\(\int \frac {e^{-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+e^{\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} (-81+54 x-81 x^2+54 x^3-27 x^4+18 x^5-3 x^6+2 x^7))}{27+27 x^2+9 x^4+x^6} \, dx\) [2168]

Optimal result

Integrand size = 169, antiderivative size = 31 \[ \int \frac {e^{-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+e^{\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (-81+54 x-81 x^2+54 x^3-27 x^4+18 x^5-3 x^6+2 x^7\right )\right )}{27+27 x^2+9 x^4+x^6} \, dx=(3-x) \left (-x+e^{2+x-\frac {4 x^2}{\left (3+x^2\right )^2}} x^2\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {e^{-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+e^{\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (-81+54 x-81 x^2+54 x^3-27 x^4+18 x^5-3 x^6+2 x^7\right )\right )}{27+27 x^2+9 x^4+x^6} \, dx=-3 x+x^2+e^{x+\frac {12}{\left (3+x^2\right )^2}-\frac {4}{3+x^2}} \left (3 e^2 x^2-e^2 x^3\right ) \] Input:

Integrate[(162*x + 63*x^3 + 24*x^4 + 51*x^5 - 8*x^6 - 3*x^7 - x^9 + E^((-1 
8 - 9*x - 8*x^2 - 6*x^3 - 2*x^4 - x^5)/(9 + 6*x^2 + x^4))*(-81 + 54*x - 81 
*x^2 + 54*x^3 - 27*x^4 + 18*x^5 - 3*x^6 + 2*x^7))/(E^((-18 - 9*x - 8*x^2 - 
 6*x^3 - 2*x^4 - x^5)/(9 + 6*x^2 + x^4))*(27 + 27*x^2 + 9*x^4 + x^6)),x]
 

Output:

-3*x + x^2 + E^(x + 12/(3 + x^2)^2 - 4/(3 + x^2))*(3*E^2*x^2 - E^2*x^3)
 

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.

\(\displaystyle \int \frac {\exp \left (-\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right ) \left (\left (2 x^7-3 x^6+18 x^5-27 x^4+54 x^3-81 x^2+54 x-81\right ) \exp \left (\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right )-x^9-3 x^7-8 x^6+51 x^5+24 x^4+63 x^3+162 x\right )}{x^6+9 x^4+27 x^2+27} \, dx\)

\(\Big \downarrow \) 2070

\(\displaystyle \int \frac {\exp \left (-\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right ) \left (\left (2 x^7-3 x^6+18 x^5-27 x^4+54 x^3-81 x^2+54 x-81\right ) \exp \left (\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right )-x^9-3 x^7-8 x^6+51 x^5+24 x^4+63 x^3+162 x\right )}{\left (x^2+3\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {51 x^5 \exp \left (-\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right )}{\left (x^2+3\right )^3}+\frac {24 x^4 \exp \left (-\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right )}{\left (x^2+3\right )^3}+\frac {63 x^3 \exp \left (-\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right )}{\left (x^2+3\right )^3}+\frac {162 x \exp \left (-\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right )}{\left (x^2+3\right )^3}-\frac {x^9 \exp \left (-\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right )}{\left (x^2+3\right )^3}-\frac {3 x^7 \exp \left (-\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right )}{\left (x^2+3\right )^3}-\frac {8 x^6 \exp \left (-\frac {-x^5-2 x^4-6 x^3-8 x^2-9 x-18}{x^4+6 x^2+9}\right )}{\left (x^2+3\right )^3}+2 x-3\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {51 x^5 \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (x^2+3\right )^3}+\frac {24 x^4 \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (x^2+3\right )^3}+\frac {63 x^3 \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (x^2+3\right )^3}+\frac {162 x \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (x^2+3\right )^3}+\frac {x^9 \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (-x^2-3\right )^3}+\frac {3 x^7 \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (-x^2-3\right )^3}+\frac {8 x^6 \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (-x^2-3\right )^3}+2 x-3\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -8 \int \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )dx-6 i \sqrt {3} \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (i \sqrt {3}-x\right )^3}dx+21 \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (i \sqrt {3}-x\right )^2}dx+9 i \sqrt {3} \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{i \sqrt {3}-x}dx-12 \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{i \sqrt {3}-x}dx+6 \int \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right ) xdx-\int \exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right ) x^3dx-6 i \sqrt {3} \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (x+i \sqrt {3}\right )^3}dx+21 \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{\left (x+i \sqrt {3}\right )^2}dx+9 i \sqrt {3} \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{x+i \sqrt {3}}dx+12 \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right )}{x+i \sqrt {3}}dx+432 \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right ) x}{\left (x^2+3\right )^3}dx-216 \int \frac {\exp \left (\frac {x^5+2 x^4+6 x^3+8 x^2+9 x+18}{\left (x^2+3\right )^2}\right ) x}{\left (x^2+3\right )^2}dx+x^2-3 x\)

