorithm="maxima")
Output:
(-1)^(2*log(2))*e^(2*I*pi*x + 2*x*log(2) - 2*log(2)^2 + 2*x*log(x - 1) - 2 *log(2)*log(x - 1) - 2*x*log(x) + 2*log(2)*log(x) + 2) + 2*x
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {-2 x+2 x^2+e^{2+2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )} \left (2 x-2 \log (2)+\left (-2 x+2 x^2\right ) \log \left (\frac {2-2 x}{x}\right )\right )}{-x+x^2} \, dx=2 \, x + e^{\left (2 \, x \log \left (\frac {2}{x} - 2\right ) - 2 \, \log \left (2\right ) \log \left (\frac {2}{x} - 2\right ) + 2\right )} \] Input:
integrate((((2*x^2-2*x)*log((2-2*x)/x)-2*log(2)+2*x)*exp((x-log(2))*log((2 -2*x)/x)+1)^2+2*x^2-2*x)/(x^2-x),x, algorithm="giac")
Output:
2*x + e^(2*x*log(2/x - 2) - 2*log(2)*log(2/x - 2) + 2)
Time = 1.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {-2 x+2 x^2+e^{2+2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )} \left (2 x-2 \log (2)+\left (-2 x+2 x^2\right ) \log \left (\frac {2-2 x}{x}\right )\right )}{-x+x^2} \, dx=2\,x+\frac {{\mathrm {e}}^2\,{\left (\frac {2}{x}-2\right )}^{2\,x}}{{\left (\frac {2}{x}-2\right )}^{2\,\ln \left (2\right )}} \] Input:
int((2*x + exp(2*log(-(2*x - 2)/x)*(x - log(2)) + 2)*(2*log(2) - 2*x + log (-(2*x - 2)/x)*(2*x - 2*x^2)) - 2*x^2)/(x - x^2),x)
Output:
2*x + (exp(2)*(2/x - 2)^(2*x))/(2/x - 2)^(2*log(2))
\[ \int \frac {-2 x+2 x^2+e^{2+2 (x-\log (2)) \log \left (\frac {2-2 x}{x}\right )} \left (2 x-2 \log (2)+\left (-2 x+2 x^2\right ) \log \left (\frac {2-2 x}{x}\right )\right )}{-x+x^2} \, dx=2 \left (\int \frac {x^{2 \,\mathrm {log}\left (2\right )} \left (-2 x +2\right )^{2 x} \mathrm {log}\left (\frac {-2 x +2}{x}\right ) x}{x^{2 x} \left (-2 x +2\right )^{2 \,\mathrm {log}\left (2\right )} x -x^{2 x} \left (-2 x +2\right )^{2 \,\mathrm {log}\left (2\right )}}d x \right ) e^{2}-2 \left (\int \frac {x^{2 \,\mathrm {log}\left (2\right )} \left (-2 x +2\right )^{2 x} \mathrm {log}\left (\frac {-2 x +2}{x}\right )}{x^{2 x} \left (-2 x +2\right )^{2 \,\mathrm {log}\left (2\right )} x -x^{2 x} \left (-2 x +2\right )^{2 \,\mathrm {log}\left (2\right )}}d x \right ) e^{2}-2 \left (\int \frac {x^{2 \,\mathrm {log}\left (2\right )} \left (-2 x +2\right )^{2 x}}{x^{2 x} \left (-2 x +2\right )^{2 \,\mathrm {log}\left (2\right )} x^{2}-x^{2 x} \left (-2 x +2\right )^{2 \,\mathrm {log}\left (2\right )} x}d x \right ) \mathrm {log}\left (2\right ) e^{2}+2 \left (\int \frac {x^{2 \,\mathrm {log}\left (2\right )} \left (-2 x +2\right )^{2 x}}{x^{2 x} \left (-2 x +2\right )^{2 \,\mathrm {log}\left (2\right )} x -x^{2 x} \left (-2 x +2\right )^{2 \,\mathrm {log}\left (2\right )}}d x \right ) e^{2}+2 x \] Input:
int((((2*x^2-2*x)*log((2-2*x)/x)-2*log(2)+2*x)*exp((x-log(2))*log((2-2*x)/ x)+1)^2+2*x^2-2*x)/(x^2-x),x)
Output:
2*(int((x**(2*log(2))*( - 2*x + 2)**(2*x)*log(( - 2*x + 2)/x)*x)/(x**(2*x) *( - 2*x + 2)**(2*log(2))*x - x**(2*x)*( - 2*x + 2)**(2*log(2))),x)*e**2 - int((x**(2*log(2))*( - 2*x + 2)**(2*x)*log(( - 2*x + 2)/x))/(x**(2*x)*( - 2*x + 2)**(2*log(2))*x - x**(2*x)*( - 2*x + 2)**(2*log(2))),x)*e**2 - int ((x**(2*log(2))*( - 2*x + 2)**(2*x))/(x**(2*x)*( - 2*x + 2)**(2*log(2))*x* *2 - x**(2*x)*( - 2*x + 2)**(2*log(2))*x),x)*log(2)*e**2 + int((x**(2*log( 2))*( - 2*x + 2)**(2*x))/(x**(2*x)*( - 2*x + 2)**(2*log(2))*x - x**(2*x)*( - 2*x + 2)**(2*log(2))),x)*e**2 + x)