\(\int \frac {e^{-\frac {3}{27+4 \log (x)}} (741+216 \log (x)+16 \log ^2(x))}{729+216 \log (x)+16 \log ^2(x)} \, dx\) [1499]

Optimal result

Integrand size = 39, antiderivative size = 16 \[ \int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx=e^{-\frac {1}{9+\frac {4 \log (x)}{3}}} x \]

[Out]

x/exp(1/(4/3*ln(x)+9))

Rubi [F]

\[ \int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx=\int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx=e^{-\frac {3}{27+4 \log (x)}} x \]

[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]))

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

method	result	size
risch	\(x \,{\mathrm e}^{-\frac {3}{4 \ln \left (x \right )+27}}\)	\(14\)
norman	\(\frac {\left (27 x +4 x \ln \left (x \right )\right ) {\mathrm e}^{-\frac {3}{4 \ln \left (x \right )+27}}}{4 \ln \left (x \right )+27}\)	\(32\)
parallelrisch	\(\frac {\left (27648 x \ln \left (x \right )+186624 x \right ) {\mathrm e}^{-\frac {3}{4 \ln \left (x \right )+27}}}{27648 \ln \left (x \right )+186624}\)	\(33\)

[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))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx=x e^{\left (-\frac {3}{4 \, \log \left (x\right ) + 27}\right )} \]

[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))

Sympy [A] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx=x e^{- \frac {3}{4 \log {\left (x \right )} + 27}} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx=x e^{\left (-\frac {3}{4 \, \log \left (x\right ) + 27}\right )} \]

[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))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx=x^{\frac {247}{9 \, {\left (4 \, \log \left (x\right ) + 27\right )}}} e^{\left (\frac {4 \, \log \left (x\right )^{2}}{4 \, \log \left (x\right ) + 27} - \frac {1}{9}\right )} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {3}{27+4 \log (x)}} \left (741+216 \log (x)+16 \log ^2(x)\right )}{729+216 \log (x)+16 \log ^2(x)} \, dx=\int \frac {{\mathrm {e}}^{-\frac {3}{4\,\ln \left (x\right )+27}}\,\left (16\,{\ln \left (x\right )}^2+216\,\ln \left (x\right )+741\right )}{16\,{\ln \left (x\right )}^2+216\,\ln \left (x\right )+729} \,d x \]

[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)