3.15.99 \(\int \frac {e^{-\frac {3}{27+4 \log (x)}} (741+216 \log (x)+16 \log ^2(x))}{729+216 \log (x)+16 \log ^2(x)} \, dx\)

Optimal. Leaf size=16 \[ e^{-\frac {1}{9+\frac {4 \log (x)}{3}}} x \]

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Rubi [F]  time = 0.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(741 + 216*Log[x] + 16*Log[x]^2)/(E^(3/(27 + 4*Log[x]))*(729 + 216*Log[x] + 16*Log[x]^2)),x]

[Out]

Defer[Int][E^(-3/(27 + 4*Log[x])), x] + 12*Defer[Int][1/(E^(3/(27 + 4*Log[x]))*(27 + 4*Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{(27+4 \log (x))^2} \, dx\\ &=\int \left (e^{-\frac {3}{27+4 \log (x)}}+\frac {12 e^{-\frac {3}{27+4 \log (x)}}}{(27+4 \log (x))^2}\right ) \, dx\\ &=12 \int \frac {e^{-\frac {3}{27+4 \log (x)}}}{(27+4 \log (x))^2} \, dx+\int e^{-\frac {3}{27+4 \log (x)}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 14, normalized size = 0.88 \begin {gather*} e^{-\frac {3}{27+4 \log (x)}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(741 + 216*Log[x] + 16*Log[x]^2)/(E^(3/(27 + 4*Log[x]))*(729 + 216*Log[x] + 16*Log[x]^2)),x]

[Out]

x/E^(3/(27 + 4*Log[x]))

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fricas [A]  time = 0.68, size = 13, normalized size = 0.81 \begin {gather*} x e^{\left (-\frac {3}{4 \, \log \relax (x) + 27}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*log(x)^2+216*log(x)+741)/(16*log(x)^2+216*log(x)+729)/exp(3/(4*log(x)+27)),x, algorithm="fricas"
)

[Out]

x*e^(-3/(4*log(x) + 27))

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giac [B]  time = 0.29, size = 30, normalized size = 1.88 \begin {gather*} x^{\frac {247}{9 \, {\left (4 \, \log \relax (x) + 27\right )}}} e^{\left (\frac {4 \, \log \relax (x)^{2}}{4 \, \log \relax (x) + 27} - \frac {1}{9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*log(x)^2+216*log(x)+741)/(16*log(x)^2+216*log(x)+729)/exp(3/(4*log(x)+27)),x, algorithm="giac")

[Out]

x^(247/9/(4*log(x) + 27))*e^(4*log(x)^2/(4*log(x) + 27) - 1/9)

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maple [A]  time = 0.03, size = 14, normalized size = 0.88

method	result	size
risch	\(x \,{\mathrm e}^{-\frac {3}{4 \ln \relax (x )+27}}\)	\(14\)
norman	\(\frac {\left (27 x +4 x \ln \relax (x )\right ) {\mathrm e}^{-\frac {3}{4 \ln \relax (x )+27}}}{4 \ln \relax (x )+27}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*ln(x)^2+216*ln(x)+741)/(16*ln(x)^2+216*ln(x)+729)/exp(3/(4*ln(x)+27)),x,method=_RETURNVERBOSE)

[Out]

x*exp(-3/(4*ln(x)+27))

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maxima [A]  time = 0.63, size = 13, normalized size = 0.81 \begin {gather*} x e^{\left (-\frac {3}{4 \, \log \relax (x) + 27}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*log(x)^2+216*log(x)+741)/(16*log(x)^2+216*log(x)+729)/exp(3/(4*log(x)+27)),x, algorithm="maxima"
)

[Out]

x*e^(-3/(4*log(x) + 27))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {3}{4\,\ln \relax (x)+27}}\,\left (16\,{\ln \relax (x)}^2+216\,\ln \relax (x)+741\right )}{16\,{\ln \relax (x)}^2+216\,\ln \relax (x)+729} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-3/(4*log(x) + 27))*(216*log(x) + 16*log(x)^2 + 741))/(216*log(x) + 16*log(x)^2 + 729),x)

[Out]

int((exp(-3/(4*log(x) + 27))*(216*log(x) + 16*log(x)^2 + 741))/(216*log(x) + 16*log(x)^2 + 729), x)

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sympy [A]  time = 2.13, size = 10, normalized size = 0.62 \begin {gather*} x e^{- \frac {3}{4 \log {\relax (x )} + 27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*ln(x)**2+216*ln(x)+741)/(16*ln(x)**2+216*ln(x)+729)/exp(3/(4*ln(x)+27)),x)

[Out]

x*exp(-3/(4*log(x) + 27))

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