\(\int \frac {e^{\frac {e^{21+x}-3 x^2-e^x x^2}{3 x+e^x x}} (e^{21+x} (18+6 e^x-18 x)+54 x^2+36 e^x x^2+6 e^{2 x} x^2)+e^{\frac {2 (e^{21+x}-3 x^2-e^x x^2)}{3 x+e^x x}} (-18 x^2-12 e^x x^2-2 e^{2 x} x^2+e^{21+x} (-6-2 e^x+6 x))}{9 x^2+6 e^x x^2+e^{2 x} x^2} \, dx\) [602]

Optimal result

Integrand size = 176, antiderivative size = 26 \[ \int \frac {e^{\frac {e^{21+x}-3 x^2-e^x x^2}{3 x+e^x x}} \left (e^{21+x} \left (18+6 e^x-18 x\right )+54 x^2+36 e^x x^2+6 e^{2 x} x^2\right )+e^{\frac {2 \left (e^{21+x}-3 x^2-e^x x^2\right )}{3 x+e^x x}} \left (-18 x^2-12 e^x x^2-2 e^{2 x} x^2+e^{21+x} \left (-6-2 e^x+6 x\right )\right )}{9 x^2+6 e^x x^2+e^{2 x} x^2} \, dx=\left (-3+e^{\frac {e^{21+x}}{\left (3+e^x\right ) x}-x}\right )^2 \] Output:

(-3+exp(exp(x+21)/x/(3+exp(x))-x))^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(26)=52\).

Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27 \[ \int \frac {e^{\frac {e^{21+x}-3 x^2-e^x x^2}{3 x+e^x x}} \left (e^{21+x} \left (18+6 e^x-18 x\right )+54 x^2+36 e^x x^2+6 e^{2 x} x^2\right )+e^{\frac {2 \left (e^{21+x}-3 x^2-e^x x^2\right )}{3 x+e^x x}} \left (-18 x^2-12 e^x x^2-2 e^{2 x} x^2+e^{21+x} \left (-6-2 e^x+6 x\right )\right )}{9 x^2+6 e^x x^2+e^{2 x} x^2} \, dx=-e^{\frac {e^{21} \left (1-\frac {6}{3+e^x}\right )}{x}-2 x} \left (-e^{\frac {e^{21}}{x}}+6 e^{\frac {3 e^{21}}{\left (3+e^x\right ) x}+x}\right ) \] Input:

Integrate[(E^((E^(21 + x) - 3*x^2 - E^x*x^2)/(3*x + E^x*x))*(E^(21 + x)*(1 
8 + 6*E^x - 18*x) + 54*x^2 + 36*E^x*x^2 + 6*E^(2*x)*x^2) + E^((2*(E^(21 + 
x) - 3*x^2 - E^x*x^2))/(3*x + E^x*x))*(-18*x^2 - 12*E^x*x^2 - 2*E^(2*x)*x^ 
2 + E^(21 + x)*(-6 - 2*E^x + 6*x)))/(9*x^2 + 6*E^x*x^2 + E^(2*x)*x^2),x]
 

Output:

-(E^((E^21*(1 - 6/(3 + E^x)))/x - 2*x)*(-E^(E^21/x) + 6*E^((3*E^21)/((3 + 
E^x)*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.