Input:

Int[(162*x + 63*x^3 + 24*x^4 + 51*x^5 - 8*x^6 - 3*x^7 - x^9 + E^((-18 - 9* 
x - 8*x^2 - 6*x^3 - 2*x^4 - x^5)/(9 + 6*x^2 + x^4))*(-81 + 54*x - 81*x^2 + 
 54*x^3 - 27*x^4 + 18*x^5 - 3*x^6 + 2*x^7))/(E^((-18 - 9*x - 8*x^2 - 6*x^3 
 - 2*x^4 - x^5)/(9 + 6*x^2 + x^4))*(27 + 27*x^2 + 9*x^4 + x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68

method	result	size
risch	\(x^{2}-3 x +\left (-x^{3}+3 x^{2}\right ) {\mathrm e}^{\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{\left (x^{2}+3\right )^{2}}}\)	\(52\)
parts	\(x^{2}-3 x +\frac {\left (-x^{7}+3 x^{6}-6 x^{5}+18 x^{4}-9 x^{3}+27 x^{2}\right ) {\mathrm e}^{-\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}}{\left (x^{2}+3\right )^{2}}\)	\(88\)
norman	\(\frac {\left (x^{6} {\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}-54 \,{\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}-27 x^{2} {\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}+27 x^{2}-9 x^{3}+18 x^{4}-6 x^{5}+3 x^{6}-x^{7}-27 x \,{\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}-18 x^{3} {\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}-3 x^{5} {\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}\right ) {\mathrm e}^{-\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}}{\left (x^{2}+3\right )^{2}}\)	\(339\)
parallelrisch	\(\frac {\left (-x^{7}+{\mathrm e}^{-\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{x^{4}+6 x^{2}+9}} x^{6}+3 x^{6}-3 \,{\mathrm e}^{-\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{x^{4}+6 x^{2}+9}} x^{5}-6 x^{5}-12 \,{\mathrm e}^{-\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{x^{4}+6 x^{2}+9}} x^{4}+18 x^{4}-18 \,{\mathrm e}^{-\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{x^{4}+6 x^{2}+9}} x^{3}-9 x^{3}-99 \,{\mathrm e}^{-\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{x^{4}+6 x^{2}+9}} x^{2}+27 x^{2}-27 \,{\mathrm e}^{-\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{x^{4}+6 x^{2}+9}} x -162 \,{\mathrm e}^{-\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{x^{4}+6 x^{2}+9}}\right ) {\mathrm e}^{\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{x^{4}+6 x^{2}+9}}}{\left (x^{2}+3\right )^{2}}\)	\(375\)
orering	\(\text {Expression too large to display}\)	\(1214\)

