\(\int (-1+e^x) \, dx\) [201]

Optimal result

Integrand size = 5, antiderivative size = 13 \[ \int \left (-1+e^x\right ) \, dx=-3+e^x-x-20 \log (\log (5)) \]

[Out]

exp(x)-3-20*ln(ln(5))-x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2225} \[ \int \left (-1+e^x\right ) \, dx=e^x-x \]

[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 = 7, normalized size of antiderivative = 0.54 \[ \int \left (-1+e^x\right ) \, dx=e^x-x \]

[In]

Integrate[-1 + E^x,x]

[Out]

E^x - x

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.54

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

[In]

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

[Out]

exp(x)-x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int \left (-1+e^x\right ) \, dx=-x + e^{x} \]

[In]

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

[Out]

-x + e^x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.23 \[ \int \left (-1+e^x\right ) \, dx=- x + e^{x} \]

[In]

integrate(exp(x)-1,x)

[Out]

-x + exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int \left (-1+e^x\right ) \, dx=-x + e^{x} \]

[In]

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

[Out]

-x + e^x

Giac [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int \left (-1+e^x\right ) \, dx=-x + e^{x} \]

[In]

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

[Out]

-x + e^x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int \left (-1+e^x\right ) \, dx={\mathrm {e}}^x-x \]

[In]

int(exp(x) - 1,x)

[Out]

exp(x) - x