Integrand size = 148, antiderivative size = 28 \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\left (e^{4+x}+2 e^{-\frac {3 e^{-\frac {e^{16}}{x}}}{x}} x\right )^2 \]
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}+2 x\right )^2 \]
Integrate[(E^(-(E^16/x) - 6/(E^(E^16/x)*x))*(-24*E^16*x + 24*x^2 + 2*E^(8 + E^16/x + 6/(E^(E^16/x)*x) + 2*x)*x^2 + 8*E^(E^16/x)*x^3 + E^(3/(E^(E^16/ x)*x))*(E^(4 + x)*(-12*E^16 + 12*x) + E^(4 + E^16/x + x)*(4*x^2 + 4*x^3))) )/x^2,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.
| \(\displaystyle \int \frac {e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}} \left (8 e^{\frac {e^{16}}{x}} x^3+2 e^{2 x+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+\frac {e^{16}}{x}+8} x^2+24 x^2+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{x+\frac {e^{16}}{x}+4} \left (4 x^3+4 x^2\right )+e^{x+4} \left (12 x-12 e^{16}\right )\right )-24 e^{16} x\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}} \left (2 x+e^{x+\frac {3 e^{-\frac {e^{16}}{x}}}{x}+4}\right ) \left (2 e^{\frac {e^{16}}{x}} x^2+e^{x+\frac {3 e^{-\frac {e^{16}}{x}}}{x}+\frac {e^{16}}{x}+4} x^2+6 x-6 e^{16}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}} \left (2 x+e^{x+4+\frac {3 e^{-\frac {e^{16}}{x}}}{x}}\right ) \left (-2 e^{\frac {e^{16}}{x}} x^2-e^{x+4+\frac {3 e^{-\frac {e^{16}}{x}}}{x}+\frac {e^{16}}{x}} x^2-6 x+6 e^{16}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}} \left (2 x+e^{x+4+\frac {3 e^{-\frac {e^{16}}{x}}}{x}}\right ) \left (-2 e^{\frac {e^{16}}{x}} x^2-e^{x+4+\frac {3 e^{-\frac {e^{16}}{x}}}{x}+\frac {e^{16}}{x}} x^2-6 x+6 e^{16}\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {4 e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}} \left (e^{\frac {e^{16}}{x}} x^2+3 x-3 e^{16}\right )}{x}-e^{2 x+8}+\frac {2 e^{x+4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}} \left (-e^{\frac {e^{16}}{x}} x^3-e^{\frac {e^{16}}{x}} x^2-3 x+3 e^{16}\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (-12 \text {Subst}\left (\int \frac {e^{-6 e^{-e^{16} x} x-e^{16} x+16}}{x}dx,x,\frac {1}{x}\right )+6 \int \frac {e^{x+20-\frac {3 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}}}{x^2}dx-12 \int e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}}dx-2 \int e^{x+4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}}dx-6 \int \frac {e^{x+4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}}}{x}dx-4 \int e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} xdx-2 \int e^{x+4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}} xdx-\frac {1}{2} e^{2 x+8}\right )\) |
Int[(E^(-(E^16/x) - 6/(E^(E^16/x)*x))*(-24*E^16*x + 24*x^2 + 2*E^(8 + E^16 /x + 6/(E^(E^16/x)*x) + 2*x)*x^2 + 8*E^(E^16/x)*x^3 + E^(3/(E^(E^16/x)*x)) *(E^(4 + x)*(-12*E^16 + 12*x) + E^(4 + E^16/x + x)*(4*x^2 + 4*x^3))))/x^2, x]
3.8.48.3.1 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 = 52.81 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86
| method | result | size |
| risch | \({\mathrm e}^{2 x +8}+4 x \,{\mathrm e}^{\frac {x^{2}-3 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}+4 x}{x}}+4 x^{2} {\mathrm e}^{-\frac {6 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}}{x}}\) | \(52\) |
| parallelrisch | \(-\frac {{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}} \left (-x \,{\mathrm e}^{2 x +8} {\mathrm e}^{\frac {{\mathrm e}^{16}}{x}} {\mathrm e}^{\frac {6 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}}{x}}-4 \,{\mathrm e}^{\frac {{\mathrm e}^{16}}{x}} {\mathrm e}^{4+x} {\mathrm e}^{\frac {3 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}}{x}} x^{2}-4 x^{3} {\mathrm e}^{\frac {{\mathrm e}^{16}}{x}}\right ) {\mathrm e}^{-\frac {6 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}}{x}}}{x}\) | \(109\) |
int((2*x^2*exp(4+x)^2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))^2+((4*x^3+4*x ^2)*exp(4+x)*exp(exp(16)/x)+(-12*exp(16)+12*x)*exp(4+x))*exp(3/x/exp(exp(1 6)/x))+8*x^3*exp(exp(16)/x)-24*x*exp(16)+24*x^2)/x^2/exp(exp(16)/x)/exp(3/ x/exp(exp(16)/x))^2,x,method=_RETURNVERBOSE)
