\(\int \frac {2-e^x+2 x}{(-1+e^x-2 x-x^2) \log (-1+e^x-2 x-x^2)} \, dx\) [1169]

Optimal result

Integrand size = 42, antiderivative size = 17 \[ \int \frac {2-e^x+2 x}{\left (-1+e^x-2 x-x^2\right ) \log \left (-1+e^x-2 x-x^2\right )} \, dx=-2-\log \left (\log \left (-1+e^x-x (2+x)\right )\right ) \]

[Out]

-2-ln(ln(exp(x)-x*(2+x)-1))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6816} \[ \int \frac {2-e^x+2 x}{\left (-1+e^x-2 x-x^2\right ) \log \left (-1+e^x-2 x-x^2\right )} \, dx=-\log \left (\log \left (-x^2-2 x+e^x-1\right )\right ) \]

[In]

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

[Out]

-Log[Log[-1 + E^x - 2*x - x^2]]

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps \begin{align*} \text {integral}& = -\log \left (\log \left (-1+e^x-2 x-x^2\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {2-e^x+2 x}{\left (-1+e^x-2 x-x^2\right ) \log \left (-1+e^x-2 x-x^2\right )} \, dx=-\log \left (\log \left (e^x-(1+x)^2\right )\right ) \]

[In]

Integrate[(2 - E^x + 2*x)/((-1 + E^x - 2*x - x^2)*Log[-1 + E^x - 2*x - x^2]),x]

[Out]

-Log[Log[E^x - (1 + x)^2]]

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

method	result	size
norman	\(-\ln \left (\ln \left ({\mathrm e}^{x}-x^{2}-2 x -1\right )\right )\)	\(17\)
risch	\(-\ln \left (\ln \left ({\mathrm e}^{x}-x^{2}-2 x -1\right )\right )\)	\(17\)
parallelrisch	\(-\ln \left (\ln \left ({\mathrm e}^{x}-x^{2}-2 x -1\right )\right )\)	\(17\)

[In]

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

[Out]

-ln(ln(exp(x)-x^2-2*x-1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {2-e^x+2 x}{\left (-1+e^x-2 x-x^2\right ) \log \left (-1+e^x-2 x-x^2\right )} \, dx=-\log \left (\log \left (-x^{2} - 2 \, x + e^{x} - 1\right )\right ) \]

[In]

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

[Out]

-log(log(-x^2 - 2*x + e^x - 1))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {2-e^x+2 x}{\left (-1+e^x-2 x-x^2\right ) \log \left (-1+e^x-2 x-x^2\right )} \, dx=- \log {\left (\log {\left (- x^{2} - 2 x + e^{x} - 1 \right )} \right )} \]

[In]

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

[Out]

-log(log(-x**2 - 2*x + exp(x) - 1))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {2-e^x+2 x}{\left (-1+e^x-2 x-x^2\right ) \log \left (-1+e^x-2 x-x^2\right )} \, dx=-\log \left (\log \left (-x^{2} - 2 \, x + e^{x} - 1\right )\right ) \]

[In]

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

[Out]

-log(log(-x^2 - 2*x + e^x - 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {2-e^x+2 x}{\left (-1+e^x-2 x-x^2\right ) \log \left (-1+e^x-2 x-x^2\right )} \, dx=-\log \left (\log \left (-x^{2} - 2 \, x + e^{x} - 1\right )\right ) \]

[In]

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

[Out]

-log(log(-x^2 - 2*x + e^x - 1))

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {2-e^x+2 x}{\left (-1+e^x-2 x-x^2\right ) \log \left (-1+e^x-2 x-x^2\right )} \, dx=-\ln \left (\ln \left ({\mathrm {e}}^x-2\,x-x^2-1\right )\right ) \]

[In]

int(-(2*x - exp(x) + 2)/(log(exp(x) - 2*x - x^2 - 1)*(2*x - exp(x) + x^2 + 1)),x)

[Out]

-log(log(exp(x) - 2*x - x^2 - 1))