\(\int \frac {9 x^2-6 x^3+x^4+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}+\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}} (-27+99 x-57 x^2+9 x^3+e^x (36 x-9 x^2))}{9 x^3-6 x^4+x^5+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}} (27 x^2-18 x^3+3 x^4)} \, dx\) [1930]

Optimal result

Integrand size = 181, antiderivative size = 33 \[ \int \frac {9 x^2-6 x^3+x^4+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}+\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}} \left (-27+99 x-57 x^2+9 x^3+e^x \left (36 x-9 x^2\right )\right )}{9 x^3-6 x^4+x^5+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}} \left (27 x^2-18 x^3+3 x^4\right )} \, dx=\log \left (3 e^{\frac {e^{3 \left (-e^3+\frac {e^x}{3-x}+x\right )}}{x}}+x\right ) \] Output:

ln(x+3*exp(1/x*exp(3*x-3*exp(3)+3*exp(x)/(3-x))))
 

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {9 x^2-6 x^3+x^4+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}+\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}} \left (-27+99 x-57 x^2+9 x^3+e^x \left (36 x-9 x^2\right )\right )}{9 x^3-6 x^4+x^5+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}} \left (27 x^2-18 x^3+3 x^4\right )} \, dx=\log \left (3 e^{\frac {e^{-3 e^3-\frac {3 e^x}{-3+x}+3 x}}{x}}+x\right ) \] Input:

Integrate[(9*x^2 - 6*x^3 + x^4 + E^(E^((-3*E^x + E^3*(9 - 3*x) - 9*x + 3*x 
^2)/(-3 + x))/x + (-3*E^x + E^3*(9 - 3*x) - 9*x + 3*x^2)/(-3 + x))*(-27 + 
99*x - 57*x^2 + 9*x^3 + E^x*(36*x - 9*x^2)))/(9*x^3 - 6*x^4 + x^5 + E^(E^( 
(-3*E^x + E^3*(9 - 3*x) - 9*x + 3*x^2)/(-3 + x))/x)*(27*x^2 - 18*x^3 + 3*x 
^4)),x]
 

Output:

Log[3*E^(E^(-3*E^3 - (3*E^x)/(-3 + x) + 3*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.

\(\displaystyle \int \frac {\left (9 x^3-57 x^2+e^x \left (36 x-9 x^2\right )+99 x-27\right ) \exp \left (\frac {3 x^2-9 x-3 e^x+e^3 (9-3 x)}{x-3}+\frac {e^{\frac {3 x^2-9 x-3 e^x+e^3 (9-3 x)}{x-3}}}{x}\right )+x^4-6 x^3+9 x^2}{\left (3 x^4-18 x^3+27 x^2\right ) \exp \left (\frac {e^{\frac {3 x^2-9 x-3 e^x+e^3 (9-3 x)}{x-3}}}{x}\right )+x^5-6 x^4+9 x^3} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (9 x^3-57 x^2+e^x \left (36 x-9 x^2\right )+99 x-27\right ) \exp \left (\frac {3 x^2-9 x-3 e^x+e^3 (9-3 x)}{x-3}+\frac {e^{\frac {3 x^2-9 x-3 e^x+e^3 (9-3 x)}{x-3}}}{x}\right )+x^4-6 x^3+9 x^2}{(3-x)^2 x^2 \left (x+3 e^{\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {3 \left (-3 x^3+3 e^x x^2+19 x^2-12 e^x x-33 x+9\right ) \exp \left (3 x-\frac {3 e^x}{x-3}+\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}-3 e^3\right )}{(x-3)^2 \left (x+3 e^{\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}}\right ) x^2}+\frac {x^2}{(x-3)^2 \left (x+3 e^{\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}}\right )}-\frac {6 x}{(x-3)^2 \left (x+3 e^{\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}}\right )}+\frac {9}{(x-3)^2 \left (x+3 e^{\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}}\right )}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (-\frac {3 \left (-3 x^3+3 e^x x^2+19 x^2-12 e^x x-33 x+9\right ) \exp \left (3 x-\frac {3 e^x}{x-3}+\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}-3 e^3\right )}{(x-3)^2 \left (x+3 e^{\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}}\right ) x^2}+\frac {x^2}{(x-3)^2 \left (x+3 e^{\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}}\right )}-\frac {6 x}{(x-3)^2 \left (x+3 e^{\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}}\right )}+\frac {9}{(x-3)^2 \left (x+3 e^{\frac {e^{3 x-\frac {3 e^x}{x-3}-3 e^3}}{x}}\right )}\right )dx\)

Input:

