3.30.78 \(\int \frac {1}{2} (-1+2 e^x) \, dx\)

Optimal. Leaf size=25 \[ -1+e^x+\frac {1}{2} (-8+x)-x-\log (3)+5 \log ^2(3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.36, number of steps used = 3, 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^x-\frac {x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*E^x)/2,x]

[Out]

E^x - x/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 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} &=\frac {1}{2} \int \left (-1+2 e^x\right ) \, dx\\ &=-\frac {x}{2}+\int e^x \, dx\\ &=e^x-\frac {x}{2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 9, normalized size = 0.36 \begin {gather*} e^x-\frac {x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*E^x)/2,x]

[Out]

E^x - x/2

________________________________________________________________________________________

fricas [A]  time = 0.73, size = 6, normalized size = 0.24 \begin {gather*} -\frac {1}{2} \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x + e^x

________________________________________________________________________________________

giac [A]  time = 0.23, size = 6, normalized size = 0.24 \begin {gather*} -\frac {1}{2} \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x + e^x

________________________________________________________________________________________

maple [A]  time = 0.03, size = 7, normalized size = 0.28

method	result	size
default	\(-\frac {x}{2}+{\mathrm e}^{x}\)	\(7\)
norman	\(-\frac {x}{2}+{\mathrm e}^{x}\)	\(7\)
risch	\(-\frac {x}{2}+{\mathrm e}^{x}\)	\(7\)
derivativedivides	\({\mathrm e}^{x}-\frac {\ln \left ({\mathrm e}^{x}\right )}{2}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x+exp(x)

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 6, normalized size = 0.24 \begin {gather*} -\frac {1}{2} \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x + e^x

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 6, normalized size = 0.24 \begin {gather*} {\mathrm {e}}^x-\frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x) - 1/2,x)

[Out]

exp(x) - x/2

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 5, normalized size = 0.20 \begin {gather*} - \frac {x}{2} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/2 + exp(x)

________________________________________________________________________________________