3.45.17 \(\int \frac {1}{32} e^{\frac {1}{32} (-16+x+16 \log (3))} \, dx\)

Optimal. Leaf size=26 \[ 3+e^{\frac {-x+\frac {x^2}{16}+x \log (3)}{2 x}} \]

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 0.58, 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*} \sqrt {3} e^{\frac {x-16}{32}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((-16 + x + 16*Log[3])/32)/32,x]

[Out]

Sqrt[3]*E^((-16 + x)/32)

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} &=\frac {1}{32} \int e^{\frac {1}{32} (-16+x+16 \log (3))} \, dx\\ &=\sqrt {3} e^{\frac {1}{32} (-16+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.58 \begin {gather*} \sqrt {3} e^{\frac {1}{32} (-16+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((-16 + x + 16*Log[3])/32)/32,x]

[Out]

Sqrt[3]*E^((-16 + x)/32)

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fricas [A]  time = 0.54, size = 10, normalized size = 0.38 \begin {gather*} e^{\left (\frac {1}{32} \, x + \frac {1}{2} \, \log \relax (3) - \frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*exp(1/2*log(3)+1/32*x-1/2),x, algorithm="fricas")

[Out]

e^(1/32*x + 1/2*log(3) - 1/2)

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giac [A]  time = 0.15, size = 10, normalized size = 0.38 \begin {gather*} e^{\left (\frac {1}{32} \, x + \frac {1}{2} \, \log \relax (3) - \frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*exp(1/2*log(3)+1/32*x-1/2),x, algorithm="giac")

[Out]

e^(1/32*x + 1/2*log(3) - 1/2)

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maple [A]  time = 0.02, size = 11, normalized size = 0.42

method	result	size
gosper	\({\mathrm e}^{\frac {\ln \relax (3)}{2}+\frac {x}{32}-\frac {1}{2}}\)	\(11\)
derivativedivides	\({\mathrm e}^{\frac {\ln \relax (3)}{2}+\frac {x}{32}-\frac {1}{2}}\)	\(11\)
default	\({\mathrm e}^{\frac {\ln \relax (3)}{2}+\frac {x}{32}-\frac {1}{2}}\)	\(11\)
norman	\({\mathrm e}^{\frac {\ln \relax (3)}{2}+\frac {x}{32}-\frac {1}{2}}\)	\(11\)
risch	\(\sqrt {3}\, {\mathrm e}^{-\frac {1}{2}+\frac {x}{32}}\)	\(11\)
meijerg	\(-{\mathrm e}^{\frac {\ln \relax (3)}{2}-\frac {1}{2}} \left (1-{\mathrm e}^{\frac {x}{32}}\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/32*exp(1/2*ln(3)+1/32*x-1/2),x,method=_RETURNVERBOSE)

[Out]

exp(1/2*ln(3)+1/32*x-1/2)

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maxima [A]  time = 0.37, size = 10, normalized size = 0.38 \begin {gather*} e^{\left (\frac {1}{32} \, x + \frac {1}{2} \, \log \relax (3) - \frac {1}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*exp(1/2*log(3)+1/32*x-1/2),x, algorithm="maxima")

[Out]

e^(1/32*x + 1/2*log(3) - 1/2)

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mupad [B]  time = 0.04, size = 10, normalized size = 0.38 \begin {gather*} \sqrt {3}\,{\mathrm {e}}^{x/32}\,{\mathrm {e}}^{-\frac {1}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x/32 + log(3)/2 - 1/2)/32,x)

[Out]

3^(1/2)*exp(x/32)*exp(-1/2)

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sympy [A]  time = 0.09, size = 12, normalized size = 0.46 \begin {gather*} \sqrt {3} e^{\frac {x}{32} - \frac {1}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*exp(1/2*ln(3)+1/32*x-1/2),x)

[Out]

sqrt(3)*exp(x/32 - 1/2)

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