\(\int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx\) [9104]

Optimal result

Integrand size = 37, antiderivative size = 14 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=1+x-\frac {12}{-e^x+x} \]

[Out]

1+x-12/(x-exp(x))

Rubi [F]

\[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx \]

[In]

Int[(12 + E^(2*x) + E^x*(-12 - 2*x) + x^2)/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

x + 12*Defer[Int][(E^x - x)^(-2), x] - 12*Defer[Int][(E^x - x)^(-1), x] - 12*Defer[Int][x/(E^x - x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{\left (e^x-x\right )^2} \, dx \\ & = \int \left (1-\frac {12}{e^x-x}-\frac {12 (-1+x)}{\left (e^x-x\right )^2}\right ) \, dx \\ & = x-12 \int \frac {1}{e^x-x} \, dx-12 \int \frac {-1+x}{\left (e^x-x\right )^2} \, dx \\ & = x-12 \int \frac {1}{e^x-x} \, dx-12 \int \left (-\frac {1}{\left (e^x-x\right )^2}+\frac {x}{\left (e^x-x\right )^2}\right ) \, dx \\ & = x+12 \int \frac {1}{\left (e^x-x\right )^2} \, dx-12 \int \frac {1}{e^x-x} \, dx-12 \int \frac {x}{\left (e^x-x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\frac {12+e^x x-x^2}{e^x-x} \]

[In]

Integrate[(12 + E^(2*x) + E^x*(-12 - 2*x) + x^2)/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

(12 + E^x*x - x^2)/(E^x - x)

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
risch	\(x -\frac {12}{x -{\mathrm e}^{x}}\)	\(13\)
norman	\(\frac {-12+x^{2}-{\mathrm e}^{x} x}{x -{\mathrm e}^{x}}\)	\(20\)
parallelrisch	\(\frac {-12+x^{2}-{\mathrm e}^{x} x}{x -{\mathrm e}^{x}}\)	\(20\)

[In]

int((exp(x)^2+(-2*x-12)*exp(x)+x^2+12)/(exp(x)^2-2*exp(x)*x+x^2),x,method=_RETURNVERBOSE)

[Out]

x-12/(x-exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\frac {x^{2} - x e^{x} - 12}{x - e^{x}} \]

[In]

integrate((exp(x)^2+(-2*x-12)*exp(x)+x^2+12)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="fricas")

[Out]

(x^2 - x*e^x - 12)/(x - e^x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=x + \frac {12}{- x + e^{x}} \]

[In]

integrate((exp(x)**2+(-2*x-12)*exp(x)+x**2+12)/(exp(x)**2-2*exp(x)*x+x**2),x)

[Out]

x + 12/(-x + exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\frac {x^{2} - x e^{x} - 12}{x - e^{x}} \]

[In]

integrate((exp(x)^2+(-2*x-12)*exp(x)+x^2+12)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="maxima")

[Out]

(x^2 - x*e^x - 12)/(x - e^x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=\frac {x^{2} - x e^{x} - 12}{x - e^{x}} \]

[In]

integrate((exp(x)^2+(-2*x-12)*exp(x)+x^2+12)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="giac")

[Out]

(x^2 - x*e^x - 12)/(x - e^x)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {12+e^{2 x}+e^x (-12-2 x)+x^2}{e^{2 x}-2 e^x x+x^2} \, dx=x-\frac {12}{x-{\mathrm {e}}^x} \]

[In]

int((exp(2*x) - exp(x)*(2*x + 12) + x^2 + 12)/(exp(2*x) - 2*x*exp(x) + x^2),x)

[Out]

x - 12/(x - exp(x))