\(\int (1+e^x) \, dx\) [5053]

Optimal result

Integrand size = 5, antiderivative size = 5 \[ \int \left (1+e^x\right ) \, dx=e^x+x \]

[Out]

exp(x)+x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2225} \[ \int \left (1+e^x\right ) \, dx=x+e^x \]

[In]

Int[1 + E^x,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \left (1+e^x\right ) \, dx=e^x+x \]

[In]

Integrate[1 + E^x,x]

[Out]

E^x + x

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

exp(x)+x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.80 \[ \int \left (1+e^x\right ) \, dx=x + e^{x} \]

[In]

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

[Out]

x + e^x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.60 \[ \int \left (1+e^x\right ) \, dx=x + e^{x} \]

[In]

integrate(exp(x)+1,x)

[Out]

x + exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.80 \[ \int \left (1+e^x\right ) \, dx=x + e^{x} \]

[In]

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

[Out]

x + e^x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.80 \[ \int \left (1+e^x\right ) \, dx=x + e^{x} \]

[In]

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

[Out]

x + e^x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.80 \[ \int \left (1+e^x\right ) \, dx=x+{\mathrm {e}}^x \]

[In]

int(exp(x) + 1,x)

[Out]

x + exp(x)