Int[(9*x^2 - 6*x^3 + x^4 + E^(E^((-3*E^x + E^3*(9 - 3*x) - 9*x + 3*x^2)/(- 
3 + x))/x + (-3*E^x + E^3*(9 - 3*x) - 9*x + 3*x^2)/(-3 + x))*(-27 + 99*x - 
 57*x^2 + 9*x^3 + E^x*(36*x - 9*x^2)))/(9*x^3 - 6*x^4 + x^5 + E^(E^((-3*E^ 
x + E^3*(9 - 3*x) - 9*x + 3*x^2)/(-3 + x))/x)*(27*x^2 - 18*x^3 + 3*x^4)),x 
]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 12.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15

method	result	size
risch	\(\ln \left (\frac {x}{3}+{\mathrm e}^{\frac {{\mathrm e}^{-\frac {3 \left (x \,{\mathrm e}^{3}-x^{2}+{\mathrm e}^{x}-3 \,{\mathrm e}^{3}+3 x \right )}{-3+x}}}{x}}\right )\)	\(38\)
parallelrisch	\(\ln \left (x +3 \,{\mathrm e}^{\frac {{\mathrm e}^{\frac {-3 \,{\mathrm e}^{x}+\left (-3 x +9\right ) {\mathrm e}^{3}+3 x^{2}-9 x}{-3+x}}}{x}}\right )\)	\(39\)

Input:

int((((-9*x^2+36*x)*exp(x)+9*x^3-57*x^2+99*x-27)*exp((-3*exp(x)+(-3*x+9)*e 
xp(3)+3*x^2-9*x)/(-3+x))*exp(exp((-3*exp(x)+(-3*x+9)*exp(3)+3*x^2-9*x)/(-3 
+x))/x)+x^4-6*x^3+9*x^2)/((3*x^4-18*x^3+27*x^2)*exp(exp((-3*exp(x)+(-3*x+9 
)*exp(3)+3*x^2-9*x)/(-3+x))/x)+x^5-6*x^4+9*x^3),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {9 x^2-6 x^3+x^4+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}+\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}} \left (-27+99 x-57 x^2+9 x^3+e^x \left (36 x-9 x^2\right )\right )}{9 x^3-6 x^4+x^5+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}} \left (27 x^2-18 x^3+3 x^4\right )} \, dx=\log \left (x + 3 \, e^{\left (\frac {e^{\left (\frac {3 \, {\left (x^{2} - {\left (x - 3\right )} e^{3} - 3 \, x - e^{x}\right )}}{x - 3}\right )}}{x}\right )}\right ) \] Input:

integrate((((-9*x^2+36*x)*exp(x)+9*x^3-57*x^2+99*x-27)*exp((-3*exp(x)+(-3* 
x+9)*exp(3)+3*x^2-9*x)/(-3+x))*exp(exp((-3*exp(x)+(-3*x+9)*exp(3)+3*x^2-9* 
x)/(-3+x))/x)+x^4-6*x^3+9*x^2)/((3*x^4-18*x^3+27*x^2)*exp(exp((-3*exp(x)+( 
-3*x+9)*exp(3)+3*x^2-9*x)/(-3+x))/x)+x^5-6*x^4+9*x^3),x, algorithm="fricas 
")
 

Output:

log(x + 3*e^(e^(3*(x^2 - (x - 3)*e^3 - 3*x - e^x)/(x - 3))/x))
 

Sympy [A] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {9 x^2-6 x^3+x^4+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}+\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}} \left (-27+99 x-57 x^2+9 x^3+e^x \left (36 x-9 x^2\right )\right )}{9 x^3-6 x^4+x^5+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}} \left (27 x^2-18 x^3+3 x^4\right )} \, dx=\log {\left (\frac {x}{3} + e^{\frac {e^{\frac {3 x^{2} - 9 x + \left (9 - 3 x\right ) e^{3} - 3 e^{x}}{x - 3}}}{x}} \right )} \] Input:

integrate((((-9*x**2+36*x)*exp(x)+9*x**3-57*x**2+99*x-27)*exp((-3*exp(x)+( 
-3*x+9)*exp(3)+3*x**2-9*x)/(-3+x))*exp(exp((-3*exp(x)+(-3*x+9)*exp(3)+3*x* 
*2-9*x)/(-3+x))/x)+x**4-6*x**3+9*x**2)/((3*x**4-18*x**3+27*x**2)*exp(exp(( 
-3*exp(x)+(-3*x+9)*exp(3)+3*x**2-9*x)/(-3+x))/x)+x**5-6*x**4+9*x**3),x)
 

