3.84.51 \(\int -8 e^{-2 x} \, dx\)

Optimal. Leaf size=9 \[ 1+4 e^{-2 x} \]

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Rubi [A]  time = 0.00, antiderivative size = 7, normalized size of antiderivative = 0.78, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {12, 2194} \begin {gather*} 4 e^{-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-8/E^(2*x),x]

[Out]

4/E^(2*x)

Rule 12

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

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (8 \int e^{-2 x} \, dx\right )\\ &=4 e^{-2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.78 \begin {gather*} 4 e^{-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-8/E^(2*x),x]

[Out]

4/E^(2*x)

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fricas [A]  time = 0.59, size = 6, normalized size = 0.67 \begin {gather*} 4 \, e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8*exp(-x)/exp(x),x, algorithm="fricas")

[Out]

4*e^(-2*x)

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giac [A]  time = 0.20, size = 6, normalized size = 0.67 \begin {gather*} 4 \, e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8*exp(-x)/exp(x),x, algorithm="giac")

[Out]

4*e^(-2*x)

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maple [A]  time = 0.02, size = 7, normalized size = 0.78

method	result	size
default	\(4 \,{\mathrm e}^{-2 x}\)	\(7\)
norman	\(4 \,{\mathrm e}^{-2 x}\)	\(7\)
risch	\(4 \,{\mathrm e}^{-2 x}\)	\(7\)
meijerg	\(-4+4 \,{\mathrm e}^{-2 x}\)	\(9\)
gosper	\(4 \left ({\mathrm e}^{-x}\right )^{2}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-8*exp(-x)/exp(x),x,method=_RETURNVERBOSE)

[Out]

4/exp(x)^2

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maxima [A]  time = 0.36, size = 6, normalized size = 0.67 \begin {gather*} 4 \, e^{\left (-2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8*exp(-x)/exp(x),x, algorithm="maxima")

[Out]

4*e^(-2*x)

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mupad [B]  time = 0.02, size = 6, normalized size = 0.67 \begin {gather*} 4\,{\mathrm {e}}^{-2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-8*exp(-2*x),x)

[Out]

4*exp(-2*x)

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sympy [A]  time = 0.07, size = 5, normalized size = 0.56 \begin {gather*} 4 e^{- 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-8*exp(-x)/exp(x),x)

[Out]

4*exp(-2*x)

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