\(\int \frac {2 x}{e^2} \, dx\) [5065]

Optimal result

Integrand size = 6, antiderivative size = 11 \[ \int \frac {2 x}{e^2} \, dx=5+\frac {x^2}{e^2}+\log (3) \]

[Out]

5+ln(3)+x^2/exp(1)^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {12, 30} \[ \int \frac {2 x}{e^2} \, dx=\frac {x^2}{e^2} \]

[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{align*} \text {integral}& = \frac {2 \int x \, dx}{e^2} \\ & = \frac {x^2}{e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int \frac {2 x}{e^2} \, dx=\frac {x^2}{e^2} \]

[In]

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

[Out]

x^2/E^2

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 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	\(x^{2} {\mathrm e}^{-2}\)	\(9\)
parallelrisch	\(x^{2} {\mathrm e}^{-2}\)	\(9\)

[In]

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

[Out]

x^2*exp(-2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {2 x}{e^2} \, dx=x^{2} e^{\left (-2\right )} \]

[In]

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

[Out]

x^2*e^(-2)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.45 \[ \int \frac {2 x}{e^2} \, dx=\frac {x^{2}}{e^{2}} \]

[In]

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

[Out]

x**2*exp(-2)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {2 x}{e^2} \, dx=x^{2} e^{\left (-2\right )} \]

[In]

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

[Out]

x^2*e^(-2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {2 x}{e^2} \, dx=x^{2} e^{\left (-2\right )} \]

[In]

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

[Out]

x^2*e^(-2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {2 x}{e^2} \, dx=x^2\,{\mathrm {e}}^{-2} \]

[In]

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

[Out]

x^2*exp(-2)