\(\int \frac {8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5+e^{e^x} (8 x^3-8 e^x x^4+e^{\frac {1+16 x^3}{16 x^2}} (1-8 x^3+8 e^x x^3))}{8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5} \, dx\) [498]

Optimal result

Integrand size = 151, antiderivative size = 29 \[ \int \frac {8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5+e^{e^x} \left (8 x^3-8 e^x x^4+e^{\frac {1+16 x^3}{16 x^2}} \left (1-8 x^3+8 e^x x^3\right )\right )}{8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5} \, dx=-\frac {7}{4}+x-\frac {e^{e^x}}{-e^{\frac {1}{16 x^2}+x}+x} \] Output:

x-7/4-exp(exp(x))/(x-exp(1/16/x^2+x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5+e^{e^x} \left (8 x^3-8 e^x x^4+e^{\frac {1+16 x^3}{16 x^2}} \left (1-8 x^3+8 e^x x^3\right )\right )}{8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5} \, dx=\frac {1}{8} \left (\frac {8 e^{e^x}}{e^{\frac {1}{16 x^2}+x}-x}+8 x\right ) \] Input:

Integrate[(8*E^((1 + 16*x^3)/(8*x^2))*x^3 - 16*E^((1 + 16*x^3)/(16*x^2))*x 
^4 + 8*x^5 + E^E^x*(8*x^3 - 8*E^x*x^4 + E^((1 + 16*x^3)/(16*x^2))*(1 - 8*x 
^3 + 8*E^x*x^3)))/(8*E^((1 + 16*x^3)/(8*x^2))*x^3 - 16*E^((1 + 16*x^3)/(16 
*x^2))*x^4 + 8*x^5),x]
 

Output:

((8*E^E^x)/(E^(1/(16*x^2) + x) - x) + 8*x)/8
 

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.

\(\displaystyle \int \frac {8 x^5+8 e^{\frac {16 x^3+1}{8 x^2}} x^3-16 e^{\frac {16 x^3+1}{16 x^2}} x^4+e^{e^x} \left (-8 e^x x^4+8 x^3+e^{\frac {16 x^3+1}{16 x^2}} \left (8 e^x x^3-8 x^3+1\right )\right )}{8 x^5+8 e^{\frac {16 x^3+1}{8 x^2}} x^3-16 e^{\frac {16 x^3+1}{16 x^2}} x^4} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {8 x^5+8 e^{\frac {16 x^3+1}{8 x^2}} x^3-16 e^{\frac {16 x^3+1}{16 x^2}} x^4+e^{e^x} \left (-8 e^x x^4+8 x^3+e^{\frac {16 x^3+1}{16 x^2}} \left (8 e^x x^3-8 x^3+1\right )\right )}{8 \left (e^{\frac {1}{16 x^2}+x}-x\right )^2 x^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \int \frac {8 x^5-16 e^{\frac {16 x^3+1}{16 x^2}} x^4+8 e^{\frac {16 x^3+1}{8 x^2}} x^3+e^{e^x} \left (-8 e^x x^4+8 x^3+e^{\frac {16 x^3+1}{16 x^2}} \left (8 e^x x^3-8 x^3+1\right )\right )}{\left (e^{x+\frac {1}{16 x^2}}-x\right )^2 x^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{8} \int \left (8 e^{-\frac {1}{16 x^2}} \left (e^{e^x}+e^{\frac {1}{16 x^2}}\right )-\frac {e^{e^x} \left (8 x^3-8 x^2-1\right )}{\left (e^{x+\frac {1}{16 x^2}}-x\right )^2 x^2}-\frac {e^{e^x-\frac {1}{16 x^2}} \left (-8 x^4+8 e^{\frac {1}{16 x^2}} x^3-e^{\frac {1}{16 x^2}}\right )}{\left (e^{x+\frac {1}{16 x^2}}-x\right ) x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{8} \left (8 \int e^{e^x-\frac {1}{16 x^2}}dx+8 \int \frac {e^{e^x}}{\left (e^{x+\frac {1}{16 x^2}}-x\right )^2}dx-8 \int \frac {e^{e^x}}{e^{x+\frac {1}{16 x^2}}-x}dx+\int \frac {e^{e^x}}{\left (e^{x+\frac {1}{16 x^2}}-x\right )^2 x^2}dx-8 \int \frac {e^{e^x} x}{\left (e^{x+\frac {1}{16 x^2}}-x\right )^2}dx+8 \int \frac {e^{e^x-\frac {1}{16 x^2}} x}{e^{x+\frac {1}{16 x^2}}-x}dx+\int \frac {e^{e^x}}{\left (e^{x+\frac {1}{16 x^2}}-x\right ) x^3}dx+8 x\right )\)

