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

Optimal. Leaf size=12 \[ e^4+\frac {e^x}{2}+x \]

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 2194} \begin {gather*} x+\frac {e^x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + E^x)/2,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.92 \begin {gather*} \frac {1}{2} \left (e^x+2 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + E^x)/2,x]

[Out]

(E^x + 2*x)/2

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fricas [A]  time = 0.53, size = 6, normalized size = 0.50 \begin {gather*} x + \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 1/2*e^x

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giac [A]  time = 0.12, size = 6, normalized size = 0.50 \begin {gather*} x + \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 1/2*e^x

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maple [A]  time = 0.02, size = 7, normalized size = 0.58

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+1/2*exp(x)

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maxima [A]  time = 0.34, size = 6, normalized size = 0.50 \begin {gather*} x + \frac {1}{2} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 1/2*e^x

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mupad [B]  time = 0.04, size = 6, normalized size = 0.50 \begin {gather*} x+\frac {{\mathrm {e}}^x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/2 + 1,x)

[Out]

x + exp(x)/2

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sympy [A]  time = 0.07, size = 5, normalized size = 0.42 \begin {gather*} x + \frac {e^{x}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(x)/2

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