3.26.18 \(\int e^{-x} (26+2 \log (\frac {9}{\log ^2(3)})) \, dx\)

Optimal. Leaf size=23 \[ \frac {\left (x-2 e^{-x} x\right ) \left (13+\log \left (\frac {9}{\log ^2(3)}\right )\right )}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 2194} \begin {gather*} -2 e^{-x} \left (13+\log \left (\frac {9}{\log ^2(3)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(26 + 2*Log[9/Log[3]^2])/E^x,x]

[Out]

(-2*(13 + Log[9/Log[3]^2]))/E^x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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} &=\left (26+2 \log \left (\frac {9}{\log ^2(3)}\right )\right ) \int e^{-x} \, dx\\ &=-2 e^{-x} \left (13+\log \left (\frac {9}{\log ^2(3)}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.70 \begin {gather*} -2 e^{-x} \left (13+\log \left (\frac {9}{\log ^2(3)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(26 + 2*Log[9/Log[3]^2])/E^x,x]

[Out]

(-2*(13 + Log[9/Log[3]^2]))/E^x

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fricas [A]  time = 0.53, size = 20, normalized size = 0.87 \begin {gather*} -2 \, e^{\left (-x\right )} \log \left (\frac {9}{\log \relax (3)^{2}}\right ) - 26 \, e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(9/log(3)^2)+26)/exp(x),x, algorithm="fricas")

[Out]

-2*e^(-x)*log(9/log(3)^2) - 26*e^(-x)

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giac [A]  time = 0.23, size = 15, normalized size = 0.65 \begin {gather*} -2 \, {\left (\log \left (\frac {9}{\log \relax (3)^{2}}\right ) + 13\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(9/log(3)^2)+26)/exp(x),x, algorithm="giac")

[Out]

-2*(log(9/log(3)^2) + 13)*e^(-x)

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maple [A]  time = 0.02, size = 16, normalized size = 0.70

method	result	size
gosper	\(-2 \left (13+\ln \left (\frac {9}{\ln \relax (3)^{2}}\right )\right ) {\mathrm e}^{-x}\)	\(16\)
norman	\(\left (-4 \ln \relax (3)+4 \ln \left (\ln \relax (3)\right )-26\right ) {\mathrm e}^{-x}\)	\(17\)
derivativedivides	\(-\left (2 \ln \left (\frac {9}{\ln \relax (3)^{2}}\right )+26\right ) {\mathrm e}^{-x}\)	\(18\)
default	\(-\left (2 \ln \left (\frac {9}{\ln \relax (3)^{2}}\right )+26\right ) {\mathrm e}^{-x}\)	\(18\)
risch	\(4 \,{\mathrm e}^{-x} \ln \left (\ln \relax (3)\right )-4 \,{\mathrm e}^{-x} \ln \relax (3)-26 \,{\mathrm e}^{-x}\)	\(25\)
meijerg	\(2 \ln \left (\frac {9}{\ln \relax (3)^{2}}\right ) \left (1-{\mathrm e}^{-x}\right )+26-26 \,{\mathrm e}^{-x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(9/ln(3)^2)+26)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-2*(13+ln(9/ln(3)^2))/exp(x)

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maxima [A]  time = 0.37, size = 15, normalized size = 0.65 \begin {gather*} -2 \, {\left (\log \left (\frac {9}{\log \relax (3)^{2}}\right ) + 13\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(9/log(3)^2)+26)/exp(x),x, algorithm="maxima")

[Out]

-2*(log(9/log(3)^2) + 13)*e^(-x)

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mupad [B]  time = 0.04, size = 15, normalized size = 0.65 \begin {gather*} -{\mathrm {e}}^{-x}\,\left (\ln \left (\frac {81}{{\ln \relax (3)}^4}\right )+26\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*(2*log(9/log(3)^2) + 26),x)

[Out]

-exp(-x)*(log(81/log(3)^4) + 26)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.65 \begin {gather*} \left (-26 - 4 \log {\relax (3 )} + 4 \log {\left (\log {\relax (3 )} \right )}\right ) e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(9/ln(3)**2)+26)/exp(x),x)

[Out]

(-26 - 4*log(3) + 4*log(log(3)))*exp(-x)

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