Output:

log(x/3 + exp(exp((3*x**2 - 9*x + (9 - 3*x)*exp(3) - 3*exp(x))/(x - 3))/x) 
)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {9 x^2-6 x^3+x^4+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}+\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}} \left (-27+99 x-57 x^2+9 x^3+e^x \left (36 x-9 x^2\right )\right )}{9 x^3-6 x^4+x^5+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}} \left (27 x^2-18 x^3+3 x^4\right )} \, dx=\log \left (\frac {1}{3} \, x + e^{\left (\frac {e^{\left (3 \, x - \frac {3 \, e^{x}}{x - 3} - 3 \, e^{3}\right )}}{x}\right )}\right ) \] Input:

integrate((((-9*x^2+36*x)*exp(x)+9*x^3-57*x^2+99*x-27)*exp((-3*exp(x)+(-3* 
x+9)*exp(3)+3*x^2-9*x)/(-3+x))*exp(exp((-3*exp(x)+(-3*x+9)*exp(3)+3*x^2-9* 
x)/(-3+x))/x)+x^4-6*x^3+9*x^2)/((3*x^4-18*x^3+27*x^2)*exp(exp((-3*exp(x)+( 
-3*x+9)*exp(3)+3*x^2-9*x)/(-3+x))/x)+x^5-6*x^4+9*x^3),x, algorithm="maxima 
")
 

Output:

log(1/3*x + e^(e^(3*x - 3*e^x/(x - 3) - 3*e^3)/x))
 

Giac [F(-1)]

Timed out. \[ \int \frac {9 x^2-6 x^3+x^4+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}+\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}} \left (-27+99 x-57 x^2+9 x^3+e^x \left (36 x-9 x^2\right )\right )}{9 x^3-6 x^4+x^5+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}} \left (27 x^2-18 x^3+3 x^4\right )} \, dx=\text {Timed out} \] Input:

integrate((((-9*x^2+36*x)*exp(x)+9*x^3-57*x^2+99*x-27)*exp((-3*exp(x)+(-3* 
x+9)*exp(3)+3*x^2-9*x)/(-3+x))*exp(exp((-3*exp(x)+(-3*x+9)*exp(3)+3*x^2-9* 
x)/(-3+x))/x)+x^4-6*x^3+9*x^2)/((3*x^4-18*x^3+27*x^2)*exp(exp((-3*exp(x)+( 
-3*x+9)*exp(3)+3*x^2-9*x)/(-3+x))/x)+x^5-6*x^4+9*x^3),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 2.38 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {9 x^2-6 x^3+x^4+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}+\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}} \left (-27+99 x-57 x^2+9 x^3+e^x \left (36 x-9 x^2\right )\right )}{9 x^3-6 x^4+x^5+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}} \left (27 x^2-18 x^3+3 x^4\right )} \, dx=\ln \left (x+3\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-\frac {9\,x-9\,{\mathrm {e}}^3+3\,{\mathrm {e}}^x+3\,x\,{\mathrm {e}}^3-3\,x^2}{x-3}}}{x}}\right ) \] Input:

int((9*x^2 - 6*x^3 + x^4 + exp(exp(-(9*x + 3*exp(x) - 3*x^2 + exp(3)*(3*x 
- 9))/(x - 3))/x)*exp(-(9*x + 3*exp(x) - 3*x^2 + exp(3)*(3*x - 9))/(x - 3) 
)*(99*x + exp(x)*(36*x - 9*x^2) - 57*x^2 + 9*x^3 - 27))/(exp(exp(-(9*x + 3 
*exp(x) - 3*x^2 + exp(3)*(3*x - 9))/(x - 3))/x)*(27*x^2 - 18*x^3 + 3*x^4) 
+ 9*x^3 - 6*x^4 + x^5),x)
 

Output:

log(x + 3*exp(exp(-(9*x - 9*exp(3) + 3*exp(x) + 3*x*exp(3) - 3*x^2)/(x - 3 
))/x))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {9 x^2-6 x^3+x^4+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}+\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}} \left (-27+99 x-57 x^2+9 x^3+e^x \left (36 x-9 x^2\right )\right )}{9 x^3-6 x^4+x^5+e^{\frac {e^{\frac {-3 e^x+e^3 (9-3 x)-9 x+3 x^2}{-3+x}}}{x}} \left (27 x^2-18 x^3+3 x^4\right )} \, dx=\mathrm {log}\left (3 e^{\frac {e^{3 x}}{e^{\frac {3 e^{x}+3 e^{3} x -9 e^{3}}{x -3}} x}}+x \right ) \] Input:

int((((-9*x^2+36*x)*exp(x)+9*x^3-57*x^2+99*x-27)*exp((-3*exp(x)+(-3*x+9)*e 
xp(3)+3*x^2-9*x)/(-3+x))*exp(exp((-3*exp(x)+(-3*x+9)*exp(3)+3*x^2-9*x)/(-3 
+x))/x)+x^4-6*x^3+9*x^2)/((3*x^4-18*x^3+27*x^2)*exp(exp((-3*exp(x)+(-3*x+9 
)*exp(3)+3*x^2-9*x)/(-3+x))/x)+x^5-6*x^4+9*x^3),x)
 

Output:

log(3*e**(e**(3*x)/(e**((3*e**x + 3*e**3*x - 9*e**3)/(x - 3))*x)) + x)