Input:

Int[(8*E^((1 + 16*x^3)/(8*x^2))*x^3 - 16*E^((1 + 16*x^3)/(16*x^2))*x^4 + 8 
*x^5 + E^E^x*(8*x^3 - 8*E^x*x^4 + E^((1 + 16*x^3)/(16*x^2))*(1 - 8*x^3 + 8 
*E^x*x^3)))/(8*E^((1 + 16*x^3)/(8*x^2))*x^3 - 16*E^((1 + 16*x^3)/(16*x^2)) 
*x^4 + 8*x^5),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(22)=44\).

Time = 3.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69

method	result	size
parallelrisch	\(\frac {8 x^{2}-8 x \,{\mathrm e}^{\frac {16 x^{3}+1}{16 x^{2}}}-8 \,{\mathrm e}^{{\mathrm e}^{x}}}{8 x -8 \,{\mathrm e}^{\frac {16 x^{3}+1}{16 x^{2}}}}\)	\(49\)

Input:

int((((8*exp(x)*x^3-8*x^3+1)*exp(1/16*(16*x^3+1)/x^2)-8*exp(x)*x^4+8*x^3)* 
exp(exp(x))+8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3+1)/x^ 
2)+8*x^5)/(8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3+1)/x^2 
)+8*x^5),x,method=_RETURNVERBOSE)
 

Output:

1/8*(8*x^2-8*x*exp(1/16*(16*x^3+1)/x^2)-8*exp(exp(x)))/(x-exp(1/16*(16*x^3 
+1)/x^2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).

Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5+e^{e^x} \left (8 x^3-8 e^x x^4+e^{\frac {1+16 x^3}{16 x^2}} \left (1-8 x^3+8 e^x x^3\right )\right )}{8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5} \, dx=\frac {x^{2} - x e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )} - e^{\left (e^{x}\right )}}{x - e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )}} \] Input:

integrate((((8*exp(x)*x^3-8*x^3+1)*exp(1/16*(16*x^3+1)/x^2)-8*exp(x)*x^4+8 
*x^3)*exp(exp(x))+8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3 
+1)/x^2)+8*x^5)/(8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3+ 
1)/x^2)+8*x^5),x, algorithm="fricas")
 

Output:

(x^2 - x*e^(1/16*(16*x^3 + 1)/x^2) - e^(e^x))/(x - e^(1/16*(16*x^3 + 1)/x^ 
2))
 

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5+e^{e^x} \left (8 x^3-8 e^x x^4+e^{\frac {1+16 x^3}{16 x^2}} \left (1-8 x^3+8 e^x x^3\right )\right )}{8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5} \, dx=x - \frac {e^{e^{x}}}{x - e^{\frac {x^{3} + \frac {1}{16}}{x^{2}}}} \] Input:

integrate((((8*exp(x)*x**3-8*x**3+1)*exp(1/16*(16*x**3+1)/x**2)-8*exp(x)*x 
**4+8*x**3)*exp(exp(x))+8*x**3*exp(1/16*(16*x**3+1)/x**2)**2-16*x**4*exp(1 
/16*(16*x**3+1)/x**2)+8*x**5)/(8*x**3*exp(1/16*(16*x**3+1)/x**2)**2-16*x** 
4*exp(1/16*(16*x**3+1)/x**2)+8*x**5),x)
 

Output:

x - exp(exp(x))/(x - exp((x**3 + 1/16)/x**2))
 

