\(\int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx\) [3009]

Optimal result

Integrand size = 24, antiderivative size = 17 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=1-x+\log \left (4-e x+e^x x\right ) \]

[Out]

ln(exp(x)*x-x*exp(1)+4)+1-x

Rubi [F]

\[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=\int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=-x+\log \left (4-e x+e^x x\right ) \]

[In]

Integrate[(-4 + E^x + E*(-1 + x))/(4 - E*x + E^x*x),x]

[Out]

-x + Log[4 - E*x + E^x*x]

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

method	result	size
norman	\(-x +\ln \left (x \,{\mathrm e}-{\mathrm e}^{x} x -4\right )\)	\(17\)
parallelrisch	\(-x +\ln \left ({\mathrm e}^{x} x -x \,{\mathrm e}+4\right )\)	\(17\)
risch	\(\ln \left (x \right )-x +\ln \left ({\mathrm e}^{x}-\frac {x \,{\mathrm e}-4}{x}\right )\)	\(22\)

[In]

int((exp(x)+(-1+x)*exp(1)-4)/(exp(x)*x-x*exp(1)+4),x,method=_RETURNVERBOSE)

[Out]

-x+ln(x*exp(1)-exp(x)*x-4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.35 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=-x + \log \left (x\right ) + \log \left (-\frac {x e - x e^{x} - 4}{x}\right ) \]

[In]

integrate((exp(x)+(-1+x)*exp(1)-4)/(exp(x)*x-x*exp(1)+4),x, algorithm="fricas")

[Out]

-x + log(x) + log(-(x*e - x*e^x - 4)/x)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=- x + \log {\left (x \right )} + \log {\left (e^{x} + \frac {- e x + 4}{x} \right )} \]

[In]

integrate((exp(x)+(-1+x)*exp(1)-4)/(exp(x)*x-x*exp(1)+4),x)

[Out]

-x + log(x) + log(exp(x) + (-E*x + 4)/x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.35 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=-x + \log \left (x\right ) + \log \left (-\frac {x e - x e^{x} - 4}{x}\right ) \]

[In]

integrate((exp(x)+(-1+x)*exp(1)-4)/(exp(x)*x-x*exp(1)+4),x, algorithm="maxima")

[Out]

-x + log(x) + log(-(x*e - x*e^x - 4)/x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=-x + \log \left (x e - x e^{x} - 4\right ) \]

[In]

integrate((exp(x)+(-1+x)*exp(1)-4)/(exp(x)*x-x*exp(1)+4),x, algorithm="giac")

[Out]

-x + log(x*e - x*e^x - 4)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-4+e^x+e (-1+x)}{4-e x+e^x x} \, dx=\ln \left (x\,{\mathrm {e}}^x-x\,\mathrm {e}+4\right )-x \]

[In]

int((exp(x) + exp(1)*(x - 1) - 4)/(x*exp(x) - x*exp(1) + 4),x)

[Out]

log(x*exp(x) - x*exp(1) + 4) - x