Integrand size = 131, antiderivative size = 28 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {x}{\frac {3}{-2+x}+\frac {e^{8 e^x}}{x^2}}-\log (x) \]
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {(-2+x) x^3}{e^{8 e^x} (-2+x)+3 x^2}-\log (x) \]
Integrate[(-9*x^4 - 6*x^5 + 6*x^6 + E^(16*E^x)*(-4 + 4*x - x^2) + E^(8*E^x )*(12*x^2 + 6*x^3 - 12*x^4 + 3*x^5 + E^x*(-32*x^4 + 32*x^5 - 8*x^6)))/(9*x ^5 + E^(16*E^x)*(4*x - 4*x^2 + x^3) + E^(8*E^x)*(-12*x^3 + 6*x^4)),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.
\(\displaystyle \int \frac {6 x^6-6 x^5-9 x^4+e^{16 e^x} \left (-x^2+4 x-4\right )+e^{8 e^x} \left (3 x^5-12 x^4+6 x^3+12 x^2+e^x \left (-8 x^6+32 x^5-32 x^4\right )\right )}{9 x^5+e^{8 e^x} \left (6 x^4-12 x^3\right )+e^{16 e^x} \left (x^3-4 x^2+4 x\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {6 x^6-6 x^5-9 x^4+e^{16 e^x} \left (-x^2+4 x-4\right )+e^{8 e^x} \left (3 x^5-12 x^4+6 x^3+12 x^2+e^x \left (-8 x^6+32 x^5-32 x^4\right )\right )}{x \left (-3 x^2-e^{8 e^x} x+2 e^{8 e^x}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {6 e^{8 e^x} x^2}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}+\frac {12 e^{8 e^x} x}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {e^{16 e^x} (x-2)^2}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2 x}+\frac {6 x^5}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}+\frac {3 e^{8 e^x} x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {6 x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {12 e^{8 e^x} x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {8 e^{x+8 e^x} (x-2)^2 x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {9 x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {e^{16 e^x}}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-4 \int \frac {e^{16 e^x}}{x \left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+12 \int \frac {e^{8 e^x} x}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-\int \frac {e^{16 e^x} x}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+6 \int \frac {e^{8 e^x} x^2}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+6 \int \frac {x^5}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-8 \int \frac {e^{x+8 e^x} x^5}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-6 \int \frac {x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+3 \int \frac {e^{8 e^x} x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+32 \int \frac {e^{x+8 e^x} x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-9 \int \frac {x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-12 \int \frac {e^{8 e^x} x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-32 \int \frac {e^{x+8 e^x} x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx\) |
Int[(-9*x^4 - 6*x^5 + 6*x^6 + E^(16*E^x)*(-4 + 4*x - x^2) + E^(8*E^x)*(12* x^2 + 6*x^3 - 12*x^4 + 3*x^5 + E^x*(-32*x^4 + 32*x^5 - 8*x^6)))/(9*x^5 + E ^(16*E^x)*(4*x - 4*x^2 + x^3) + E^(8*E^x)*(-12*x^3 + 6*x^4)),x]
3.11.50.3.1 Defintions of rubi rules used
Time = 0.88 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25
method | result | size |
risch | \(-\ln \left (x \right )+\frac {\left (-2+x \right ) x^{3}}{{\mathrm e}^{8 \,{\mathrm e}^{x}} x +3 x^{2}-2 \,{\mathrm e}^{8 \,{\mathrm e}^{x}}}\) | \(35\) |
parallelrisch | \(-\frac {-3 x^{4}+9 x^{2} \ln \left (x \right )+3 \ln \left (x \right ) {\mathrm e}^{8 \,{\mathrm e}^{x}} x +6 x^{3}-6 \ln \left (x \right ) {\mathrm e}^{8 \,{\mathrm e}^{x}}}{3 \left ({\mathrm e}^{8 \,{\mathrm e}^{x}} x +3 x^{2}-2 \,{\mathrm e}^{8 \,{\mathrm e}^{x}}\right )}\) | \(62\) |
int(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x^5-12* x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)*exp(8* exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {x^{4} - 2 \, x^{3} - 3 \, x^{2} \log \left (x\right ) - {\left (x - 2\right )} e^{\left (8 \, e^{x}\right )} \log \left (x\right )}{3 \, x^{2} + {\left (x - 2\right )} e^{\left (8 \, e^{x}\right )}} \]
integrate(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x ^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)* exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x, algorithm=\
Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=- \log {\left (x \right )} + \frac {x^{4} - 2 x^{3}}{3 x^{2} + \left (x - 2\right ) e^{8 e^{x}}} \]
integrate(((-x**2+4*x-4)*exp(8*exp(x))**2+((-8*x**6+32*x**5-32*x**4)*exp(x )+3*x**5-12*x**4+6*x**3+12*x**2)*exp(8*exp(x))+6*x**6-6*x**5-9*x**4)/((x** 3-4*x**2+4*x)*exp(8*exp(x))**2+(6*x**4-12*x**3)*exp(8*exp(x))+9*x**5),x)
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {x^{4} - 2 \, x^{3}}{3 \, x^{2} + {\left (x - 2\right )} e^{\left (8 \, e^{x}\right )}} - \log \left (x\right ) \]
integrate(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x ^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)* exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {x^{4} - 2 \, x^{3} - 3 \, x^{2} \log \left (x\right ) - x e^{\left (8 \, e^{x}\right )} \log \left (x\right ) + 2 \, e^{\left (8 \, e^{x}\right )} \log \left (x\right )}{3 \, x^{2} + x e^{\left (8 \, e^{x}\right )} - 2 \, e^{\left (8 \, e^{x}\right )}} \]
integrate(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x ^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)* exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x, algorithm=\
(x^4 - 2*x^3 - 3*x^2*log(x) - x*e^(8*e^x)*log(x) + 2*e^(8*e^x)*log(x))/(3* x^2 + x*e^(8*e^x) - 2*e^(8*e^x))
Time = 10.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=-\frac {3\,x^2\,\ln \left (x\right )+2\,x^3-x^4-2\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}\,\ln \left (x\right )+x\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}\,\ln \left (x\right )}{x\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}-2\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}+3\,x^2} \]
int(-(exp(16*exp(x))*(x^2 - 4*x + 4) - exp(8*exp(x))*(12*x^2 - exp(x)*(32* x^4 - 32*x^5 + 8*x^6) + 6*x^3 - 12*x^4 + 3*x^5) + 9*x^4 + 6*x^5 - 6*x^6)/( exp(16*exp(x))*(4*x - 4*x^2 + x^3) - exp(8*exp(x))*(12*x^3 - 6*x^4) + 9*x^ 5),x)