\(\int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} (e^{4+x} (-12 e^{16}+12 x)+e^{4+\frac {e^{16}}{x}+x} (4 x^2+4 x^3)))}{x^2} \, dx\) [748]

Optimal result

Integrand size = 148, antiderivative size = 28 \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\left (e^{4+x}+2 e^{-\frac {3 e^{-\frac {e^{16}}{x}}}{x}} x\right )^2 \] Output:

(exp(4+x)+2*x/exp(3/x/exp(exp(16)/x)))^2
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}+2 x\right )^2 \] Input:

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]
 

Output:

(E^(4 + 3/(E^(E^16/x)*x) + x) + 2*x)^2/E^(6/(E^(E^16/x)*x))
 

Rubi [F]

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

\(\displaystyle -2 \left (-12 \text {Subst}\left (\int \frac {e^{-6 e^{-e^{16} x} x-e^{16} x+16}}{x}dx,x,\frac {1}{x}\right )+6 \int \frac {e^{x+20-\frac {3 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}}}{x^2}dx-12 \int e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}}dx-2 \int e^{x+4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}}dx-6 \int \frac {e^{x+4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}-\frac {e^{16}}{x}}}{x}dx-4 \int e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} xdx-2 \int e^{x+4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}} xdx-\frac {1}{2} e^{2 x+8}\right )\)

Input:

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]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 83.61 (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	\(-\frac {{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}} \left (-x \,{\mathrm e}^{2 x +8} {\mathrm e}^{\frac {{\mathrm e}^{16}}{x}} {\mathrm e}^{\frac {6 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}}{x}}-4 \,{\mathrm e}^{\frac {{\mathrm e}^{16}}{x}} {\mathrm e}^{4+x} {\mathrm e}^{\frac {3 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}}{x}} x^{2}-4 x^{3} {\mathrm e}^{\frac {{\mathrm e}^{16}}{x}}\right ) {\mathrm e}^{-\frac {6 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}}{x}}}{x}\)	\(109\)

Input:

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)
 

Output:

exp(2*x+8)+4*x*exp((x^2-3*exp(-exp(16)/x)+4*x)/x)+4*x^2*exp(-6/x*exp(-exp( 
16)/x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (24) = 48\).

Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.68 \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx={\left (4 \, x^{2} e^{\left (\frac {2 \, e^{16}}{x}\right )} + 4 \, x e^{\left (\frac {x^{2} + 4 \, x + e^{16}}{x} + \frac {e^{16}}{x} + \frac {3 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )} + e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x + e^{16}\right )}}{x} + \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}\right )} e^{\left (-\frac {2 \, e^{16}}{x} - \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )} \] Input:

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="fricas")
 

Output:

(4*x^2*e^(2*e^16/x) + 4*x*e^((x^2 + 4*x + e^16)/x + e^16/x + 3*e^(-e^16/x) 
/x) + e^(2*(x^2 + 4*x + e^16)/x + 6*e^(-e^16/x)/x))*e^(-2*e^16/x - 6*e^(-e 
^16/x)/x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).

Time = 106.55 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=4 x^{2} e^{- \frac {6 e^{- \frac {e^{16}}{x}}}{x}} + 4 x e^{- \frac {3 e^{- \frac {e^{16}}{x}}}{x}} e^{x + 4} + e^{2 x + 8} \] Input:

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)
 

Output:

4*x**2*exp(-6*exp(-exp(16)/x)/x) + 4*x*exp(-3*exp(-exp(16)/x)/x)*exp(x + 4 
) + exp(2*x + 8)
 

Maxima [F]

\[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\int { \frac {2 \, {\left (4 \, x^{3} e^{\left (\frac {e^{16}}{x}\right )} + x^{2} e^{\left (2 \, x + \frac {e^{16}}{x} + \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x} + 8\right )} + 12 \, x^{2} - 12 \, x e^{16} + 2 \, {\left ({\left (x^{3} + x^{2}\right )} e^{\left (x + \frac {e^{16}}{x} + 4\right )} + 3 \, {\left (x - e^{16}\right )} e^{\left (x + 4\right )}\right )} e^{\left (\frac {3 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}\right )} e^{\left (-\frac {e^{16}}{x} - \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}}{x^{2}} \,d x } \] Input:

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="maxima")
 

Output:

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)
 

Giac [F]

\[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\int { \frac {2 \, {\left (4 \, x^{3} e^{\left (\frac {e^{16}}{x}\right )} + x^{2} e^{\left (2 \, x + \frac {e^{16}}{x} + \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x} + 8\right )} + 12 \, x^{2} - 12 \, x e^{16} + 2 \, {\left ({\left (x^{3} + x^{2}\right )} e^{\left (x + \frac {e^{16}}{x} + 4\right )} + 3 \, {\left (x - e^{16}\right )} e^{\left (x + 4\right )}\right )} e^{\left (\frac {3 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}\right )} e^{\left (-\frac {e^{16}}{x} - \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}}{x^{2}} \,d x } \] Input:

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="giac")
 

Output:

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)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\int \frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}\,{\mathrm {e}}^{-\frac {6\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}}{x}}\,\left (8\,x^3\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x}}-24\,x\,{\mathrm {e}}^{16}+{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}}{x}}\,\left ({\mathrm {e}}^{x+4}\,\left (12\,x-12\,{\mathrm {e}}^{16}\right )+{\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x}}\,{\mathrm {e}}^{x+4}\,\left (4\,x^3+4\,x^2\right )\right )+24\,x^2+2\,x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x}}\,{\mathrm {e}}^{2\,x+8}\,{\mathrm {e}}^{\frac {6\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}}{x}}\right )}{x^2} \,d x \] Input:

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)
 

Output:

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)
 

Reduce [B] (verification not implemented)

Time = 32.01 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.64 \[ \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx=\frac {e^{\frac {2 e^{\frac {e^{16}}{x}} x^{2}+6}{e^{\frac {e^{16}}{x}} x}} e^{8}+4 e^{\frac {e^{\frac {e^{16}}{x}} x^{2}+3}{e^{\frac {e^{16}}{x}} x}} e^{4} x +4 x^{2}}{e^{\frac {6}{e^{\frac {e^{16}}{x}} x}}} \] Input:

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)
 

Output:

(e**((2*e**(e**16/x)*x**2 + 6)/(e**(e**16/x)*x))*e**8 + 4*e**((e**(e**16/x 
)*x**2 + 3)/(e**(e**16/x)*x))*e**4*x + 4*x**2)/e**(6/(e**(e**16/x)*x))