\(\int \frac {-20+5 x}{(-2 x+x^2) \log ^2(\frac {2-x}{x^2})} \, dx\) [9805]

Optimal result

Integrand size = 27, antiderivative size = 16 \[ \int \frac {-20+5 x}{\left (-2 x+x^2\right ) \log ^2\left (\frac {2-x}{x^2}\right )} \, dx=\frac {5}{\log \left (\frac {2}{x^2}-\frac {1}{x}\right )} \]

[Out]

5/ln(2/x^2-1/x)

Rubi [F]

\[ \int \frac {-20+5 x}{\left (-2 x+x^2\right ) \log ^2\left (\frac {2-x}{x^2}\right )} \, dx=\int \frac {-20+5 x}{\left (-2 x+x^2\right ) \log ^2\left (\frac {2-x}{x^2}\right )} \, dx \]

[In]

Int[(-20 + 5*x)/((-2*x + x^2)*Log[(2 - x)/x^2]^2),x]

[Out]

Defer[Int][(-20 + 5*x)/((-2 + x)*x*Log[(2 - x)/x^2]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-20+5 x}{(-2+x) x \log ^2\left (\frac {2-x}{x^2}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {-20+5 x}{\left (-2 x+x^2\right ) \log ^2\left (\frac {2-x}{x^2}\right )} \, dx=\frac {5}{\log \left (\frac {2-x}{x^2}\right )} \]

[In]

Integrate[(-20 + 5*x)/((-2*x + x^2)*Log[(2 - x)/x^2]^2),x]

[Out]

5/Log[(2 - x)/x^2]

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

method	result	size
parallelrisch	\(\frac {5}{\ln \left (-\frac {-2+x}{x^{2}}\right )}\)	\(14\)
norman	\(\frac {5}{\ln \left (\frac {2-x}{x^{2}}\right )}\)	\(15\)
risch	\(\frac {5}{\ln \left (\frac {2-x}{x^{2}}\right )}\)	\(15\)
derivativedivides	\(\frac {5}{\ln \left (\frac {\frac {2}{x}-1}{x}\right )}\)	\(17\)
default	\(\frac {5}{\ln \left (\frac {\frac {2}{x}-1}{x}\right )}\)	\(17\)

[In]

int((5*x-20)/(x^2-2*x)/ln((2-x)/x^2)^2,x,method=_RETURNVERBOSE)

[Out]

5/ln(-(-2+x)/x^2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {-20+5 x}{\left (-2 x+x^2\right ) \log ^2\left (\frac {2-x}{x^2}\right )} \, dx=\frac {5}{\log \left (-\frac {x - 2}{x^{2}}\right )} \]

[In]

integrate((5*x-20)/(x^2-2*x)/log((2-x)/x^2)^2,x, algorithm="fricas")

[Out]

5/log(-(x - 2)/x^2)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.50 \[ \int \frac {-20+5 x}{\left (-2 x+x^2\right ) \log ^2\left (\frac {2-x}{x^2}\right )} \, dx=\frac {5}{\log {\left (\frac {2 - x}{x^{2}} \right )}} \]

[In]

integrate((5*x-20)/(x**2-2*x)/ln((2-x)/x**2)**2,x)

[Out]

5/log((2 - x)/x**2)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {-20+5 x}{\left (-2 x+x^2\right ) \log ^2\left (\frac {2-x}{x^2}\right )} \, dx=-\frac {5}{2 \, \log \left (x\right ) - \log \left (-x + 2\right )} \]

[In]

integrate((5*x-20)/(x^2-2*x)/log((2-x)/x^2)^2,x, algorithm="maxima")

[Out]

-5/(2*log(x) - log(-x + 2))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {-20+5 x}{\left (-2 x+x^2\right ) \log ^2\left (\frac {2-x}{x^2}\right )} \, dx=\frac {5}{\log \left (-\frac {x - 2}{x^{2}}\right )} \]

[In]

integrate((5*x-20)/(x^2-2*x)/log((2-x)/x^2)^2,x, algorithm="giac")

[Out]

5/log(-(x - 2)/x^2)

Mupad [B] (verification not implemented)

Time = 15.93 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {-20+5 x}{\left (-2 x+x^2\right ) \log ^2\left (\frac {2-x}{x^2}\right )} \, dx=\frac {5}{\ln \left (-\frac {x-2}{x^2}\right )} \]

[In]

int(-(5*x - 20)/(log(-(x - 2)/x^2)^2*(2*x - x^2)),x)

[Out]

5/log(-(x - 2)/x^2)