\(\int -\frac {2}{3} e^{2 x/3} \, dx\) [666]

Optimal result

Integrand size = 11, antiderivative size = 11 \[ \int -\frac {2}{3} e^{2 x/3} \, dx=5-e^{2 x/3} \]

[Out]

5-exp(2/3*x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2225} \[ \int -\frac {2}{3} e^{2 x/3} \, dx=-e^{2 x/3} \]

[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 2225

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{align*} \text {integral}& = -\left (\frac {2}{3} \int e^{2 x/3} \, dx\right ) \\ & = -e^{2 x/3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int -\frac {2}{3} e^{2 x/3} \, dx=-e^{2 x/3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 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	\(-{\mathrm e}^{\frac {2 x}{3}}\)	\(7\)
parallelrisch	\(-{\mathrm e}^{\frac {2 x}{3}}\)	\(7\)
parts	\(-{\mathrm e}^{\frac {2 x}{3}}\)	\(7\)
meijerg	\(1-{\mathrm e}^{\frac {2 x}{3}}\)	\(9\)

[In]

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

[Out]

-exp(2/3*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int -\frac {2}{3} e^{2 x/3} \, dx=-e^{\left (\frac {2}{3} \, x\right )} \]

[In]

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

[Out]

-e^(2/3*x)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int -\frac {2}{3} e^{2 x/3} \, dx=- e^{\frac {2 x}{3}} \]

[In]

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

[Out]

-exp(2*x/3)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int -\frac {2}{3} e^{2 x/3} \, dx=-e^{\left (\frac {2}{3} \, x\right )} \]

[In]

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

[Out]

-e^(2/3*x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int -\frac {2}{3} e^{2 x/3} \, dx=-e^{\left (\frac {2}{3} \, x\right )} \]

[In]

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

[Out]

-e^(2/3*x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int -\frac {2}{3} e^{2 x/3} \, dx=-{\mathrm {e}}^{\frac {2\,x}{3}} \]

[In]

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

[Out]

-exp((2*x)/3)