\(\int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx\) [3380]

Optimal result

Integrand size = 26, antiderivative size = 23 \[ \int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx=\frac {-2+2 x-\log \left (2+e^x-2 x\right )}{6 e} \]

[Out]

1/6/exp(1)*(2*x-2-ln(exp(x)-2*x+2))

Rubi [F]

\[ \int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx=\int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx \]

[In]

Int[(6 + E^x - 4*x)/(6*E^(1 + x) + E*(12 - 12*x)),x]

[Out]

x/(6*E) + (2*Defer[Int][(2 + E^x - 2*x)^(-1), x])/(3*E) - Defer[Int][x/(2 + E^x - 2*x), x]/(3*E)

Rubi steps \begin{align*} \text {integral}& = \int \frac {6+e^x-4 x}{6 e \left (2+e^x-2 x\right )} \, dx \\ & = \frac {\int \frac {6+e^x-4 x}{2+e^x-2 x} \, dx}{6 e} \\ & = \frac {\int \left (1+\frac {2 (-2+x)}{-2-e^x+2 x}\right ) \, dx}{6 e} \\ & = \frac {x}{6 e}+\frac {\int \frac {-2+x}{-2-e^x+2 x} \, dx}{3 e} \\ & = \frac {x}{6 e}+\frac {\int \left (\frac {2}{2+e^x-2 x}-\frac {x}{2+e^x-2 x}\right ) \, dx}{3 e} \\ & = \frac {x}{6 e}-\frac {\int \frac {x}{2+e^x-2 x} \, dx}{3 e}+\frac {2 \int \frac {1}{2+e^x-2 x} \, dx}{3 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx=\frac {2 x-\log \left (2+e^x-2 x\right )}{6 e} \]

[In]

Integrate[(6 + E^x - 4*x)/(6*E^(1 + x) + E*(12 - 12*x)),x]

[Out]

(2*x - Log[2 + E^x - 2*x])/(6*E)

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

method	result	size
risch	\(\frac {{\mathrm e}^{-1} x}{3}-\frac {{\mathrm e}^{-1} \ln \left ({\mathrm e}^{x}-2 x +2\right )}{6}\)	\(19\)
parallelrisch	\(-\frac {\left (-2 x +\ln \left (-\frac {{\mathrm e}^{x}}{2}+x -1\right )\right ) {\mathrm e}^{-1}}{6}\)	\(19\)
norman	\(\frac {{\mathrm e}^{-1} x}{3}-\frac {{\mathrm e}^{-1} \ln \left (2 x -2-{\mathrm e}^{x}\right )}{6}\)	\(25\)

[In]

int((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x,method=_RETURNVERBOSE)

[Out]

1/3*exp(-1)*x-1/6*exp(-1)*ln(exp(x)-2*x+2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx=\frac {1}{6} \, {\left (2 \, x - \log \left (-2 \, {\left (x - 1\right )} e + e^{\left (x + 1\right )}\right )\right )} e^{\left (-1\right )} \]

[In]

integrate((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x, algorithm="fricas")

[Out]

1/6*(2*x - log(-2*(x - 1)*e + e^(x + 1)))*e^(-1)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx=\frac {x}{3 e} - \frac {\log {\left (- 2 x + e^{x} + 2 \right )}}{6 e} \]

[In]

integrate((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x)

[Out]

x*exp(-1)/3 - exp(-1)*log(-2*x + exp(x) + 2)/6

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx=\frac {1}{3} \, x e^{\left (-1\right )} - \frac {1}{6} \, e^{\left (-1\right )} \log \left (-2 \, x + e^{x} + 2\right ) \]

[In]

integrate((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x, algorithm="maxima")

[Out]

1/3*x*e^(-1) - 1/6*e^(-1)*log(-2*x + e^x + 2)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx=\frac {1}{6} \, {\left (2 \, x - \log \left (2 \, x - e^{x} - 2\right )\right )} e^{\left (-1\right )} \]

[In]

integrate((exp(x)+6-4*x)/(6*exp(1)*exp(x)+(-12*x+12)*exp(1)),x, algorithm="giac")

[Out]

1/6*(2*x - log(2*x - e^x - 2))*e^(-1)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {6+e^x-4 x}{6 e^{1+x}+e (12-12 x)} \, dx=\frac {{\mathrm {e}}^{-1}\,\left (2\,x-\ln \left (2\,x-{\mathrm {e}}^x-2\right )\right )}{6} \]

[In]

int((exp(x) - 4*x + 6)/(6*exp(1)*exp(x) - exp(1)*(12*x - 12)),x)

[Out]

(exp(-1)*(2*x - log(2*x - exp(x) - 2)))/6