Timed out. \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\text {Timed out} \]
integrate((2*x^2*exp(4+x)^2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))^2+((4*x ^3+4*x^2)*exp(4+x)*exp(exp(16)/x)+(-12*exp(16)+12*x)*exp(4+x))*exp(3/x/exp (exp(16)/x))+8*x^3*exp(exp(16)/x)-24*x*exp(16)+24*x^2)/x^2/exp(exp(16)/x)/ exp(3/x/exp(exp(16)/x))^2,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 113.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=4 x^{2} e^{- \frac {6 e^{- \frac {e^{16}}{x}}}{x}} + 4 x e^{- \frac {3 e^{- \frac {e^{16}}{x}}}{x}} e^{x + 4} + e^{2 x + 8} \]
integrate((2*x**2*exp(4+x)**2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))**2+(( 4*x**3+4*x**2)*exp(4+x)*exp(exp(16)/x)+(-12*exp(16)+12*x)*exp(4+x))*exp(3/ x/exp(exp(16)/x))+8*x**3*exp(exp(16)/x)-24*x*exp(16)+24*x**2)/x**2/exp(exp (16)/x)/exp(3/x/exp(exp(16)/x))**2,x)
\[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\int { \frac {2 \, {\left (4 \, x^{3} e^{\left (\frac {e^{16}}{x}\right )} + x^{2} e^{\left (2 \, x + \frac {e^{16}}{x} + \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x} + 8\right )} + 12 \, x^{2} - 12 \, x e^{16} + 2 \, {\left ({\left (x^{3} + x^{2}\right )} e^{\left (x + \frac {e^{16}}{x} + 4\right )} + 3 \, {\left (x - e^{16}\right )} e^{\left (x + 4\right )}\right )} e^{\left (\frac {3 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}\right )} e^{\left (-\frac {e^{16}}{x} - \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}}{x^{2}} \,d x } \]
integrate((2*x^2*exp(4+x)^2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))^2+((4*x ^3+4*x^2)*exp(4+x)*exp(exp(16)/x)+(-12*exp(16)+12*x)*exp(4+x))*exp(3/x/exp (exp(16)/x))+8*x^3*exp(exp(16)/x)-24*x*exp(16)+24*x^2)/x^2/exp(exp(16)/x)/ exp(3/x/exp(exp(16)/x))^2,x, algorithm=\
4*x*e^(x - 3*e^(-e^16/x)/x + 4) + e^(2*x + 8) + 2*integrate(4*(x^2*e^(e^16 /x) + 3*x - 3*e^16)*e^(-e^16/x - 6*e^(-e^16/x)/x)/x, x)
\[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\int { \frac {2 \, {\left (4 \, x^{3} e^{\left (\frac {e^{16}}{x}\right )} + x^{2} e^{\left (2 \, x + \frac {e^{16}}{x} + \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x} + 8\right )} + 12 \, x^{2} - 12 \, x e^{16} + 2 \, {\left ({\left (x^{3} + x^{2}\right )} e^{\left (x + \frac {e^{16}}{x} + 4\right )} + 3 \, {\left (x - e^{16}\right )} e^{\left (x + 4\right )}\right )} e^{\left (\frac {3 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}\right )} e^{\left (-\frac {e^{16}}{x} - \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}}{x^{2}} \,d x } \]
integrate((2*x^2*exp(4+x)^2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))^2+((4*x ^3+4*x^2)*exp(4+x)*exp(exp(16)/x)+(-12*exp(16)+12*x)*exp(4+x))*exp(3/x/exp (exp(16)/x))+8*x^3*exp(exp(16)/x)-24*x*exp(16)+24*x^2)/x^2/exp(exp(16)/x)/ exp(3/x/exp(exp(16)/x))^2,x, algorithm=\
integrate(2*(4*x^3*e^(e^16/x) + x^2*e^(2*x + e^16/x + 6*e^(-e^16/x)/x + 8) + 12*x^2 - 12*x*e^16 + 2*((x^3 + x^2)*e^(x + e^16/x + 4) + 3*(x - e^16)*e ^(x + 4))*e^(3*e^(-e^16/x)/x))*e^(-e^16/x - 6*e^(-e^16/x)/x)/x^2, x)
Timed out. \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\int \frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}\,{\mathrm {e}}^{-\frac {6\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}}{x}}\,\left (8\,x^3\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x}}-24\,x\,{\mathrm {e}}^{16}+{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}}{x}}\,\left ({\mathrm {e}}^{x+4}\,\left (12\,x-12\,{\mathrm {e}}^{16}\right )+{\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x}}\,{\mathrm {e}}^{x+4}\,\left (4\,x^3+4\,x^2\right )\right )+24\,x^2+2\,x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x}}\,{\mathrm {e}}^{2\,x+8}\,{\mathrm {e}}^{\frac {6\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}}{x}}\right )}{x^2} \,d x \]
int((exp(-exp(16)/x)*exp(-(6*exp(-exp(16)/x))/x)*(8*x^3*exp(exp(16)/x) - 2 4*x*exp(16) + exp((3*exp(-exp(16)/x))/x)*(exp(x + 4)*(12*x - 12*exp(16)) + exp(exp(16)/x)*exp(x + 4)*(4*x^2 + 4*x^3)) + 24*x^2 + 2*x^2*exp(exp(16)/x )*exp(2*x + 8)*exp((6*exp(-exp(16)/x))/x)))/x^2,x)