Integrand size = 179, antiderivative size = 26 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=x^{\frac {2}{-x+\frac {1}{2} \log ^4\left (3+e^{3-x}\right )}} \]
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=x^{-\frac {4}{2 x-\log ^4\left (3+e^{3-x}\right )}} \]
Integrate[(x^(4/(-2*x + Log[3 + E^(3 - x)]^4))*(-24*x - 8*E^(3 - x)*x + (1 2 + 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 + (-1 2*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]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^{\frac {4}{\log ^4\left (e^{3-x}+3\right )-2 x}} \left (-8 e^{3-x} x-24 x+\left (4 e^{3-x}+12\right ) \log ^4\left (e^{3-x}+3\right )+16 e^{3-x} x \log (x) \log ^3\left (e^{3-x}+3\right )+\left (8 e^{3-x} x+24 x\right ) \log (x)\right )}{4 e^{3-x} x^3+12 x^3+\left (-4 e^{3-x} x^2-12 x^2\right ) \log ^4\left (e^{3-x}+3\right )+\left (e^{3-x} x+3 x\right ) \log ^8\left (e^{3-x}+3\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^x x^{\frac {4}{\log ^4\left (e^{3-x}+3\right )-2 x}-1} \left (-8 e^{3-x} x-24 x+\left (4 e^{3-x}+12\right ) \log ^4\left (e^{3-x}+3\right )+16 e^{3-x} x \log (x) \log ^3\left (e^{3-x}+3\right )+\left (8 e^{3-x} x+24 x\right ) \log (x)\right )}{\left (3 e^x+e^3\right ) \left (2 x-\log ^4\left (e^{3-x}+3\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 x^{\frac {4}{\log ^4\left (e^{3-x}+3\right )-2 x}-1} \left (-2 x+\log ^4\left (e^{3-x}+3\right )+4 x \log (x) \log ^3\left (e^{3-x}+3\right )+2 x \log (x)\right )}{\left (2 x-\log ^4\left (e^{3-x}+3\right )\right )^2}-\frac {48 e^x x^{\frac {4}{\log ^4\left (e^{3-x}+3\right )-2 x}} \log ^3\left (e^{3-x}+3\right ) \log (x)}{\left (3 e^x+e^3\right ) \left (2 x-\log ^4\left (e^{3-x}+3\right )\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {x^{\frac {4}{\log ^4\left (3+e^{3-x}\right )-2 x}-1}}{\log ^4\left (3+e^{3-x}\right )-2 x}dx+8 \int \frac {x^{\frac {4}{\log ^4\left (3+e^{3-x}\right )-2 x}} \log (x)}{\left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2}dx-48 \int \frac {e^x x^{\frac {4}{\log ^4\left (3+e^{3-x}\right )-2 x}} \log ^3\left (3+e^{3-x}\right ) \log (x)}{\left (e^3+3 e^x\right ) \left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2}dx+16 \int \frac {x^{\frac {4}{\log ^4\left (3+e^{3-x}\right )-2 x}} \log ^3\left (3+e^{3-x}\right ) \log (x)}{\left (\log ^4\left (3+e^{3-x}\right )-2 x\right )^2}dx\) |
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 + (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]
3.15.7.3.1 Defintions of rubi rules used
Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
\[x^{\frac {4}{\ln \left ({\mathrm e}^{-x +3}+3\right )^{4}-2 x}}\]
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)
Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=x^{\frac {4}{\log \left (e^{\left (-x + 3\right )} + 3\right )^{4} - 2 \, x}} \]
integrate(((4*exp(-x+3)+12)*log(exp(-x+3)+3)^4+16*x*exp(-x+3)*log(x)*log(e xp(-x+3)+3)^3+(8*x*exp(-x+3)+24*x)*log(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*e xp(-x+3)-12*x^2)*log(exp(-x+3)+3)^4+4*x^3*exp(-x+3)+12*x^3),x, algorithm=\
Timed out. \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=\text {Timed out} \]
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)/(l n(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)
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 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=x^{\frac {4}{x^{4} - 4 \, x^{3} \log \left (e^{3} + 3 \, e^{x}\right ) + 6 \, x^{2} \log \left (e^{3} + 3 \, e^{x}\right )^{2} + \log \left (e^{3} + 3 \, e^{x}\right )^{4} - 2 \, {\left (2 \, \log \left (e^{3} + 3 \, e^{x}\right )^{3} + 1\right )} x}} \]
integrate(((4*exp(-x+3)+12)*log(exp(-x+3)+3)^4+16*x*exp(-x+3)*log(x)*log(e xp(-x+3)+3)^3+(8*x*exp(-x+3)+24*x)*log(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*e xp(-x+3)-12*x^2)*log(exp(-x+3)+3)^4+4*x^3*exp(-x+3)+12*x^3),x, algorithm=\
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))
Timed out. \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=\text {Timed out} \]
integrate(((4*exp(-x+3)+12)*log(exp(-x+3)+3)^4+16*x*exp(-x+3)*log(x)*log(e xp(-x+3)+3)^3+(8*x*exp(-x+3)+24*x)*log(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*e xp(-x+3)-12*x^2)*log(exp(-x+3)+3)^4+4*x^3*exp(-x+3)+12*x^3),x, algorithm=\
Time = 12.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx=\frac {1}{x^{\frac {4}{2\,x-{\ln \left ({\mathrm {e}}^{-x}\,{\mathrm {e}}^3+3\right )}^4}}} \]
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)