\(\displaystyle \int \frac {\left (-12 e^x x^2-2 e^{2 x} x^2-18 x^2+e^{x+21} \left (6 x-2 e^x-6\right )\right ) \exp \left (\frac {2 \left (-e^x x^2-3 x^2+e^{x+21}\right )}{e^x x+3 x}\right )+e^{\frac {-e^x x^2-3 x^2+e^{x+21}}{e^x x+3 x}} \left (36 e^x x^2+6 e^{2 x} x^2+54 x^2+e^{x+21} \left (-18 x+6 e^x+18\right )\right )}{6 e^x x^2+e^{2 x} x^2+9 x^2} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {2 e^{\frac {e^{x+21}}{e^x x+3 x}-2 x} \left (-6 e^x x^2-e^{2 x} x^2-9 x^2+3 e^{x+21} x-3 e^{x+21}-e^{2 x+21}\right ) \left (e^{\frac {e^{x+21}}{e^x x+3 x}}-3 \exp \left (\frac {e^x x^2}{e^x x+3 x}+\frac {3 x^2}{e^x x+3 x}\right )\right )}{\left (e^x+3\right )^2 x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int -\frac {e^{\frac {e^{x+21}}{e^x x+3 x}-2 x} \left (e^{\frac {e^{x+21}}{e^x x+3 x}}-3 \exp \left (\frac {e^x x^2}{e^x x+3 x}+\frac {3 x^2}{e^x x+3 x}\right )\right ) \left (6 e^x x^2+e^{2 x} x^2+9 x^2-3 e^{x+21} x+3 e^{x+21}+e^{2 x+21}\right )}{\left (3+e^x\right )^2 x^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {e^{\frac {e^{x+21}}{e^x x+3 x}-2 x} \left (e^{\frac {e^{x+21}}{e^x x+3 x}}-3 \exp \left (\frac {e^x x^2}{e^x x+3 x}+\frac {3 x^2}{e^x x+3 x}\right )\right ) \left (6 e^x x^2+e^{2 x} x^2+9 x^2-3 e^{x+21} x+3 e^{x+21}+e^{2 x+21}\right )}{\left (3+e^x\right )^2 x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {e^{\frac {2 e^{x+21}}{e^x x+3 x}-2 x} \left (6 e^x x^2+e^{2 x} x^2+9 x^2-3 e^{x+21} x+3 e^{x+21}+e^{2 x+21}\right )}{\left (3+e^x\right )^2 x^2}-\frac {3 e^{\frac {e^{x+21}}{e^x x+3 x}-x} \left (6 e^x x^2+e^{2 x} x^2+9 x^2-3 e^{x+21} x+3 e^{x+21}+e^{2 x+21}\right )}{\left (3+e^x\right )^2 x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (-3 \int \frac {e^{-x+\frac {e^{x+21}}{e^x x+3 x}+21}}{x^2}dx+\int \frac {e^{-2 x+\frac {2 e^{x+21}}{e^x x+3 x}+21}}{x^2}dx+9 \int \frac {e^{-x+\frac {e^{x+21}}{e^x x+3 x}+21}}{\left (3+e^x\right ) x^2}dx-3 \int \frac {e^{-2 x+\frac {2 e^{x+21}}{e^x x+3 x}+21}}{\left (3+e^x\right ) x^2}dx-3 \int e^{\frac {e^{x+21}}{e^x x+3 x}-x}dx+\int e^{\frac {2 e^{x+21}}{e^x x+3 x}-2 x}dx-27 \int \frac {e^{-x+\frac {e^{x+21}}{e^x x+3 x}+21}}{\left (3+e^x\right )^2 x}dx+9 \int \frac {e^{-2 x+\frac {2 e^{x+21}}{e^x x+3 x}+21}}{\left (3+e^x\right )^2 x}dx+9 \int \frac {e^{-x+\frac {e^{x+21}}{e^x x+3 x}+21}}{\left (3+e^x\right ) x}dx-3 \int \frac {e^{-2 x+\frac {2 e^{x+21}}{e^x x+3 x}+21}}{\left (3+e^x\right ) x}dx\right )\)

Input:

Int[(E^((E^(21 + x) - 3*x^2 - E^x*x^2)/(3*x + E^x*x))*(E^(21 + x)*(18 + 6* 
E^x - 18*x) + 54*x^2 + 36*E^x*x^2 + 6*E^(2*x)*x^2) + E^((2*(E^(21 + x) - 3 
*x^2 - E^x*x^2))/(3*x + E^x*x))*(-18*x^2 - 12*E^x*x^2 - 2*E^(2*x)*x^2 + E^ 
(21 + x)*(-6 - 2*E^x + 6*x)))/(9*x^2 + 6*E^x*x^2 + E^(2*x)*x^2),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(23)=46\).

