\(\int (4-2 e^x) \, dx\) [1799]

Optimal result

Integrand size = 7, antiderivative size = 11 \[ \int \left (4-2 e^x\right ) \, dx=1-2 \left (e^x-2 x\right ) \]

[Out]

4*x-2*exp(x)+1

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2225} \[ \int \left (4-2 e^x\right ) \, dx=4 x-2 e^x \]

[In]

Int[4 - 2*E^x,x]

[Out]

-2*E^x + 4*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}& = 4 x-2 \int e^x \, dx \\ & = -2 e^x+4 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \left (4-2 e^x\right ) \, dx=-2 e^x+4 x \]

[In]

Integrate[4 - 2*E^x,x]

[Out]

-2*E^x + 4*x

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82

method	result	size
default	\(4 x -2 \,{\mathrm e}^{x}\)	\(9\)
norman	\(4 x -2 \,{\mathrm e}^{x}\)	\(9\)
risch	\(4 x -2 \,{\mathrm e}^{x}\)	\(9\)
parallelrisch	\(4 x -2 \,{\mathrm e}^{x}\)	\(9\)
parts	\(4 x -2 \,{\mathrm e}^{x}\)	\(9\)
derivativedivides	\(-2 \,{\mathrm e}^{x}+4 \ln \left ({\mathrm e}^{x}\right )\)	\(11\)

[In]

int(-2*exp(x)+4,x,method=_RETURNVERBOSE)

[Out]

4*x-2*exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \left (4-2 e^x\right ) \, dx=4 \, x - 2 \, e^{x} \]

[In]

integrate(-2*exp(x)+4,x, algorithm="fricas")

[Out]

4*x - 2*e^x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int \left (4-2 e^x\right ) \, dx=4 x - 2 e^{x} \]

[In]

integrate(-2*exp(x)+4,x)

[Out]

4*x - 2*exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \left (4-2 e^x\right ) \, dx=4 \, x - 2 \, e^{x} \]

[In]

integrate(-2*exp(x)+4,x, algorithm="maxima")

[Out]

4*x - 2*e^x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \left (4-2 e^x\right ) \, dx=4 \, x - 2 \, e^{x} \]

[In]

integrate(-2*exp(x)+4,x, algorithm="giac")

[Out]

4*x - 2*e^x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \left (4-2 e^x\right ) \, dx=4\,x-2\,{\mathrm {e}}^x \]

[In]

int(4 - 2*exp(x),x)

[Out]

4*x - 2*exp(x)