\(\int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx\) [3263]

Optimal result

Integrand size = 47, antiderivative size = 16 \[ \int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx=\frac {1}{\frac {4 e^{-1+12 x}}{x}+x} \]

[Out]

1/(x+4*exp(12*x-1)/x)

Rubi [F]

\[ \int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx=\int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx \]

[In]

Int[(E^(-1 + 12*x)*(4 - 48*x) - x^2)/(16*E^(-2 + 24*x) + 8*E^(-1 + 12*x)*x^2 + x^4),x]

[Out]

-2*E^2*Defer[Int][x^2/(4*E^(12*x) + E*x^2)^2, x] + 12*E^2*Defer[Int][x^3/(4*E^(12*x) + E*x^2)^2, x] + E*Defer[
Int][(4*E^(12*x) + E*x^2)^(-1), x] - 12*E*Defer[Int][x/(4*E^(12*x) + E*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^2 \left (e^{-1+12 x} (4-48 x)-x^2\right )}{\left (4 e^{12 x}+e x^2\right )^2} \, dx \\ & = e^2 \int \frac {e^{-1+12 x} (4-48 x)-x^2}{\left (4 e^{12 x}+e x^2\right )^2} \, dx \\ & = e^2 \int \left (\frac {2 x^2 (-1+6 x)}{\left (4 e^{12 x}+e x^2\right )^2}-\frac {-1+12 x}{e \left (4 e^{12 x}+e x^2\right )}\right ) \, dx \\ & = -\left (e \int \frac {-1+12 x}{4 e^{12 x}+e x^2} \, dx\right )+\left (2 e^2\right ) \int \frac {x^2 (-1+6 x)}{\left (4 e^{12 x}+e x^2\right )^2} \, dx \\ & = -\left (e \int \left (-\frac {1}{4 e^{12 x}+e x^2}+\frac {12 x}{4 e^{12 x}+e x^2}\right ) \, dx\right )+\left (2 e^2\right ) \int \left (-\frac {x^2}{\left (4 e^{12 x}+e x^2\right )^2}+\frac {6 x^3}{\left (4 e^{12 x}+e x^2\right )^2}\right ) \, dx \\ & = e \int \frac {1}{4 e^{12 x}+e x^2} \, dx-(12 e) \int \frac {x}{4 e^{12 x}+e x^2} \, dx-\left (2 e^2\right ) \int \frac {x^2}{\left (4 e^{12 x}+e x^2\right )^2} \, dx+\left (12 e^2\right ) \int \frac {x^3}{\left (4 e^{12 x}+e x^2\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx=\frac {e x}{4 e^{12 x}+e x^2} \]

[In]

Integrate[(E^(-1 + 12*x)*(4 - 48*x) - x^2)/(16*E^(-2 + 24*x) + 8*E^(-1 + 12*x)*x^2 + x^4),x]

[Out]

(E*x)/(4*E^(12*x) + E*x^2)

Maple [A] (verified)

Time = 1.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
norman	\(\frac {x}{x^{2}+4 \,{\mathrm e}^{12 x -1}}\)	\(17\)
risch	\(\frac {x}{x^{2}+4 \,{\mathrm e}^{12 x -1}}\)	\(17\)
parallelrisch	\(\frac {x}{x^{2}+4 \,{\mathrm e}^{12 x -1}}\)	\(17\)

[In]

int(((-48*x+4)*exp(12*x-1)-x^2)/(16*exp(12*x-1)^2+8*x^2*exp(12*x-1)+x^4),x,method=_RETURNVERBOSE)

[Out]

x/(x^2+4*exp(12*x-1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx=\frac {x}{x^{2} + 4 \, e^{\left (12 \, x - 1\right )}} \]

[In]

integrate(((-48*x+4)*exp(12*x-1)-x^2)/(16*exp(12*x-1)^2+8*x^2*exp(12*x-1)+x^4),x, algorithm="fricas")

[Out]

x/(x^2 + 4*e^(12*x - 1))

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx=\frac {x}{x^{2} + 4 e^{12 x - 1}} \]

[In]

integrate(((-48*x+4)*exp(12*x-1)-x**2)/(16*exp(12*x-1)**2+8*x**2*exp(12*x-1)+x**4),x)

[Out]

x/(x**2 + 4*exp(12*x - 1))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx=\frac {x e}{x^{2} e + 4 \, e^{\left (12 \, x\right )}} \]

[In]

integrate(((-48*x+4)*exp(12*x-1)-x^2)/(16*exp(12*x-1)^2+8*x^2*exp(12*x-1)+x^4),x, algorithm="maxima")

[Out]

x*e/(x^2*e + 4*e^(12*x))

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx=\frac {x e}{x^{2} e + 4 \, e^{\left (12 \, x\right )}} \]

[In]

integrate(((-48*x+4)*exp(12*x-1)-x^2)/(16*exp(12*x-1)^2+8*x^2*exp(12*x-1)+x^4),x, algorithm="giac")

[Out]

x*e/(x^2*e + 4*e^(12*x))

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-1+12 x} (4-48 x)-x^2}{16 e^{-2+24 x}+8 e^{-1+12 x} x^2+x^4} \, dx=\frac {x}{4\,{\mathrm {e}}^{12\,x-1}+x^2} \]

[In]

int(-(exp(12*x - 1)*(48*x - 4) + x^2)/(16*exp(24*x - 2) + 8*x^2*exp(12*x - 1) + x^4),x)

[Out]

x/(4*exp(12*x - 1) + x^2)