Integrand size = 107, antiderivative size = 27 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 e^{2+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \]
Time = 1.85 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 e^{2+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \]
Integrate[(E^((8*x^2)/(9*E^(2*x)))*(-72*E^(2 + 2*x) + E^2*(64*x^2 - 64*x^3 ) + E^(2*x)*(-144*E^(2 + 2*x) + E^2*(64*x - 64*x^2))))/(9*E^(8*x) + 27*E^( 6*x)*x + 27*E^(4*x)*x^2 + 9*E^(2*x)*x^3),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.
\(\displaystyle \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (e^{2 x} \left (e^2 \left (64 x-64 x^2\right )-144 e^{2 x+2}\right )+e^2 \left (64 x^2-64 x^3\right )-72 e^{2 x+2}\right )}{9 e^{2 x} x^3+27 e^{4 x} x^2+27 e^{6 x} x+9 e^{8 x}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {8 e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} \left (-8 x^3-8 e^{2 x} x^2+8 x^2+8 e^{2 x} x-9 e^{2 x}-18 e^{4 x}\right )}{9 \left (x+e^{2 x}\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {8}{9} \int -\frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} \left (8 x^3+8 e^{2 x} x^2-8 x^2-8 e^{2 x} x+9 e^{2 x}+18 e^{4 x}\right )}{\left (x+e^{2 x}\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {8}{9} \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} \left (8 x^3+8 e^{2 x} x^2-8 x^2-8 e^{2 x} x+9 e^{2 x}+18 e^{4 x}\right )}{\left (x+e^{2 x}\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {8}{9} \int \left (\frac {9 e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x (2 x-1)}{\left (x+e^{2 x}\right )^3}+\frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} \left (8 x^2-44 x+9\right )}{\left (x+e^{2 x}\right )^2}+\frac {18 e^{\frac {8}{9} e^{-2 x} x^2-2 x+2}}{x+e^{2 x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {8}{9} \left (-9 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x}{\left (x+e^{2 x}\right )^3}dx+18 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x^2}{\left (x+e^{2 x}\right )^3}dx+9 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2}}{\left (x+e^{2 x}\right )^2}dx-44 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x}{\left (x+e^{2 x}\right )^2}dx+8 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x^2}{\left (x+e^{2 x}\right )^2}dx+18 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2}}{x+e^{2 x}}dx\right )\) |
Int[(E^((8*x^2)/(9*E^(2*x)))*(-72*E^(2 + 2*x) + E^2*(64*x^2 - 64*x^3) + E^ (2*x)*(-144*E^(2 + 2*x) + E^2*(64*x - 64*x^2))))/(9*E^(8*x) + 27*E^(6*x)*x + 27*E^(4*x)*x^2 + 9*E^(2*x)*x^3),x]
3.17.21.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 5.85 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {4 \,{\mathrm e}^{2+\frac {8 x^{2} {\mathrm e}^{-2 x}}{9}}}{\left ({\mathrm e}^{2 x}+x \right )^{2}}\) | \(23\) |
parallelrisch | \(\frac {4 \,{\mathrm e}^{2} {\mathrm e}^{\frac {8 x^{2} {\mathrm e}^{-2 x}}{9}}}{x^{2}+2 x \,{\mathrm e}^{2 x}+{\mathrm e}^{4 x}}\) | \(38\) |
int(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*exp(1)^2 *exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp(x)^2*e xp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x)^2*x^3 ),x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 \, e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )} + 6\right )}}{x^{2} e^{4} + 2 \, x e^{\left (2 \, x + 4\right )} + e^{\left (4 \, x + 4\right )}} \]
integrate(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*ex p(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp( x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x) ^2*x^3),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 e^{2} e^{\frac {8 x^{2} e^{- 2 x}}{9}}}{x^{2} + 2 x e^{2 x} + e^{4 x}} \]
integrate(((-144*exp(1)**2*exp(x)**2+(-64*x**2+64*x)*exp(1)**2)*exp(2*x)-7 2*exp(1)**2*exp(x)**2+(-64*x**3+64*x**2)*exp(1)**2)*exp(4/9*x**2/exp(x)**2 )**2/(9*exp(x)**2*exp(2*x)**3+27*x*exp(x)**2*exp(2*x)**2+27*x**2*exp(x)**2 *exp(2*x)+9*exp(x)**2*x**3),x)
Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 \, e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )} + 2\right )}}{x^{2} + 2 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}} \]
integrate(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*ex p(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp( x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x) ^2*x^3),x, algorithm=\
\[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\int { -\frac {8 \, {\left (8 \, {\left (x^{3} - x^{2}\right )} e^{2} + 2 \, {\left (4 \, {\left (x^{2} - x\right )} e^{2} + 9 \, e^{\left (2 \, x + 2\right )}\right )} e^{\left (2 \, x\right )} + 9 \, e^{\left (2 \, x + 2\right )}\right )} e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )}\right )}}{9 \, {\left (x^{3} e^{\left (2 \, x\right )} + 3 \, x^{2} e^{\left (4 \, x\right )} + 3 \, x e^{\left (6 \, x\right )} + e^{\left (8 \, x\right )}\right )}} \,d x } \]
integrate(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*ex p(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp( x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x) ^2*x^3),x, algorithm=\
integrate(-8/9*(8*(x^3 - x^2)*e^2 + 2*(4*(x^2 - x)*e^2 + 9*e^(2*x + 2))*e^ (2*x) + 9*e^(2*x + 2))*e^(8/9*x^2*e^(-2*x))/(x^3*e^(2*x) + 3*x^2*e^(4*x) + 3*x*e^(6*x) + e^(8*x)), x)
Time = 10.97 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4\,{\mathrm {e}}^2\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^{-2\,x}}{9}}}{{\mathrm {e}}^{4\,x}+2\,x\,{\mathrm {e}}^{2\,x}+x^2} \]