\(\int \frac {x^{\frac {4}{-2 x+\log ^4(3+e^{3-x})}} (-24 x-8 e^{3-x} x+(12+4 e^{3-x}) \log ^4(3+e^{3-x})+(24 x+8 e^{3-x} x) \log (x)+16 e^{3-x} x \log ^3(3+e^{3-x}) \log (x))}{12 x^3+4 e^{3-x} x^3+(-12 x^2-4 e^{3-x} x^2) \log ^4(3+e^{3-x})+(3 x+e^{3-x} x) \log ^8(3+e^{3-x})} \, dx\) [7407]

Optimal result

Integrand size = 179, antiderivative size = 26 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=x^{\frac {2}{-x+\frac {1}{2} \log ^4\left (3+e^{3-x}\right )}} \]

[Out]

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

Rubi [F]

\[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=\int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx \]

[In]

Int[(x^(4/(-2*x + Log[3 + E^(3 - x)]^4))*(-24*x - 8*E^(3 - x)*x + (12 + 4*E^(3 - x))*Log[3 + E^(3 - x)]^4 + (2
4*x + 8*E^(3 - x)*x)*Log[x] + 16*E^(3 - x)*x*Log[3 + E^(3 - x)]^3*Log[x]))/(12*x^3 + 4*E^(3 - x)*x^3 + (-12*x^
2 - 4*E^(3 - x)*x^2)*Log[3 + E^(3 - x)]^4 + (3*x + E^(3 - x)*x)*Log[3 + E^(3 - x)]^8),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=x^{-\frac {4}{2 x-\log ^4\left (3+e^{3-x}\right )}} \]

[In]

Integrate[(x^(4/(-2*x + Log[3 + E^(3 - x)]^4))*(-24*x - 8*E^(3 - x)*x + (12 + 4*E^(3 - x))*Log[3 + E^(3 - x)]^
4 + (24*x + 8*E^(3 - x)*x)*Log[x] + 16*E^(3 - x)*x*Log[3 + E^(3 - x)]^3*Log[x]))/(12*x^3 + 4*E^(3 - x)*x^3 + (
-12*x^2 - 4*E^(3 - x)*x^2)*Log[3 + E^(3 - x)]^4 + (3*x + E^(3 - x)*x)*Log[3 + E^(3 - x)]^8),x]

[Out]

x^(-4/(2*x - Log[3 + E^(3 - x)]^4))

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

\[x^{\frac {4}{\ln \left ({\mathrm e}^{-x +3}+3\right )^{4}-2 x}}\]

[In]

int(((4*exp(-x+3)+12)*ln(exp(-x+3)+3)^4+16*x*exp(-x+3)*ln(x)*ln(exp(-x+3)+3)^3+(8*x*exp(-x+3)+24*x)*ln(x)-8*x*
exp(-x+3)-24*x)*exp(4*ln(x)/(ln(exp(-x+3)+3)^4-2*x))/((x*exp(-x+3)+3*x)*ln(exp(-x+3)+3)^8+(-4*x^2*exp(-x+3)-12
*x^2)*ln(exp(-x+3)+3)^4+4*x^3*exp(-x+3)+12*x^3),x)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=x^{\frac {4}{\log \left (e^{\left (-x + 3\right )} + 3\right )^{4} - 2 \, x}} \]

[In]

integrate(((4*exp(-x+3)+12)*log(exp(-x+3)+3)^4+16*x*exp(-x+3)*log(x)*log(exp(-x+3)+3)^3+(8*x*exp(-x+3)+24*x)*l
og(x)-8*x*exp(-x+3)-24*x)*exp(4*log(x)/(log(exp(-x+3)+3)^4-2*x))/((x*exp(-x+3)+3*x)*log(exp(-x+3)+3)^8+(-4*x^2
*exp(-x+3)-12*x^2)*log(exp(-x+3)+3)^4+4*x^3*exp(-x+3)+12*x^3),x, algorithm="fricas")

[Out]

x^(4/(log(e^(-x + 3) + 3)^4 - 2*x))

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=\text {Timed out} \]

[In]

integrate(((4*exp(-x+3)+12)*ln(exp(-x+3)+3)**4+16*x*exp(-x+3)*ln(x)*ln(exp(-x+3)+3)**3+(8*x*exp(-x+3)+24*x)*ln
(x)-8*x*exp(-x+3)-24*x)*exp(4*ln(x)/(ln(exp(-x+3)+3)**4-2*x))/((x*exp(-x+3)+3*x)*ln(exp(-x+3)+3)**8+(-4*x**2*e
xp(-x+3)-12*x**2)*ln(exp(-x+3)+3)**4+4*x**3*exp(-x+3)+12*x**3),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (21) = 42\).

Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=x^{\frac {4}{x^{4} - 4 \, x^{3} \log \left (e^{3} + 3 \, e^{x}\right ) + 6 \, x^{2} \log \left (e^{3} + 3 \, e^{x}\right )^{2} + \log \left (e^{3} + 3 \, e^{x}\right )^{4} - 2 \, {\left (2 \, \log \left (e^{3} + 3 \, e^{x}\right )^{3} + 1\right )} x}} \]

[In]

integrate(((4*exp(-x+3)+12)*log(exp(-x+3)+3)^4+16*x*exp(-x+3)*log(x)*log(exp(-x+3)+3)^3+(8*x*exp(-x+3)+24*x)*l
og(x)-8*x*exp(-x+3)-24*x)*exp(4*log(x)/(log(exp(-x+3)+3)^4-2*x))/((x*exp(-x+3)+3*x)*log(exp(-x+3)+3)^8+(-4*x^2
*exp(-x+3)-12*x^2)*log(exp(-x+3)+3)^4+4*x^3*exp(-x+3)+12*x^3),x, algorithm="maxima")

[Out]

x^(4/(x^4 - 4*x^3*log(e^3 + 3*e^x) + 6*x^2*log(e^3 + 3*e^x)^2 + log(e^3 + 3*e^x)^4 - 2*(2*log(e^3 + 3*e^x)^3 +
1)*x))

Giac [F(-1)]

Timed out. \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=\text {Timed out} \]

[In]

integrate(((4*exp(-x+3)+12)*log(exp(-x+3)+3)^4+16*x*exp(-x+3)*log(x)*log(exp(-x+3)+3)^3+(8*x*exp(-x+3)+24*x)*l
og(x)-8*x*exp(-x+3)-24*x)*exp(4*log(x)/(log(exp(-x+3)+3)^4-2*x))/((x*exp(-x+3)+3*x)*log(exp(-x+3)+3)^8+(-4*x^2
*exp(-x+3)-12*x^2)*log(exp(-x+3)+3)^4+4*x^3*exp(-x+3)+12*x^3),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 12.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=\frac {1}{x^{\frac {4}{2\,x-{\ln \left ({\mathrm {e}}^{-x}\,{\mathrm {e}}^3+3\right )}^4}}} \]

[In]

int((exp(-(4*log(x))/(2*x - log(exp(3 - x) + 3)^4))*(log(exp(3 - x) + 3)^4*(4*exp(3 - x) + 12) - 8*x*exp(3 - x
) - 24*x + log(x)*(24*x + 8*x*exp(3 - x)) + 16*x*exp(3 - x)*log(exp(3 - x) + 3)^3*log(x)))/(log(exp(3 - x) + 3
)^8*(3*x + x*exp(3 - x)) - log(exp(3 - x) + 3)^4*(4*x^2*exp(3 - x) + 12*x^2) + 4*x^3*exp(3 - x) + 12*x^3),x)

[Out]

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