\(\int e^{2-x^2} x \, dx\) [645]

Optimal result

Integrand size = 11, antiderivative size = 13 \[ \int e^{2-x^2} x \, dx=-\frac {1}{2} e^{2-x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2240} \[ \int e^{2-x^2} x \, dx=-\frac {1}{2} e^{2-x^2} \]

[In]

Int[E^(2 - x^2)*x,x]

[Out]

-1/2*E^(2 - x^2)

Rule 2240

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*} \text {integral}& = -\frac {1}{2} e^{2-x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int e^{2-x^2} x \, dx=-\frac {1}{2} e^{2-x^2} \]

[In]

Integrate[E^(2 - x^2)*x,x]

[Out]

-1/2*E^(2 - x^2)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85

method	result	size
gosper	\(-\frac {{\mathrm e}^{-x^{2}+2}}{2}\)	\(11\)
derivativedivides	\(-\frac {{\mathrm e}^{-x^{2}+2}}{2}\)	\(11\)
default	\(-\frac {{\mathrm e}^{-x^{2}+2}}{2}\)	\(11\)
norman	\(-\frac {{\mathrm e}^{-x^{2}+2}}{2}\)	\(11\)
risch	\(-\frac {{\mathrm e}^{-x^{2}+2}}{2}\)	\(11\)
parallelrisch	\(-\frac {{\mathrm e}^{-x^{2}+2}}{2}\)	\(11\)
meijerg	\(\frac {{\mathrm e}^{2} \left (1-{\mathrm e}^{-x^{2}}\right )}{2}\)	\(15\)
parts	\(\frac {{\mathrm e}^{2} \operatorname {erf}\left (x \right ) \sqrt {\pi }\, x}{2}-\frac {{\mathrm e}^{2} \left (x \,\operatorname {erf}\left (x \right ) \sqrt {\pi }+{\mathrm e}^{-x^{2}}\right )}{2}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int e^{2-x^2} x \, dx=-\frac {1}{2} \, e^{\left (-x^{2} + 2\right )} \]

[In]

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

[Out]

-1/2*e^(-x^2 + 2)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int e^{2-x^2} x \, dx=- \frac {e^{2 - x^{2}}}{2} \]

[In]

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

[Out]

-exp(2 - x**2)/2

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int e^{2-x^2} x \, dx=-\frac {1}{2} \, e^{\left (-x^{2} + 2\right )} \]

[In]

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

[Out]

-1/2*e^(-x^2 + 2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int e^{2-x^2} x \, dx=-\frac {1}{2} \, e^{\left (-x^{2} + 2\right )} \]

[In]

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

[Out]

-1/2*e^(-x^2 + 2)

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int e^{2-x^2} x \, dx=-\frac {{\mathrm {e}}^2\,{\mathrm {e}}^{-x^2}}{2} \]

[In]

int(x*exp(2 - x^2),x)

[Out]

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