Maxima [F]

\[ \int \frac {8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5+e^{e^x} \left (8 x^3-8 e^x x^4+e^{\frac {1+16 x^3}{16 x^2}} \left (1-8 x^3+8 e^x x^3\right )\right )}{8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5} \, dx=\int { \frac {8 \, x^{5} - 16 \, x^{4} e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )} + 8 \, x^{3} e^{\left (\frac {16 \, x^{3} + 1}{8 \, x^{2}}\right )} - {\left (8 \, x^{4} e^{x} - 8 \, x^{3} - {\left (8 \, x^{3} e^{x} - 8 \, x^{3} + 1\right )} e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )}\right )} e^{\left (e^{x}\right )}}{8 \, {\left (x^{5} - 2 \, x^{4} e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )} + x^{3} e^{\left (\frac {16 \, x^{3} + 1}{8 \, x^{2}}\right )}\right )}} \,d x } \] Input:

integrate((((8*exp(x)*x^3-8*x^3+1)*exp(1/16*(16*x^3+1)/x^2)-8*exp(x)*x^4+8 
*x^3)*exp(exp(x))+8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3 
+1)/x^2)+8*x^5)/(8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3+ 
1)/x^2)+8*x^5),x, algorithm="maxima")
 

Output:

x + 1/8*integrate(((8*x^3*e^(2*x) - (8*x^3 - 1)*e^x)*e^(1/16/x^2 + e^x) - 
8*(x^4*e^x - x^3)*e^(e^x))/(x^5 - 2*x^4*e^(x + 1/16/x^2) + x^3*e^(2*x + 1/ 
8/x^2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (22) = 44\).

Time = 0.19 (sec) , antiderivative size = 326, normalized size of antiderivative = 11.24 \[ \int \frac {8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5+e^{e^x} \left (8 x^3-8 e^x x^4+e^{\frac {1+16 x^3}{16 x^2}} \left (1-8 x^3+8 e^x x^3\right )\right )}{8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5} \, dx=\frac {8 \, x^{5} e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )} - 8 \, x^{4} e^{\left (\frac {16 \, x^{3} + 1}{8 \, x^{2}}\right )} - 8 \, x^{4} e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )} + 8 \, x^{3} e^{\left (x + \frac {16 \, x^{3} + 16 \, x^{2} e^{x} + 1}{16 \, x^{2}}\right )} - 8 \, x^{3} e^{\left (x + \frac {16 \, x^{3} + 1}{16 \, x^{2}} + e^{x}\right )} - 8 \, x^{3} e^{\left (\frac {16 \, x^{3} + 16 \, x^{2} e^{x} + 1}{16 \, x^{2}}\right )} + 8 \, x^{3} e^{\left (\frac {16 \, x^{3} + 1}{8 \, x^{2}}\right )} + 8 \, x^{2} e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}} + e^{x}\right )} - x^{2} e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )} + x e^{\left (\frac {16 \, x^{3} + 1}{8 \, x^{2}}\right )} + e^{\left (\frac {16 \, x^{3} + 16 \, x^{2} e^{x} + 1}{16 \, x^{2}}\right )}}{8 \, x^{4} e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )} - 8 \, x^{3} e^{\left (\frac {16 \, x^{3} + 1}{8 \, x^{2}}\right )} - 8 \, x^{3} e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )} + 8 \, x^{2} e^{\left (\frac {16 \, x^{3} + 1}{8 \, x^{2}}\right )} - x e^{\left (\frac {16 \, x^{3} + 1}{16 \, x^{2}}\right )} + e^{\left (\frac {16 \, x^{3} + 1}{8 \, x^{2}}\right )}} \] Input:

integrate((((8*exp(x)*x^3-8*x^3+1)*exp(1/16*(16*x^3+1)/x^2)-8*exp(x)*x^4+8 
*x^3)*exp(exp(x))+8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3 
+1)/x^2)+8*x^5)/(8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3+ 
1)/x^2)+8*x^5),x, algorithm="giac")
 

Output:

