3.22.8 \(\int e^{-x} (-3-e^x) \, dx\)

Optimal. Leaf size=12 \[ 4+3 e^{-x}-x \]

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2248, 43} \begin {gather*} 3 e^{-x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - E^x)/E^x,x]

[Out]

3/E^x - x

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2248

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(g*h*Log[G])/(d*e*Log[F])]}, Dist[(Denominator[m]*G^(f*h - (c*g*h)/d))/(d*e*Log[F]), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^((e*(c + d*x))/Denominator[m])], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {-3-x}{x^2} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {3}{x^2}-\frac {1}{x}\right ) \, dx,x,e^x\right )\\ &=3 e^{-x}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.92 \begin {gather*} 3 e^{-x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - E^x)/E^x,x]

[Out]

3/E^x - x

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fricas [A]  time = 0.51, size = 12, normalized size = 1.00 \begin {gather*} -{\left (x e^{x} - 3\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)-3)/exp(x),x, algorithm="fricas")

[Out]

-(x*e^x - 3)*e^(-x)

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giac [A]  time = 0.22, size = 10, normalized size = 0.83 \begin {gather*} -x + 3 \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)-3)/exp(x),x, algorithm="giac")

[Out]

-x + 3*e^(-x)

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maple [A]  time = 0.02, size = 11, normalized size = 0.92

method	result	size
risch	\(-x +3 \,{\mathrm e}^{-x}\)	\(11\)
derivativedivides	\(-\ln \left ({\mathrm e}^{x}\right )+3 \,{\mathrm e}^{-x}\)	\(13\)
default	\(-\ln \left ({\mathrm e}^{x}\right )+3 \,{\mathrm e}^{-x}\)	\(13\)
norman	\(\left (-{\mathrm e}^{x} x +3\right ) {\mathrm e}^{-x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(x)-3)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-x+3*exp(-x)

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maxima [A]  time = 0.39, size = 10, normalized size = 0.83 \begin {gather*} -x + 3 \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)-3)/exp(x),x, algorithm="maxima")

[Out]

-x + 3*e^(-x)

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mupad [B]  time = 0.04, size = 10, normalized size = 0.83 \begin {gather*} 3\,{\mathrm {e}}^{-x}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*(exp(x) + 3),x)

[Out]

3*exp(-x) - x

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sympy [A]  time = 0.07, size = 5, normalized size = 0.42 \begin {gather*} - x + 3 e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)-3)/exp(x),x)

[Out]

-x + 3*exp(-x)

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