\(\int \frac {2-e^x+e^{\frac {x^2}{2}} (-3-3 x^2)}{e^{2 x}-4 e^x x+4 x^2+9 e^{x^2} x^2+e^{\frac {x^2}{2}} (6 e^x x-12 x^2)} \, dx\) [2535]

Optimal result

Integrand size = 76, antiderivative size = 28 \[ \int \frac {2-e^x+e^{\frac {x^2}{2}} \left (-3-3 x^2\right )}{e^{2 x}-4 e^x x+4 x^2+9 e^{x^2} x^2+e^{\frac {x^2}{2}} \left (6 e^x x-12 x^2\right )} \, dx=\frac {x}{x^2+x \left (e^x+3 \left (-1+e^{\frac {x^2}{2}}\right ) x\right )} \]

[Out]

x/(x^2+x*(x*(3*exp(1/4*x^2)^2-3)+exp(x)))

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6820, 6818} \[ \int \frac {2-e^x+e^{\frac {x^2}{2}} \left (-3-3 x^2\right )}{e^{2 x}-4 e^x x+4 x^2+9 e^{x^2} x^2+e^{\frac {x^2}{2}} \left (6 e^x x-12 x^2\right )} \, dx=\frac {1}{3 e^{\frac {x^2}{2}} x-2 x+e^x} \]

[In]

Int[(2 - E^x + E^(x^2/2)*(-3 - 3*x^2))/(E^(2*x) - 4*E^x*x + 4*x^2 + 9*E^x^2*x^2 + E^(x^2/2)*(6*E^x*x - 12*x^2)
),x]

[Out]

(E^x - 2*x + 3*E^(x^2/2)*x)^(-1)

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {2-e^x+e^{\frac {x^2}{2}} \left (-3-3 x^2\right )}{e^{2 x}-4 e^x x+4 x^2+9 e^{x^2} x^2+e^{\frac {x^2}{2}} \left (6 e^x x-12 x^2\right )} \, dx=\frac {1}{e^x-2 x+3 e^{\frac {x^2}{2}} x} \]

[In]

Integrate[(2 - E^x + E^(x^2/2)*(-3 - 3*x^2))/(E^(2*x) - 4*E^x*x + 4*x^2 + 9*E^x^2*x^2 + E^(x^2/2)*(6*E^x*x - 1
2*x^2)),x]

[Out]

(E^x - 2*x + 3*E^(x^2/2)*x)^(-1)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64

method	result	size
risch	\(\frac {1}{3 x \,{\mathrm e}^{\frac {x^{2}}{2}}+{\mathrm e}^{x}-2 x}\)	\(18\)
parallelrisch	\(\frac {1}{3 x \,{\mathrm e}^{\frac {x^{2}}{2}}+{\mathrm e}^{x}-2 x}\)	\(20\)

[In]

int(((-3*x^2-3)*exp(1/4*x^2)^2-exp(x)+2)/(9*x^2*exp(1/4*x^2)^4+(6*exp(x)*x-12*x^2)*exp(1/4*x^2)^2+exp(x)^2-4*e
xp(x)*x+4*x^2),x,method=_RETURNVERBOSE)

[Out]

1/(3*x*exp(1/2*x^2)+exp(x)-2*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {2-e^x+e^{\frac {x^2}{2}} \left (-3-3 x^2\right )}{e^{2 x}-4 e^x x+4 x^2+9 e^{x^2} x^2+e^{\frac {x^2}{2}} \left (6 e^x x-12 x^2\right )} \, dx=\frac {1}{3 \, x e^{\left (\frac {1}{2} \, x^{2}\right )} - 2 \, x + e^{x}} \]

[In]

integrate(((-3*x^2-3)*exp(1/4*x^2)^2-exp(x)+2)/(9*x^2*exp(1/4*x^2)^4+(6*exp(x)*x-12*x^2)*exp(1/4*x^2)^2+exp(x)
^2-4*exp(x)*x+4*x^2),x, algorithm="fricas")

[Out]

1/(3*x*e^(1/2*x^2) - 2*x + e^x)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {2-e^x+e^{\frac {x^2}{2}} \left (-3-3 x^2\right )}{e^{2 x}-4 e^x x+4 x^2+9 e^{x^2} x^2+e^{\frac {x^2}{2}} \left (6 e^x x-12 x^2\right )} \, dx=\frac {1}{3 x e^{\frac {x^{2}}{2}} - 2 x + e^{x}} \]

[In]

integrate(((-3*x**2-3)*exp(1/4*x**2)**2-exp(x)+2)/(9*x**2*exp(1/4*x**2)**4+(6*exp(x)*x-12*x**2)*exp(1/4*x**2)*
*2+exp(x)**2-4*exp(x)*x+4*x**2),x)

[Out]

1/(3*x*exp(x**2/2) - 2*x + exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {2-e^x+e^{\frac {x^2}{2}} \left (-3-3 x^2\right )}{e^{2 x}-4 e^x x+4 x^2+9 e^{x^2} x^2+e^{\frac {x^2}{2}} \left (6 e^x x-12 x^2\right )} \, dx=\frac {1}{3 \, x e^{\left (\frac {1}{2} \, x^{2}\right )} - 2 \, x + e^{x}} \]

[In]

integrate(((-3*x^2-3)*exp(1/4*x^2)^2-exp(x)+2)/(9*x^2*exp(1/4*x^2)^4+(6*exp(x)*x-12*x^2)*exp(1/4*x^2)^2+exp(x)
^2-4*exp(x)*x+4*x^2),x, algorithm="maxima")

[Out]

1/(3*x*e^(1/2*x^2) - 2*x + e^x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {2-e^x+e^{\frac {x^2}{2}} \left (-3-3 x^2\right )}{e^{2 x}-4 e^x x+4 x^2+9 e^{x^2} x^2+e^{\frac {x^2}{2}} \left (6 e^x x-12 x^2\right )} \, dx=-\frac {e^{\left (\frac {1}{2} \, x^{2}\right )}}{2 \, x e^{\left (\frac {1}{2} \, x^{2}\right )} - 3 \, x e^{\left (x^{2}\right )} - e^{\left (\frac {1}{2} \, x^{2} + x\right )}} \]

[In]

integrate(((-3*x^2-3)*exp(1/4*x^2)^2-exp(x)+2)/(9*x^2*exp(1/4*x^2)^4+(6*exp(x)*x-12*x^2)*exp(1/4*x^2)^2+exp(x)
^2-4*exp(x)*x+4*x^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.42 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {2-e^x+e^{\frac {x^2}{2}} \left (-3-3 x^2\right )}{e^{2 x}-4 e^x x+4 x^2+9 e^{x^2} x^2+e^{\frac {x^2}{2}} \left (6 e^x x-12 x^2\right )} \, dx=\frac {1}{{\mathrm {e}}^x-2\,x+3\,x\,{\mathrm {e}}^{\frac {x^2}{2}}} \]

[In]

int(-(exp(x) + exp(x^2/2)*(3*x^2 + 3) - 2)/(exp(2*x) + exp(x^2/2)*(6*x*exp(x) - 12*x^2) + 9*x^2*exp(x^2) - 4*x
*exp(x) + 4*x^2),x)

[Out]

1/(exp(x) - 2*x + 3*x*exp(x^2/2))