\(\int \frac {3^{-\frac {2 (x^3-x^4)}{1+4 x^3 \log (\frac {x}{3})}} x^{\frac {2 (x^3-x^4)}{1+4 x^3 \log (\frac {x}{3})}} (2 x^2-2 x^3+(6 x^2-8 x^3) \log (\frac {x}{3})-8 x^6 \log ^2(\frac {x}{3}))}{1+8 x^3 \log (\frac {x}{3})+16 x^6 \log ^2(\frac {x}{3})} \, dx\) [7762]

Optimal result

Integrand size = 127, antiderivative size = 27 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=1+e^{\frac {2 (1-x)}{4+\frac {1}{x^3 \log \left (\frac {x}{3}\right )}}} \]

[Out]

exp((1-x)/(1/x^3/ln(1/3*x)+4))^2+1

Rubi [F]

\[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=\int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx \]

[In]

Int[(x^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(2*x^2 - 2*x^3 + (6*x^2 - 8*x^3)*Log[x/3] - 8*x^6*Log[x/3]^2))/(
3^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(1 + 8*x^3*Log[x/3] + 16*x^6*Log[x/3]^2)),x]

[Out]

-1/2*Defer[Int][x^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))/3^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3])), x] + (3*Def
er[Int][x^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))/(3^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))*(1 + 4*x^3*Log[x/3]
)^2), x])/2 + 2*(1 - Log[27])*Defer[Int][x^(2 + (2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))/(3^((2*(1 - x)*x^3)/(1 +
4*x^3*Log[x/3]))*(1 + 4*x^3*Log[x/3])^2), x] - 2*Defer[Int][x^(3 + (2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))/(3^(
(2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))*(1 + 4*x^3*Log[x/3])^2), x] - Defer[Int][x^((2*(1 - x)*x^3)/(1 + 4*x^3*L
og[x/3]))/(3^((2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))*(1 + 4*x^3*Log[x/3])), x] + 2*Defer[Int][(3^(1 - (2*(1 - x
)*x^3)/(1 + 4*x^3*Log[x/3]))*x^(2 + (2*(1 - x)*x^3)/(1 + 4*x^3*Log[x/3]))*Log[x])/(1 + 4*x^3*Log[x/3])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2\ 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (1-x-\log (27)-4 x \log \left (\frac {x}{3}\right )-4 x^4 \log ^2\left (\frac {x}{3}\right )+3 \log (x)\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx \\ & = 2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (1-x-\log (27)-4 x \log \left (\frac {x}{3}\right )-4 x^4 \log ^2\left (\frac {x}{3}\right )+3 \log (x)\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx \\ & = 2 \int \left (-\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}+\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} (1-\log (27))}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}-\frac {4\ 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log \left (\frac {x}{3}\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}-\frac {4\ 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{6+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log ^2\left (\frac {x}{3}\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}+\frac {3^{1-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log (x)}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}\right ) \, dx \\ & = -\left (2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx\right )+2 \int \frac {3^{1-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log (x)}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-8 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log \left (\frac {x}{3}\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-8 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{6+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log ^2\left (\frac {x}{3}\right )}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx+(2 (1-\log (27))) \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx \\ & = -\left (2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx\right )+2 \int \frac {3^{1-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log (x)}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-8 \int \left (\frac {1}{16} 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}+\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{16 \left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}-\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{8 \left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )}\right ) \, dx-8 \int \left (-\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{4 \left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2}+\frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{4 \left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )}\right ) \, dx+(2 (1-\log (27))) \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx \\ & = -\left (\frac {1}{2} \int 3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \, dx\right )-\frac {1}{2} \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx+2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{3+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx-2 \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{1+4 x^3 \log \left (\frac {x}{3}\right )} \, dx+2 \int \frac {3^{1-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \log (x)}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx+(2 (1-\log (27))) \int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{2+\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{\left (1+4 x^3 \log \left (\frac {x}{3}\right )\right )^2} \, dx+\int \frac {3^{-\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 (1-x) x^3}{1+4 x^3 \log \left (\frac {x}{3}\right )}}}{1+4 x^3 \log \left (\frac {x}{3}\right )} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=\int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx \]

[In]

Integrate[(x^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(2*x^2 - 2*x^3 + (6*x^2 - 8*x^3)*Log[x/3] - 8*x^6*Log[x/3]
^2))/(3^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(1 + 8*x^3*Log[x/3] + 16*x^6*Log[x/3]^2)),x]

[Out]

Integrate[(x^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(2*x^2 - 2*x^3 + (6*x^2 - 8*x^3)*Log[x/3] - 8*x^6*Log[x/3]
^2))/(3^((2*(x^3 - x^4))/(1 + 4*x^3*Log[x/3]))*(1 + 8*x^3*Log[x/3] + 16*x^6*Log[x/3]^2)), x]

Maple [A] (verified)

Time = 7.60 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\({\mathrm e}^{-\frac {2 x^{3} \left (-1+x \right ) \ln \left (\frac {x}{3}\right )}{4 x^{3} \ln \left (\frac {x}{3}\right )+1}}\)	\(29\)
risch	\(\left (\frac {x}{3}\right )^{-\frac {2 x^{3} \left (-1+x \right )}{4 x^{3} \ln \left (\frac {x}{3}\right )+1}}\)	\(33\)