Time = 9.42 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.38

method	result	size
parallelrisch	\({\mathrm e}^{\frac {-2 \,{\mathrm e}^{x} x^{2}-6 x^{2}+2 \,{\mathrm e}^{x +21}}{x \left (3+{\mathrm e}^{x}\right )}}-6 \,{\mathrm e}^{\frac {{\mathrm e}^{x +21}-{\mathrm e}^{x} x^{2}-3 x^{2}}{x \left (3+{\mathrm e}^{x}\right )}}\)	\(62\)
risch	\({\mathrm e}^{-\frac {2 \left (-{\mathrm e}^{x +21}+{\mathrm e}^{x} x^{2}+3 x^{2}\right )}{x \left (3+{\mathrm e}^{x}\right )}}-6 \,{\mathrm e}^{-\frac {-{\mathrm e}^{x +21}+{\mathrm e}^{x} x^{2}+3 x^{2}}{x \left (3+{\mathrm e}^{x}\right )}}\)	\(64\)
norman	\(\frac {{\mathrm e}^{x} x \,{\mathrm e}^{\frac {-2 \,{\mathrm e}^{x} x^{2}-6 x^{2}+2 \,{\mathrm e}^{x +21}}{{\mathrm e}^{x} x +3 x}}-18 x \,{\mathrm e}^{\frac {{\mathrm e}^{x} {\mathrm e}^{21}-{\mathrm e}^{x} x^{2}-3 x^{2}}{{\mathrm e}^{x} x +3 x}}+3 x \,{\mathrm e}^{\frac {-2 \,{\mathrm e}^{x} x^{2}-6 x^{2}+2 \,{\mathrm e}^{x +21}}{{\mathrm e}^{x} x +3 x}}-6 \,{\mathrm e}^{x} x \,{\mathrm e}^{\frac {{\mathrm e}^{x} {\mathrm e}^{21}-{\mathrm e}^{x} x^{2}-3 x^{2}}{{\mathrm e}^{x} x +3 x}}}{x \left (3+{\mathrm e}^{x}\right )}\)	\(151\)

Input:

int((((-2*exp(x)+6*x-6)*exp(x+21)-2*exp(x)^2*x^2-12*exp(x)*x^2-18*x^2)*exp 
((exp(x+21)-exp(x)*x^2-3*x^2)/(exp(x)*x+3*x))^2+((6*exp(x)-18*x+18)*exp(x+ 
21)+6*exp(x)^2*x^2+36*exp(x)*x^2+54*x^2)*exp((exp(x+21)-exp(x)*x^2-3*x^2)/ 
(exp(x)*x+3*x)))/(exp(x)^2*x^2+6*exp(x)*x^2+9*x^2),x,method=_RETURNVERBOSE 
)
 

Output:

exp((exp(x+21)-exp(x)*x^2-3*x^2)/x/(3+exp(x)))^2-6*exp((exp(x+21)-exp(x)*x 
^2-3*x^2)/x/(3+exp(x)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (23) = 46\).

Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.04 \[ \int \frac {e^{\frac {e^{21+x}-3 x^2-e^x x^2}{3 x+e^x x}} \left (e^{21+x} \left (18+6 e^x-18 x\right )+54 x^2+36 e^x x^2+6 e^{2 x} x^2\right )+e^{\frac {2 \left (e^{21+x}-3 x^2-e^x x^2\right )}{3 x+e^x x}} \left (-18 x^2-12 e^x x^2-2 e^{2 x} x^2+e^{21+x} \left (-6-2 e^x+6 x\right )\right )}{9 x^2+6 e^x x^2+e^{2 x} x^2} \, dx=-6 \, e^{\left (-\frac {3 \, x^{2} e^{21} + {\left (x^{2} - e^{21}\right )} e^{\left (x + 21\right )}}{3 \, x e^{21} + x e^{\left (x + 21\right )}}\right )} + e^{\left (-\frac {2 \, {\left (3 \, x^{2} e^{21} + {\left (x^{2} - e^{21}\right )} e^{\left (x + 21\right )}\right )}}{3 \, x e^{21} + x e^{\left (x + 21\right )}}\right )} \] Input:

integrate((((-2*exp(x)+6*x-6)*exp(x+21)-2*exp(x)^2*x^2-12*exp(x)*x^2-18*x^ 
2)*exp((exp(x+21)-exp(x)*x^2-3*x^2)/(exp(x)*x+3*x))^2+((6*exp(x)-18*x+18)* 
exp(x+21)+6*exp(x)^2*x^2+36*exp(x)*x^2+54*x^2)*exp((exp(x+21)-exp(x)*x^2-3 
*x^2)/(exp(x)*x+3*x)))/(exp(x)^2*x^2+6*exp(x)*x^2+9*x^2),x, algorithm="fri 
cas")
 

Output:

-6*e^(-(3*x^2*e^21 + (x^2 - e^21)*e^(x + 21))/(3*x*e^21 + x*e^(x + 21))) + 
 e^(-2*(3*x^2*e^21 + (x^2 - e^21)*e^(x + 21))/(3*x*e^21 + x*e^(x + 21)))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (17) = 34\).

Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {e^{\frac {e^{21+x}-3 x^2-e^x x^2}{3 x+e^x x}} \left (e^{21+x} \left (18+6 e^x-18 x\right )+54 x^2+36 e^x x^2+6 e^{2 x} x^2\right )+e^{\frac {2 \left (e^{21+x}-3 x^2-e^x x^2\right )}{3 x+e^x x}} \left (-18 x^2-12 e^x x^2-2 e^{2 x} x^2+e^{21+x} \left (-6-2 e^x+6 x\right )\right )}{9 x^2+6 e^x x^2+e^{2 x} x^2} \, dx=e^{\frac {2 \left (- x^{2} e^{x} - 3 x^{2} + e^{21} e^{x}\right )}{x e^{x} + 3 x}} - 6 e^{\frac {- x^{2} e^{x} - 3 x^{2} + e^{21} e^{x}}{x e^{x} + 3 x}} \] Input:

integrate((((-2*exp(x)+6*x-6)*exp(x+21)-2*exp(x)**2*x**2-12*exp(x)*x**2-18 
*x**2)*exp((exp(x+21)-exp(x)*x**2-3*x**2)/(exp(x)*x+3*x))**2+((6*exp(x)-18 
*x+18)*exp(x+21)+6*exp(x)**2*x**2+36*exp(x)*x**2+54*x**2)*exp((exp(x+21)-e 
xp(x)*x**2-3*x**2)/(exp(x)*x+3*x)))/(exp(x)**2*x**2+6*exp(x)*x**2+9*x**2), 
x)
 

Output:

exp(2*(-x**2*exp(x) - 3*x**2 + exp(21)*exp(x))/(x*exp(x) + 3*x)) - 6*exp(( 
-x**2*exp(x) - 3*x**2 + exp(21)*exp(x))/(x*exp(x) + 3*x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (23) = 46\).

Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \frac {e^{\frac {e^{21+x}-3 x^2-e^x x^2}{3 x+e^x x}} \left (e^{21+x} \left (18+6 e^x-18 x\right )+54 x^2+36 e^x x^2+6 e^{2 x} x^2\right )+e^{\frac {2 \left (e^{21+x}-3 x^2-e^x x^2\right )}{3 x+e^x x}} \left (-18 x^2-12 e^x x^2-2 e^{2 x} x^2+e^{21+x} \left (-6-2 e^x+6 x\right )\right )}{9 x^2+6 e^x x^2+e^{2 x} x^2} \, dx=-{\left (6 \, e^{\left (\frac {x e^{x}}{e^{x} + 3} + \frac {3 \, x}{e^{x} + 3} + \frac {e^{\left (x + 21\right )}}{x e^{x} + 3 \, x}\right )} - e^{\left (\frac {2 \, e^{\left (x + 21\right )}}{x e^{x} + 3 \, x}\right )}\right )} e^{\left (-\frac {2 \, x e^{x}}{e^{x} + 3} - \frac {6 \, x}{e^{x} + 3}\right )} \] Input:

integrate((((-2*exp(x)+6*x-6)*exp(x+21)-2*exp(x)^2*x^2-12*exp(x)*x^2-18*x^ 
2)*exp((exp(x+21)-exp(x)*x^2-3*x^2)/(exp(x)*x+3*x))^2+((6*exp(x)-18*x+18)* 
exp(x+21)+6*exp(x)^2*x^2+36*exp(x)*x^2+54*x^2)*exp((exp(x+21)-exp(x)*x^2-3 
*x^2)/(exp(x)*x+3*x)))/(exp(x)^2*x^2+6*exp(x)*x^2+9*x^2),x, algorithm="max 
ima")
 

Output:

-(6*e^(x*e^x/(e^x + 3) + 3*x/(e^x + 3) + e^(x + 21)/(x*e^x + 3*x)) - e^(2* 
e^(x + 21)/(x*e^x + 3*x)))*e^(-2*x*e^x/(e^x + 3) - 6*x/(e^x + 3))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {e^{21+x}-3 x^2-e^x x^2}{3 x+e^x x}} \left (e^{21+x} \left (18+6 e^x-18 x\right )+54 x^2+36 e^x x^2+6 e^{2 x} x^2\right )+e^{\frac {2 \left (e^{21+x}-3 x^2-e^x x^2\right )}{3 x+e^x x}} \left (-18 x^2-12 e^x x^2-2 e^{2 x} x^2+e^{21+x} \left (-6-2 e^x+6 x\right )\right )}{9 x^2+6 e^x x^2+e^{2 x} x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((((-2*exp(x)+6*x-6)*exp(x+21)-2*exp(x)^2*x^2-12*exp(x)*x^2-18*x^ 
2)*exp((exp(x+21)-exp(x)*x^2-3*x^2)/(exp(x)*x+3*x))^2+((6*exp(x)-18*x+18)* 
exp(x+21)+6*exp(x)^2*x^2+36*exp(x)*x^2+54*x^2)*exp((exp(x+21)-exp(x)*x^2-3 
*x^2)/(exp(x)*x+3*x)))/(exp(x)^2*x^2+6*exp(x)*x^2+9*x^2),x, algorithm="gia 
c")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{-7776,[2,0,26,24]%%%}+%%%{-606528,[2,0,25,24]%%%}+%%%{-227 
44800,[2,
 

