3.43.25 \(\int e^{-x} (-2+2 x) \, dx\)

Optimal. Leaf size=15 \[ -2 e^{-x} x-\log ^2(2) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.33, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2176, 2194} \begin {gather*} 2 e^{-x} (1-x)-2 e^{-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + 2*x)/E^x,x]

[Out]

-2/E^x + (2*(1 - x))/E^x

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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} &=2 e^{-x} (1-x)+2 \int e^{-x} \, dx\\ &=-2 e^{-x}+2 e^{-x} (1-x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 8, normalized size = 0.53 \begin {gather*} -2 e^{-x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 2*x)/E^x,x]

[Out]

(-2*x)/E^x

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 7, normalized size = 0.47 \begin {gather*} -2 \, x e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*e^(-x)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 7, normalized size = 0.47 \begin {gather*} -2 \, x e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*e^(-x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 8, normalized size = 0.53

method	result	size
gosper	\(-2 x \,{\mathrm e}^{-x}\)	\(8\)
default	\(-2 x \,{\mathrm e}^{-x}\)	\(8\)
norman	\(-2 x \,{\mathrm e}^{-x}\)	\(8\)
risch	\(-2 x \,{\mathrm e}^{-x}\)	\(8\)
meijerg	\(-\left (2 x +2\right ) {\mathrm e}^{-x}+2 \,{\mathrm e}^{-x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x/exp(x)

________________________________________________________________________________________

maxima [A]  time = 0.34, size = 16, normalized size = 1.07 \begin {gather*} -2 \, {\left (x + 1\right )} e^{\left (-x\right )} + 2 \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(x + 1)*e^(-x) + 2*e^(-x)

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 7, normalized size = 0.47 \begin {gather*} -2\,x\,{\mathrm {e}}^{-x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*(2*x - 2),x)

[Out]

-2*x*exp(-x)

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 7, normalized size = 0.47 \begin {gather*} - 2 x e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x-2)/exp(x),x)

[Out]

-2*x*exp(-x)

________________________________________________________________________________________