3.51.65 \(\int \frac {2 x}{e^2} \, dx\)

Optimal. Leaf size=11 \[ 5+\frac {x^2}{e^2}+\log (3) \]

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Rubi [A]  time = 0.00, antiderivative size = 7, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {12, 30} \begin {gather*} \frac {x^2}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/E^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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {2 \int x \, dx}{e^2}\\ &=\frac {x^2}{e^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.64 \begin {gather*} \frac {x^2}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/E^2

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fricas [A]  time = 0.68, size = 6, normalized size = 0.55 \begin {gather*} x^{2} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x/exp(1)^2,x, algorithm="fricas")

[Out]

x^2*e^(-2)

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giac [A]  time = 0.14, size = 6, normalized size = 0.55 \begin {gather*} x^{2} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x/exp(1)^2,x, algorithm="giac")

[Out]

x^2*e^(-2)

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maple [A]  time = 0.03, size = 7, normalized size = 0.64

method	result	size
risch	\(x^{2} {\mathrm e}^{-2}\)	\(7\)
gosper	\(x^{2} {\mathrm e}^{-2}\)	\(9\)
default	\(x^{2} {\mathrm e}^{-2}\)	\(9\)
norman	\(x^{2} {\mathrm e}^{-2}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x/exp(1)^2,x,method=_RETURNVERBOSE)

[Out]

x^2*exp(-2)

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maxima [A]  time = 0.34, size = 6, normalized size = 0.55 \begin {gather*} x^{2} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x/exp(1)^2,x, algorithm="maxima")

[Out]

x^2*e^(-2)

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mupad [B]  time = 0.03, size = 6, normalized size = 0.55 \begin {gather*} x^2\,{\mathrm {e}}^{-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*exp(-2)

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sympy [A]  time = 0.02, size = 5, normalized size = 0.45 \begin {gather*} \frac {x^{2}}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x/exp(1)**2,x)

[Out]

x**2*exp(-2)

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