3.75.8 \(\int e^x (-2-x) \, dx\)

Optimal. Leaf size=20 \[ -10-\frac {e^2}{5}+x-\left (1+e^x\right ) (1+x) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.60, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2176, 2194} \begin {gather*} e^x-e^x (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x*(-2 - x),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 8, normalized size = 0.40 \begin {gather*} -e^x (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x*(-2 - x),x]

[Out]

-(E^x*(1 + x))

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fricas [A]  time = 0.48, size = 7, normalized size = 0.35 \begin {gather*} -{\left (x + 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*e^x

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giac [A]  time = 2.18, size = 7, normalized size = 0.35 \begin {gather*} -{\left (x + 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*e^x

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maple [A]  time = 0.02, size = 8, normalized size = 0.40

method	result	size
gosper	\(-\left (x +1\right ) {\mathrm e}^{x}\)	\(8\)
risch	\(\left (-x -1\right ) {\mathrm e}^{x}\)	\(9\)
default	\(-{\mathrm e}^{x} x -{\mathrm e}^{x}\)	\(11\)
norman	\(-{\mathrm e}^{x} x -{\mathrm e}^{x}\)	\(11\)
meijerg	\(1+\frac {\left (-2 x +2\right ) {\mathrm e}^{x}}{2}-2 \,{\mathrm e}^{x}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x+1)*exp(x)

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maxima [A]  time = 0.37, size = 12, normalized size = 0.60 \begin {gather*} -{\left (x - 1\right )} e^{x} - 2 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x - 1)*e^x - 2*e^x

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mupad [B]  time = 6.34, size = 7, normalized size = 0.35 \begin {gather*} -{\mathrm {e}}^x\,\left (x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x)*(x + 2),x)

[Out]

-exp(x)*(x + 1)

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sympy [A]  time = 0.09, size = 7, normalized size = 0.35 \begin {gather*} \left (- x - 1\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-x - 1)*exp(x)

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