Input:

int(((2*x^7-3*x^6+18*x^5-27*x^4+54*x^3-81*x^2+54*x-81)*exp((-x^5-2*x^4-6*x 
^3-8*x^2-9*x-18)/(x^4+6*x^2+9))-x^9-3*x^7-8*x^6+51*x^5+24*x^4+63*x^3+162*x 
)/(x^6+9*x^4+27*x^2+27)/exp((-x^5-2*x^4-6*x^3-8*x^2-9*x-18)/(x^4+6*x^2+9)) 
,x,method=_RETURNVERBOSE)
 

Output:

x^2-3*x+(-x^3+3*x^2)*exp((x^5+2*x^4+6*x^3+8*x^2+9*x+18)/(x^2+3)^2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {e^{-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+e^{\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (-81+54 x-81 x^2+54 x^3-27 x^4+18 x^5-3 x^6+2 x^7\right )\right )}{27+27 x^2+9 x^4+x^6} \, dx=x^{2} - {\left (x^{3} - 3 \, x^{2}\right )} e^{\left (\frac {x^{5} + 2 \, x^{4} + 6 \, x^{3} + 8 \, x^{2} + 9 \, x + 18}{x^{4} + 6 \, x^{2} + 9}\right )} - 3 \, x \] Input:

integrate(((2*x^7-3*x^6+18*x^5-27*x^4+54*x^3-81*x^2+54*x-81)*exp((-x^5-2*x 
^4-6*x^3-8*x^2-9*x-18)/(x^4+6*x^2+9))-x^9-3*x^7-8*x^6+51*x^5+24*x^4+63*x^3 
+162*x)/(x^6+9*x^4+27*x^2+27)/exp((-x^5-2*x^4-6*x^3-8*x^2-9*x-18)/(x^4+6*x 
^2+9)),x, algorithm="fricas")
 

Output:

x^2 - (x^3 - 3*x^2)*e^((x^5 + 2*x^4 + 6*x^3 + 8*x^2 + 9*x + 18)/(x^4 + 6*x 
^2 + 9)) - 3*x
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).

Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+e^{\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (-81+54 x-81 x^2+54 x^3-27 x^4+18 x^5-3 x^6+2 x^7\right )\right )}{27+27 x^2+9 x^4+x^6} \, dx=x^{2} - 3 x + \left (- x^{3} + 3 x^{2}\right ) e^{- \frac {- x^{5} - 2 x^{4} - 6 x^{3} - 8 x^{2} - 9 x - 18}{x^{4} + 6 x^{2} + 9}} \] Input:

integrate(((2*x**7-3*x**6+18*x**5-27*x**4+54*x**3-81*x**2+54*x-81)*exp((-x 
**5-2*x**4-6*x**3-8*x**2-9*x-18)/(x**4+6*x**2+9))-x**9-3*x**7-8*x**6+51*x* 
*5+24*x**4+63*x**3+162*x)/(x**6+9*x**4+27*x**2+27)/exp((-x**5-2*x**4-6*x** 
3-8*x**2-9*x-18)/(x**4+6*x**2+9)),x)
 

Output:

x**2 - 3*x + (-x**3 + 3*x**2)*exp(-(-x**5 - 2*x**4 - 6*x**3 - 8*x**2 - 9*x 
 - 18)/(x**4 + 6*x**2 + 9))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (29) = 58\).

