3.645 \(\int e^{2-x^2} x \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{2} e^{2-x^2} \]

[Out]

-E^(2 - x^2)/2

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Rubi [A]  time = 0.0093858, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2209} \[ -\frac{1}{2} e^{2-x^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(2 - x^2)*x,x]

[Out]

-E^(2 - x^2)/2

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int e^{2-x^2} x \, dx &=-\frac{1}{2} e^{2-x^2}\\ \end{align*}

Mathematica [A]  time = 0.0020969, size = 13, normalized size = 1. \[ -\frac{1}{2} e^{2-x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2 - x^2)*x,x]

[Out]

-E^(2 - x^2)/2

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Maple [A]  time = 0.018, size = 11, normalized size = 0.9 \begin{align*} -{\frac{{{\rm e}^{-{x}^{2}+2}}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.962511, size = 14, normalized size = 1.08 \begin{align*} -\frac{1}{2} \, e^{\left (-x^{2} + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(-x^2 + 2)

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Fricas [A]  time = 0.762508, size = 26, normalized size = 2. \begin{align*} -\frac{1}{2} \, e^{\left (-x^{2} + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(-x^2 + 2)

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Sympy [A]  time = 0.083658, size = 8, normalized size = 0.62 \begin{align*} - \frac{e^{2 - x^{2}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(2 - x**2)/2

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Giac [A]  time = 1.26808, size = 14, normalized size = 1.08 \begin{align*} -\frac{1}{2} \, e^{\left (-x^{2} + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(-x^2 + 2)