3.87.13 \(\int -\frac {4}{5} e^{-x^2} x \, dx\)

Optimal. Leaf size=11 \[ \frac {2 e^{-x^2}}{5} \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 2209} \begin {gather*} \frac {2 e^{-x^2}}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x)/(5*E^x^2),x]

[Out]

2/(5*E^x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 {gather*} \begin {aligned} \text {integral} &=-\left (\frac {4}{5} \int e^{-x^2} x \, dx\right )\\ &=\frac {2 e^{-x^2}}{5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {2 e^{-x^2}}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x)/(5*E^x^2),x]

[Out]

2/(5*E^x^2)

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fricas [A]  time = 0.61, size = 8, normalized size = 0.73 \begin {gather*} \frac {2}{5} \, e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4/5*x/exp(x^2),x, algorithm="fricas")

[Out]

2/5*e^(-x^2)

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giac [A]  time = 0.14, size = 8, normalized size = 0.73 \begin {gather*} \frac {2}{5} \, e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4/5*x/exp(x^2),x, algorithm="giac")

[Out]

2/5*e^(-x^2)

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maple [A]  time = 0.03, size = 9, normalized size = 0.82

method	result	size
gosper	\(\frac {2 \,{\mathrm e}^{-x^{2}}}{5}\)	\(9\)
derivativedivides	\(\frac {2 \,{\mathrm e}^{-x^{2}}}{5}\)	\(9\)
default	\(\frac {2 \,{\mathrm e}^{-x^{2}}}{5}\)	\(9\)
norman	\(\frac {2 \,{\mathrm e}^{-x^{2}}}{5}\)	\(9\)
risch	\(\frac {2 \,{\mathrm e}^{-x^{2}}}{5}\)	\(9\)
meijerg	\(-\frac {2}{5}+\frac {2 \,{\mathrm e}^{-x^{2}}}{5}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-4/5*x/exp(x^2),x,method=_RETURNVERBOSE)

[Out]

2/5/exp(x^2)

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maxima [A]  time = 0.36, size = 8, normalized size = 0.73 \begin {gather*} \frac {2}{5} \, e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4/5*x/exp(x^2),x, algorithm="maxima")

[Out]

2/5*e^(-x^2)

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mupad [B]  time = 0.02, size = 8, normalized size = 0.73 \begin {gather*} \frac {2\,{\mathrm {e}}^{-x^2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x*exp(-x^2))/5,x)

[Out]

(2*exp(-x^2))/5

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sympy [A]  time = 0.08, size = 7, normalized size = 0.64 \begin {gather*} \frac {2 e^{- x^{2}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4/5*x/exp(x**2),x)

[Out]

2*exp(-x**2)/5

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