3.12.51 \(\int \frac {-2 x+2 x^2+e^{2+2 (x-\log (2)) \log (\frac {2-2 x}{x})} (2 x-2 \log (2)+(-2 x+2 x^2) \log (\frac {2-2 x}{x}))}{-x+x^2} \, dx\)

Optimal. Leaf size=25 \[ -5+e^{2-2 (-x+\log (2)) \log \left (-2+\frac {2}{x}\right )}+2 x \]

________________________________________________________________________________________

Rubi [F]  time = 4.93, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

2*x + 2*E^2*Log[-2 + 2/x]*Defer[Int][(-2 + 2/x)^(2*x - Log[4]), x] + 2*E^2*(1 - Log[2])*Defer[Int][(-2 + 2/x)^
(2*x - Log[4])/(-1 + x), x] + 2*E^2*Log[2]*Defer[Int][(-2 + 2/x)^(2*x - Log[4])/x, x] - 2*E^2*Defer[Int][Defer
[Int][(-2 + 2/x)^(2*x - Log[4]), x]/(-1 + x), x] + 2*E^2*Defer[Int][Defer[Int][(-2 + 2/x)^(2*x - Log[4]), x]/x
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\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 )}{(-1+x) x} \, dx\\ &=\int \left (2+\frac {2 e^2 \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \left (x-\log (2)-x \log \left (-2+\frac {2}{x}\right )+x^2 \log \left (-2+\frac {2}{x}\right )\right )}{(-1+x) x}\right ) \, dx\\ &=2 x+\left (2 e^2\right ) \int \frac {\left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \left (x-\log (2)-x \log \left (-2+\frac {2}{x}\right )+x^2 \log \left (-2+\frac {2}{x}\right )\right )}{(-1+x) x} \, dx\\ &=2 x+\left (2 e^2\right ) \int \left (\frac {\left (-2+\frac {2}{x}\right )^{2 x-\log (4)} (x-\log (2))}{(-1+x) x}+\left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \log \left (-2+\frac {2}{x}\right )\right ) \, dx\\ &=2 x+\left (2 e^2\right ) \int \frac {\left (-2+\frac {2}{x}\right )^{2 x-\log (4)} (x-\log (2))}{(-1+x) x} \, dx+\left (2 e^2\right ) \int \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \log \left (-2+\frac {2}{x}\right ) \, dx\\ &=2 x+\left (2 e^2\right ) \int \left (\frac {\left (-2+\frac {2}{x}\right )^{2 x-\log (4)} (1-\log (2))}{-1+x}+\frac {\left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \log (2)}{x}\right ) \, dx-\left (2 e^2\right ) \int \frac {\int \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \, dx}{(-1+x) x} \, dx+\left (2 e^2 \log \left (-2+\frac {2}{x}\right )\right ) \int \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \, dx\\ &=2 x-\left (2 e^2\right ) \int \left (\frac {\int \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \, dx}{-1+x}-\frac {\int \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \, dx}{x}\right ) \, dx+\left (2 e^2 (1-\log (2))\right ) \int \frac {\left (-2+\frac {2}{x}\right )^{2 x-\log (4)}}{-1+x} \, dx+\left (2 e^2 \log (2)\right ) \int \frac {\left (-2+\frac {2}{x}\right )^{2 x-\log (4)}}{x} \, dx+\left (2 e^2 \log \left (-2+\frac {2}{x}\right )\right ) \int \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \, dx\\ &=2 x-\left (2 e^2\right ) \int \frac {\int \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \, dx}{-1+x} \, dx+\left (2 e^2\right ) \int \frac {\int \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \, dx}{x} \, dx+\left (2 e^2 (1-\log (2))\right ) \int \frac {\left (-2+\frac {2}{x}\right )^{2 x-\log (4)}}{-1+x} \, dx+\left (2 e^2 \log (2)\right ) \int \frac {\left (-2+\frac {2}{x}\right )^{2 x-\log (4)}}{x} \, dx+\left (2 e^2 \log \left (-2+\frac {2}{x}\right )\right ) \int \left (-2+\frac {2}{x}\right )^{2 x-\log (4)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 5.18, size = 24, normalized size = 0.96 \begin {gather*} e^2 \left (-2+\frac {2}{x}\right )^{2 x-\log (4)}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