Time = 0.26 (sec) , antiderivative size = 214, normalized size of antiderivative = 6.90 \[ \int \frac {e^{-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+e^{\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (-81+54 x-81 x^2+54 x^3-27 x^4+18 x^5-3 x^6+2 x^7\right )\right )}{27+27 x^2+9 x^4+x^6} \, dx=x^{2} - {\left (x^{3} e^{2} - 3 \, x^{2} e^{2}\right )} e^{\left (x + \frac {12}{x^{4} + 6 \, x^{2} + 9} - \frac {4}{x^{2} + 3}\right )} - 3 \, x + \frac {27 \, {\left (5 \, x^{3} + 9 \, x\right )}}{8 \, {\left (x^{4} + 6 \, x^{2} + 9\right )}} - \frac {27 \, {\left (3 \, x^{3} + 7 \, x\right )}}{8 \, {\left (x^{4} + 6 \, x^{2} + 9\right )}} - \frac {27 \, {\left (x^{3} + 5 \, x\right )}}{8 \, {\left (x^{4} + 6 \, x^{2} + 9\right )}} - \frac {27 \, {\left (x^{3} - 3 \, x\right )}}{8 \, {\left (x^{4} + 6 \, x^{2} + 9\right )}} + \frac {27 \, {\left (4 \, x^{2} + 9\right )}}{2 \, {\left (x^{4} + 6 \, x^{2} + 9\right )}} - \frac {27 \, {\left (2 \, x^{2} + 5\right )}}{2 \, {\left (x^{4} + 6 \, x^{2} + 9\right )}} - \frac {27 \, {\left (2 \, x^{2} + 3\right )}}{2 \, {\left (x^{4} + 6 \, x^{2} + 9\right )}} - \frac {27}{2 \, {\left (x^{4} + 6 \, x^{2} + 9\right )}} \] Input:

integrate(((2*x^7-3*x^6+18*x^5-27*x^4+54*x^3-81*x^2+54*x-81)*exp((-x^5-2*x 
^4-6*x^3-8*x^2-9*x-18)/(x^4+6*x^2+9))-x^9-3*x^7-8*x^6+51*x^5+24*x^4+63*x^3 
+162*x)/(x^6+9*x^4+27*x^2+27)/exp((-x^5-2*x^4-6*x^3-8*x^2-9*x-18)/(x^4+6*x 
^2+9)),x, algorithm="maxima")
 

Output:

x^2 - (x^3*e^2 - 3*x^2*e^2)*e^(x + 12/(x^4 + 6*x^2 + 9) - 4/(x^2 + 3)) - 3 
*x + 27/8*(5*x^3 + 9*x)/(x^4 + 6*x^2 + 9) - 27/8*(3*x^3 + 7*x)/(x^4 + 6*x^ 
2 + 9) - 27/8*(x^3 + 5*x)/(x^4 + 6*x^2 + 9) - 27/8*(x^3 - 3*x)/(x^4 + 6*x^ 
2 + 9) + 27/2*(4*x^2 + 9)/(x^4 + 6*x^2 + 9) - 27/2*(2*x^2 + 5)/(x^4 + 6*x^ 
2 + 9) - 27/2*(2*x^2 + 3)/(x^4 + 6*x^2 + 9) - 27/2/(x^4 + 6*x^2 + 9)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (29) = 58\).

Time = 0.76 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.68 \[ \int \frac {e^{-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+e^{\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (-81+54 x-81 x^2+54 x^3-27 x^4+18 x^5-3 x^6+2 x^7\right )\right )}{27+27 x^2+9 x^4+x^6} \, dx=-x^{3} e^{\left (\frac {x^{5} + 6 \, x^{3} - 4 \, x^{2} + 9 \, x}{x^{4} + 6 \, x^{2} + 9} + 2\right )} + 3 \, x^{2} e^{\left (\frac {x^{5} + 6 \, x^{3} - 4 \, x^{2} + 9 \, x}{x^{4} + 6 \, x^{2} + 9} + 2\right )} + x^{2} - 3 \, x \] Input:

integrate(((2*x^7-3*x^6+18*x^5-27*x^4+54*x^3-81*x^2+54*x-81)*exp((-x^5-2*x 
^4-6*x^3-8*x^2-9*x-18)/(x^4+6*x^2+9))-x^9-3*x^7-8*x^6+51*x^5+24*x^4+63*x^3 
+162*x)/(x^6+9*x^4+27*x^2+27)/exp((-x^5-2*x^4-6*x^3-8*x^2-9*x-18)/(x^4+6*x 
^2+9)),x, algorithm="giac")
 

Output:

