3.58.26 \(\int (1+e^x) \, dx\)

Optimal. Leaf size=13 \[ e^x+x+\log \left (\frac {2}{x}\right )+\log (x) \]

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Rubi [A]  time = 0.00, antiderivative size = 5, normalized size of antiderivative = 0.38, number of steps used = 2, number of rules used = 1, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2194} \begin {gather*} x+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^x,x]

[Out]

E^x + 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} &=x+\int e^x \, dx\\ &=e^x+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 5, normalized size = 0.38 \begin {gather*} e^x+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^x,x]

[Out]

E^x + x

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fricas [A]  time = 0.73, size = 4, normalized size = 0.31 \begin {gather*} x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + e^x

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giac [A]  time = 0.16, size = 4, normalized size = 0.31 \begin {gather*} x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + e^x

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maple [A]  time = 0.01, size = 5, normalized size = 0.38

method	result	size
default	\({\mathrm e}^{x}+x\)	\(5\)
norman	\({\mathrm e}^{x}+x\)	\(5\)
risch	\({\mathrm e}^{x}+x\)	\(5\)
derivativedivides	\({\mathrm e}^{x}+\ln \left ({\mathrm e}^{x}\right )\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)+1,x,method=_RETURNVERBOSE)

[Out]

exp(x)+x

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maxima [A]  time = 0.36, size = 4, normalized size = 0.31 \begin {gather*} x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + e^x

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mupad [B]  time = 0.03, size = 4, normalized size = 0.31 \begin {gather*} x+{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x) + 1,x)

[Out]

x + exp(x)

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sympy [A]  time = 0.06, size = 3, normalized size = 0.23 \begin {gather*} x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)+1,x)

[Out]

x + exp(x)

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