E^2*(-2 + 2/x)^(2*x - Log[4]) + 2*x

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 24, normalized size = 0.96 \begin {gather*} 2 \, x + e^{\left (2 \, {\left (x - \log \relax (2)\right )} \log \left (-\frac {2 \, {\left (x - 1\right )}}{x}\right ) + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-2*x)*log((-2*x+2)/x)-2*log(2)+2*x)*exp((x-log(2))*log((-2*x+2)/x)+1)^2+2*x^2-2*x)/(x^2-x),x
, algorithm="fricas")

[Out]

2*x + e^(2*(x - log(2))*log(-2*(x - 1)/x) + 2)

________________________________________________________________________________________

giac [A]  time = 1.79, size = 30, normalized size = 1.20 \begin {gather*} 2 \, x + e^{\left (2 \, x \log \left (\frac {2}{x} - 2\right ) - 2 \, \log \relax (2) \log \left (\frac {2}{x} - 2\right ) + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-2*x)*log((-2*x+2)/x)-2*log(2)+2*x)*exp((x-log(2))*log((-2*x+2)/x)+1)^2+2*x^2-2*x)/(x^2-x),x
, algorithm="giac")

[Out]

2*x + e^(2*x*log(2/x - 2) - 2*log(2)*log(2/x - 2) + 2)

________________________________________________________________________________________

maple [A]  time = 0.25, size = 26, normalized size = 1.04

method	result	size
risch	\(2 x +\left (\frac {-2 x +2}{x}\right )^{-2 \ln \relax (2)+2 x} {\mathrm e}^{2}\)	\(26\)
default	\(2 x +{\mathrm e}^{\left (-2 \ln \relax (2)+2 x \right ) \ln \left (\frac {-2 x +2}{x}\right )+2}\)	\(27\)
norman	\(2 x +{\mathrm e}^{\left (-2 \ln \relax (2)+2 x \right ) \ln \left (\frac {-2 x +2}{x}\right )+2}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2-2*x)*ln((-2*x+2)/x)-2*ln(2)+2*x)*exp((x-ln(2))*ln((-2*x+2)/x)+1)^2+2*x^2-2*x)/(x^2-x),x,method=_R
ETURNVERBOSE)

[Out]

2*x+(((-2*x+2)/x)^(x-ln(2)))^2*exp(2)

________________________________________________________________________________________

maxima [C]  time = 0.72, size = 55, normalized size = 2.20 \begin {gather*} \left (-1\right )^{2 \, \log \relax (2)} e^{\left (2 i \, \pi x + 2 \, x \log \relax (2) - 2 \, \log \relax (2)^{2} + 2 \, x \log \left (x - 1\right ) - 2 \, \log \relax (2) \log \left (x - 1\right ) - 2 \, x \log \relax (x) + 2 \, \log \relax (2) \log \relax (x) + 2\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-2*x)*log((-2*x+2)/x)-2*log(2)+2*x)*exp((x-log(2))*log((-2*x+2)/x)+1)^2+2*x^2-2*x)/(x^2-x),x
, algorithm="maxima")

[Out]

(-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

________________________________________________________________________________________

mupad [B]  time = 1.22, size = 32, normalized size = 1.28 \begin {gather*} 2\,x+\frac {{\mathrm {e}}^2\,{\left (\frac {2}{x}-2\right )}^{2\,x}}{{\left (\frac {2}{x}-2\right )}^{2\,\ln \relax (2)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

2*x + (exp(2)*(2/x - 2)^(2*x))/(2/x - 2)^(2*log(2))

________________________________________________________________________________________

sympy [A]  time = 0.43, size = 20, normalized size = 0.80 \begin {gather*} 2 x + e^{2 \left (x - \log {\relax (2 )}\right ) \log {\left (\frac {2 - 2 x}{x} \right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2-2*x)*ln((-2*x+2)/x)-2*ln(2)+2*x)*exp((x-ln(2))*ln((-2*x+2)/x)+1)**2+2*x**2-2*x)/(x**2-x),x
)

[Out]

2*x + exp(2*(x - log(2))*log((2 - 2*x)/x) + 2)

________________________________________________________________________________________