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.96 \[ \int \frac {e^{\frac {e^{21+x}-3 x^2-e^x x^2}{3 x+e^x x}} \left (e^{21+x} \left (18+6 e^x-18 x\right )+54 x^2+36 e^x x^2+6 e^{2 x} x^2\right )+e^{\frac {2 \left (e^{21+x}-3 x^2-e^x x^2\right )}{3 x+e^x x}} \left (-18 x^2-12 e^x x^2-2 e^{2 x} x^2+e^{21+x} \left (-6-2 e^x+6 x\right )\right )}{9 x^2+6 e^x x^2+e^{2 x} x^2} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{21}\,{\mathrm {e}}^x}{3\,x+x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {2\,x^2\,{\mathrm {e}}^x}{3\,x+x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {6\,x^2}{3\,x+x\,{\mathrm {e}}^x}}\,\left ({\mathrm {e}}^{\frac {{\mathrm {e}}^{21}\,{\mathrm {e}}^x}{3\,x+x\,{\mathrm {e}}^x}}-6\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^x}{3\,x+x\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {3\,x^2}{3\,x+x\,{\mathrm {e}}^x}}\right ) \] Input:

int(-(exp(-(2*(x^2*exp(x) - exp(x + 21) + 3*x^2))/(3*x + x*exp(x)))*(exp(x 
 + 21)*(2*exp(x) - 6*x + 6) + 12*x^2*exp(x) + 2*x^2*exp(2*x) + 18*x^2) - e 
xp(-(x^2*exp(x) - exp(x + 21) + 3*x^2)/(3*x + x*exp(x)))*(exp(x + 21)*(6*e 
xp(x) - 18*x + 18) + 36*x^2*exp(x) + 6*x^2*exp(2*x) + 54*x^2))/(6*x^2*exp( 
x) + x^2*exp(2*x) + 9*x^2),x)
 

Output:

exp((exp(21)*exp(x))/(3*x + x*exp(x)))*exp(-(2*x^2*exp(x))/(3*x + x*exp(x) 
))*exp(-(6*x^2)/(3*x + x*exp(x)))*(exp((exp(21)*exp(x))/(3*x + x*exp(x))) 
- 6*exp((x^2*exp(x))/(3*x + x*exp(x)))*exp((3*x^2)/(3*x + x*exp(x))))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {e^{\frac {e^{21+x}-3 x^2-e^x x^2}{3 x+e^x x}} \left (e^{21+x} \left (18+6 e^x-18 x\right )+54 x^2+36 e^x x^2+6 e^{2 x} x^2\right )+e^{\frac {2 \left (e^{21+x}-3 x^2-e^x x^2\right )}{3 x+e^x x}} \left (-18 x^2-12 e^x x^2-2 e^{2 x} x^2+e^{21+x} \left (-6-2 e^x+6 x\right )\right )}{9 x^2+6 e^x x^2+e^{2 x} x^2} \, dx=\frac {e^{\frac {e^{x} e^{21}}{e^{x} x +3 x}} \left (e^{\frac {e^{x} e^{21}}{e^{x} x +3 x}}-6 e^{x}\right )}{e^{2 x}} \] Input:

int((((-2*exp(x)+6*x-6)*exp(x+21)-2*exp(x)^2*x^2-12*exp(x)*x^2-18*x^2)*exp 
((exp(x+21)-exp(x)*x^2-3*x^2)/(exp(x)*x+3*x))^2+((6*exp(x)-18*x+18)*exp(x+ 
21)+6*exp(x)^2*x^2+36*exp(x)*x^2+54*x^2)*exp((exp(x+21)-exp(x)*x^2-3*x^2)/ 
(exp(x)*x+3*x)))/(exp(x)^2*x^2+6*exp(x)*x^2+9*x^2),x)
 

Output:

(e**((e**x*e**21)/(e**x*x + 3*x))*(e**((e**x*e**21)/(e**x*x + 3*x)) - 6*e* 
*x))/e**(2*x)