(8*x^5*e^(1/16*(16*x^3 + 1)/x^2) - 8*x^4*e^(1/8*(16*x^3 + 1)/x^2) - 8*x^4* 
e^(1/16*(16*x^3 + 1)/x^2) + 8*x^3*e^(x + 1/16*(16*x^3 + 16*x^2*e^x + 1)/x^ 
2) - 8*x^3*e^(x + 1/16*(16*x^3 + 1)/x^2 + e^x) - 8*x^3*e^(1/16*(16*x^3 + 1 
6*x^2*e^x + 1)/x^2) + 8*x^3*e^(1/8*(16*x^3 + 1)/x^2) + 8*x^2*e^(1/16*(16*x 
^3 + 1)/x^2 + e^x) - x^2*e^(1/16*(16*x^3 + 1)/x^2) + x*e^(1/8*(16*x^3 + 1) 
/x^2) + e^(1/16*(16*x^3 + 16*x^2*e^x + 1)/x^2))/(8*x^4*e^(1/16*(16*x^3 + 1 
)/x^2) - 8*x^3*e^(1/8*(16*x^3 + 1)/x^2) - 8*x^3*e^(1/16*(16*x^3 + 1)/x^2) 
+ 8*x^2*e^(1/8*(16*x^3 + 1)/x^2) - x*e^(1/16*(16*x^3 + 1)/x^2) + e^(1/8*(1 
6*x^3 + 1)/x^2))
 

Mupad [B] (verification not implemented)

Time = 2.57 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5+e^{e^x} \left (8 x^3-8 e^x x^4+e^{\frac {1+16 x^3}{16 x^2}} \left (1-8 x^3+8 e^x x^3\right )\right )}{8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5} \, dx=x-\frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{x-{\mathrm {e}}^{x+\frac {1}{16\,x^2}}} \] Input:

int((8*x^3*exp((2*(x^3 + 1/16))/x^2) - 16*x^4*exp((x^3 + 1/16)/x^2) + exp( 
exp(x))*(8*x^3 - 8*x^4*exp(x) + exp((x^3 + 1/16)/x^2)*(8*x^3*exp(x) - 8*x^ 
3 + 1)) + 8*x^5)/(8*x^3*exp((2*(x^3 + 1/16))/x^2) - 16*x^4*exp((x^3 + 1/16 
)/x^2) + 8*x^5),x)
 

Output:

x - exp(exp(x))/(x - exp(x + 1/(16*x^2)))
 

Reduce [F]

\[ \int \frac {8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5+e^{e^x} \left (8 x^3-8 e^x x^4+e^{\frac {1+16 x^3}{16 x^2}} \left (1-8 x^3+8 e^x x^3\right )\right )}{8 e^{\frac {1+16 x^3}{8 x^2}} x^3-16 e^{\frac {1+16 x^3}{16 x^2}} x^4+8 x^5} \, dx=\int \frac {\left (\left (8 \,{\mathrm e}^{x} x^{3}-8 x^{3}+1\right ) {\mathrm e}^{\frac {16 x^{3}+1}{16 x^{2}}}-8 \,{\mathrm e}^{x} x^{4}+8 x^{3}\right ) {\mathrm e}^{{\mathrm e}^{x}}+8 x^{3} \left ({\mathrm e}^{\frac {16 x^{3}+1}{16 x^{2}}}\right )^{2}-16 x^{4} {\mathrm e}^{\frac {16 x^{3}+1}{16 x^{2}}}+8 x^{5}}{8 x^{3} \left ({\mathrm e}^{\frac {16 x^{3}+1}{16 x^{2}}}\right )^{2}-16 x^{4} {\mathrm e}^{\frac {16 x^{3}+1}{16 x^{2}}}+8 x^{5}}d x \] Input:

int((((8*exp(x)*x^3-8*x^3+1)*exp(1/16*(16*x^3+1)/x^2)-8*exp(x)*x^4+8*x^3)* 
exp(exp(x))+8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3+1)/x^ 
2)+8*x^5)/(8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3+1)/x^2 
)+8*x^5),x)
 

Output:

int((((8*exp(x)*x^3-8*x^3+1)*exp(1/16*(16*x^3+1)/x^2)-8*exp(x)*x^4+8*x^3)* 
exp(exp(x))+8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3+1)/x^ 
2)+8*x^5)/(8*x^3*exp(1/16*(16*x^3+1)/x^2)^2-16*x^4*exp(1/16*(16*x^3+1)/x^2 
)+8*x^5),x)