3.7.66 \(\int -\frac {2}{3} e^{2 x/3} \, dx\)

Optimal. Leaf size=11 \[ 5-e^{2 x/3} \]

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2194} \begin {gather*} -e^{2 x/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*E^((2*x)/3))/3,x]

[Out]

-E^((2*x)/3)

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 (\frac {2}{3} \int e^{2 x/3} \, dx\right )\\ &=-e^{2 x/3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.82 \begin {gather*} -e^{2 x/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^((2*x)/3))/3,x]

[Out]

-E^((2*x)/3)

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fricas [A]  time = 0.82, size = 6, normalized size = 0.55 \begin {gather*} -e^{\left (\frac {2}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*exp(2/3*x),x, algorithm="fricas")

[Out]

-e^(2/3*x)

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giac [A]  time = 0.28, size = 6, normalized size = 0.55 \begin {gather*} -e^{\left (\frac {2}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*exp(2/3*x),x, algorithm="giac")

[Out]

-e^(2/3*x)

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

method	result	size
gosper	\(-{\mathrm e}^{\frac {2 x}{3}}\)	\(7\)
derivativedivides	\(-{\mathrm e}^{\frac {2 x}{3}}\)	\(7\)
default	\(-{\mathrm e}^{\frac {2 x}{3}}\)	\(7\)
norman	\(-{\mathrm e}^{\frac {2 x}{3}}\)	\(7\)
risch	\(-{\mathrm e}^{\frac {2 x}{3}}\)	\(7\)
meijerg	\(1-{\mathrm e}^{\frac {2 x}{3}}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/3*exp(2/3*x),x,method=_RETURNVERBOSE)

[Out]

-exp(2/3*x)

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maxima [A]  time = 0.46, size = 6, normalized size = 0.55 \begin {gather*} -e^{\left (\frac {2}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*exp(2/3*x),x, algorithm="maxima")

[Out]

-e^(2/3*x)

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mupad [B]  time = 0.02, size = 6, normalized size = 0.55 \begin {gather*} -{\mathrm {e}}^{\frac {2\,x}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*exp((2*x)/3))/3,x)

[Out]

-exp((2*x)/3)

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sympy [A]  time = 0.06, size = 7, normalized size = 0.64 \begin {gather*} - e^{\frac {2 x}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*exp(2/3*x),x)

[Out]

-exp(2*x/3)

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