3.3.1 \(\int (-1+e^x) \, dx\)

Optimal. Leaf size=13 \[ -3+e^x-x-20 \log (\log (5)) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 7, normalized size of antiderivative = 0.54, 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*} e^x-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*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 7, normalized size = 0.54 \begin {gather*} e^x-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + E^x,x]

[Out]

E^x - x

________________________________________________________________________________________

fricas [A]  time = 0.97, size = 6, normalized size = 0.46 \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

________________________________________________________________________________________

giac [A]  time = 0.25, size = 6, normalized size = 0.46 \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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 7, normalized size = 0.54

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x)-x

________________________________________________________________________________________

maxima [A]  time = 0.74, size = 6, normalized size = 0.46 \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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 6, normalized size = 0.46 \begin {gather*} {\mathrm {e}}^x-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x) - 1,x)

[Out]

exp(x) - x

________________________________________________________________________________________

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)

________________________________________________________________________________________