3.52.28 \(\int e^{-x} (21-9 x-4 \log (3)) \, dx\)

Optimal. Leaf size=29 \[ x \left (9 e^{-x}+\frac {\frac {9}{4}+4 e^{-x} (-3+\log (3))}{x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2176, 2194} \begin {gather*} 9 e^{-x}-e^{-x} (-9 x+21-4 \log (3)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(21 - 9*x - 4*Log[3])/E^x,x]

[Out]

9/E^x - (21 - 9*x - 4*Log[3])/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} &=-e^{-x} (21-9 x-4 \log (3))-9 \int e^{-x} \, dx\\ &=9 e^{-x}-e^{-x} (21-9 x-4 \log (3))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 13, normalized size = 0.45 \begin {gather*} e^{-x} (-12+9 x+\log (81)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(21 - 9*x - 4*Log[3])/E^x,x]

[Out]

(-12 + 9*x + Log[81])/E^x

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fricas [A]  time = 0.82, size = 14, normalized size = 0.48 \begin {gather*} {\left (9 \, x + 4 \, \log \relax (3) - 12\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(3)-9*x+21)/exp(x),x, algorithm="fricas")

[Out]

(9*x + 4*log(3) - 12)*e^(-x)

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giac [A]  time = 0.21, size = 14, normalized size = 0.48 \begin {gather*} {\left (9 \, x + 4 \, \log \relax (3) - 12\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(3)-9*x+21)/exp(x),x, algorithm="giac")

[Out]

(9*x + 4*log(3) - 12)*e^(-x)

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maple [A]  time = 0.03, size = 15, normalized size = 0.52

method	result	size
gosper	\(\left (9 x +4 \ln \relax (3)-12\right ) {\mathrm e}^{-x}\)	\(15\)
norman	\(\left (9 x +4 \ln \relax (3)-12\right ) {\mathrm e}^{-x}\)	\(15\)
risch	\(\left (9 x +4 \ln \relax (3)-12\right ) {\mathrm e}^{-x}\)	\(15\)
default	\(-12 \,{\mathrm e}^{-x}+9 x \,{\mathrm e}^{-x}+4 \,{\mathrm e}^{-x} \ln \relax (3)\)	\(23\)
meijerg	\(-4 \ln \relax (3) \left (1-{\mathrm e}^{-x}\right )+12+\frac {9 \left (2 x +2\right ) {\mathrm e}^{-x}}{2}-21 \,{\mathrm e}^{-x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*ln(3)-9*x+21)/exp(x),x,method=_RETURNVERBOSE)

[Out]

(9*x+4*ln(3)-12)/exp(x)

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maxima [A]  time = 0.36, size = 24, normalized size = 0.83 \begin {gather*} 9 \, {\left (x + 1\right )} e^{\left (-x\right )} + 4 \, e^{\left (-x\right )} \log \relax (3) - 21 \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*log(3)-9*x+21)/exp(x),x, algorithm="maxima")

[Out]

9*(x + 1)*e^(-x) + 4*e^(-x)*log(3) - 21*e^(-x)

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mupad [B]  time = 3.20, size = 12, normalized size = 0.41 \begin {gather*} {\mathrm {e}}^{-x}\,\left (9\,x+\ln \left (81\right )-12\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*(9*x + 4*log(3) - 21),x)

[Out]

exp(-x)*(9*x + log(81) - 12)

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sympy [A]  time = 0.09, size = 12, normalized size = 0.41 \begin {gather*} \left (9 x - 12 + 4 \log {\relax (3 )}\right ) e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*ln(3)-9*x+21)/exp(x),x)

[Out]

(9*x - 12 + 4*log(3))*exp(-x)

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