[In]

int((-8*x^6*ln(1/3*x)^2+(-8*x^3+6*x^2)*ln(1/3*x)-2*x^3+2*x^2)*exp((-x^4+x^3)*ln(1/3*x)/(4*x^3*ln(1/3*x)+1))^2/
(16*x^6*ln(1/3*x)^2+8*x^3*ln(1/3*x)+1),x,method=_RETURNVERBOSE)

[Out]

exp(-x^3*(-1+x)*ln(1/3*x)/(4*x^3*ln(1/3*x)+1))^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=\frac {1}{\left (\frac {1}{3} \, x\right )^{\frac {2 \, {\left (x^{4} - x^{3}\right )}}{4 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + 1}}} \]

[In]

integrate((-8*x^6*log(1/3*x)^2+(-8*x^3+6*x^2)*log(1/3*x)-2*x^3+2*x^2)*exp((-x^4+x^3)*log(1/3*x)/(4*x^3*log(1/3
*x)+1))^2/(16*x^6*log(1/3*x)^2+8*x^3*log(1/3*x)+1),x, algorithm="fricas")

[Out]

1/((1/3*x)^(2*(x^4 - x^3)/(4*x^3*log(1/3*x) + 1)))

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=e^{\frac {2 \left (- x^{4} + x^{3}\right ) \log {\left (\frac {x}{3} \right )}}{4 x^{3} \log {\left (\frac {x}{3} \right )} + 1}} \]

[In]

integrate((-8*x**6*ln(1/3*x)**2+(-8*x**3+6*x**2)*ln(1/3*x)-2*x**3+2*x**2)*exp((-x**4+x**3)*ln(1/3*x)/(4*x**3*l
n(1/3*x)+1))**2/(16*x**6*ln(1/3*x)**2+8*x**3*ln(1/3*x)+1),x)

[Out]

exp(2*(-x**4 + x**3)*log(x/3)/(4*x**3*log(x/3) + 1))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).

Time = 0.43 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=e^{\left (-\frac {1}{2} \, x - \frac {x}{2 \, {\left (4 \, x^{3} \log \left (3\right ) - 4 \, x^{3} \log \left (x\right ) - 1\right )}} + \frac {1}{2 \, {\left (4 \, x^{3} \log \left (3\right ) - 4 \, x^{3} \log \left (x\right ) - 1\right )}} + \frac {1}{2}\right )} \]

[In]

integrate((-8*x^6*log(1/3*x)^2+(-8*x^3+6*x^2)*log(1/3*x)-2*x^3+2*x^2)*exp((-x^4+x^3)*log(1/3*x)/(4*x^3*log(1/3
*x)+1))^2/(16*x^6*log(1/3*x)^2+8*x^3*log(1/3*x)+1),x, algorithm="maxima")

[Out]

e^(-1/2*x - 1/2*x/(4*x^3*log(3) - 4*x^3*log(x) - 1) + 1/2/(4*x^3*log(3) - 4*x^3*log(x) - 1) + 1/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).

Time = 1.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx=\frac {\left (\frac {1}{3} \, x\right )^{\frac {2 \, x^{3}}{4 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + 1}}}{\left (\frac {1}{3} \, x\right )^{\frac {2 \, x^{4}}{4 \, x^{3} \log \left (\frac {1}{3} \, x\right ) + 1}}} \]

[In]

integrate((-8*x^6*log(1/3*x)^2+(-8*x^3+6*x^2)*log(1/3*x)-2*x^3+2*x^2)*exp((-x^4+x^3)*log(1/3*x)/(4*x^3*log(1/3
*x)+1))^2/(16*x^6*log(1/3*x)^2+8*x^3*log(1/3*x)+1),x, algorithm="giac")

[Out]

(1/3*x)^(2*x^3/(4*x^3*log(1/3*x) + 1))/(1/3*x)^(2*x^4/(4*x^3*log(1/3*x) + 1))

Mupad [B] (verification not implemented)

Time = 13.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \[ \int \frac {3^{-\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} x^{\frac {2 \left (x^3-x^4\right )}{1+4 x^3 \log \left (\frac {x}{3}\right )}} \left (2 x^2-2 x^3+\left (6 x^2-8 x^3\right ) \log \left (\frac {x}{3}\right )-8 x^6 \log ^2\left (\frac {x}{3}\right )\right )}{1+8 x^3 \log \left (\frac {x}{3}\right )+16 x^6 \log ^2\left (\frac {x}{3}\right )} \, dx={\left (\frac {1}{9}\right )}^{\frac {x^3-x^4}{4\,x^3\,\ln \left (x\right )-4\,x^3\,\ln \left (3\right )+1}}\,x^{\frac {2\,\left (x^3-x^4\right )}{4\,x^3\,\ln \left (x\right )-4\,x^3\,\ln \left (3\right )+1}} \]

[In]

int((exp((2*log(x/3)*(x^3 - x^4))/(4*x^3*log(x/3) + 1))*(log(x/3)*(6*x^2 - 8*x^3) + 2*x^2 - 2*x^3 - 8*x^6*log(
x/3)^2))/(8*x^3*log(x/3) + 16*x^6*log(x/3)^2 + 1),x)

[Out]

(1/9)^((x^3 - x^4)/(4*x^3*log(x) - 4*x^3*log(3) + 1))*x^((2*(x^3 - x^4))/(4*x^3*log(x) - 4*x^3*log(3) + 1))