-x^3*e^((x^5 + 6*x^3 - 4*x^2 + 9*x)/(x^4 + 6*x^2 + 9) + 2) + 3*x^2*e^((x^5 
 + 6*x^3 - 4*x^2 + 9*x)/(x^4 + 6*x^2 + 9) + 2) + x^2 - 3*x
 

Mupad [B] (verification not implemented)

Time = 4.24 (sec) , antiderivative size = 221, normalized size of antiderivative = 7.13 \[ \int \frac {e^{-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+e^{\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (-81+54 x-81 x^2+54 x^3-27 x^4+18 x^5-3 x^6+2 x^7\right )\right )}{27+27 x^2+9 x^4+x^6} \, dx=x^2-3\,x+3\,x^2\,{\mathrm {e}}^{\frac {9\,x}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {x^5}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {2\,x^4}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {6\,x^3}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {8\,x^2}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {18}{x^4+6\,x^2+9}}-x^3\,{\mathrm {e}}^{\frac {9\,x}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {x^5}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {2\,x^4}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {6\,x^3}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {8\,x^2}{x^4+6\,x^2+9}}\,{\mathrm {e}}^{\frac {18}{x^4+6\,x^2+9}} \] Input:

int((exp((9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5 + 18)/(6*x^2 + x^4 + 9))*(162* 
x + exp(-(9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5 + 18)/(6*x^2 + x^4 + 9))*(54*x 
 - 81*x^2 + 54*x^3 - 27*x^4 + 18*x^5 - 3*x^6 + 2*x^7 - 81) + 63*x^3 + 24*x 
^4 + 51*x^5 - 8*x^6 - 3*x^7 - x^9))/(27*x^2 + 9*x^4 + x^6 + 27),x)
 

Output:

x^2 - 3*x + 3*x^2*exp((9*x)/(6*x^2 + x^4 + 9))*exp(x^5/(6*x^2 + x^4 + 9))* 
exp((2*x^4)/(6*x^2 + x^4 + 9))*exp((6*x^3)/(6*x^2 + x^4 + 9))*exp((8*x^2)/ 
(6*x^2 + x^4 + 9))*exp(18/(6*x^2 + x^4 + 9)) - x^3*exp((9*x)/(6*x^2 + x^4 
+ 9))*exp(x^5/(6*x^2 + x^4 + 9))*exp((2*x^4)/(6*x^2 + x^4 + 9))*exp((6*x^3 
)/(6*x^2 + x^4 + 9))*exp((8*x^2)/(6*x^2 + x^4 + 9))*exp(18/(6*x^2 + x^4 + 
9))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.77 \[ \int \frac {e^{-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+e^{\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}} \left (-81+54 x-81 x^2+54 x^3-27 x^4+18 x^5-3 x^6+2 x^7\right )\right )}{27+27 x^2+9 x^4+x^6} \, dx=\frac {x \left (e^{\frac {4 x^{2}}{x^{4}+6 x^{2}+9}} x -3 e^{\frac {4 x^{2}}{x^{4}+6 x^{2}+9}}-e^{x} e^{2} x^{2}+3 e^{x} e^{2} x \right )}{e^{\frac {4 x^{2}}{x^{4}+6 x^{2}+9}}} \] Input:

int(((2*x^7-3*x^6+18*x^5-27*x^4+54*x^3-81*x^2+54*x-81)*exp((-x^5-2*x^4-6*x 
^3-8*x^2-9*x-18)/(x^4+6*x^2+9))-x^9-3*x^7-8*x^6+51*x^5+24*x^4+63*x^3+162*x 
)/(x^6+9*x^4+27*x^2+27)/exp((-x^5-2*x^4-6*x^3-8*x^2-9*x-18)/(x^4+6*x^2+9)) 
,x)
 

Output:

(x*(e**((4*x**2)/(x**4 + 6*x**2 + 9))*x - 3*e**((4*x**2)/(x**4 + 6*x**2 + 
9)) - e**x*e**2*x**2 + 3*e**x*e**2*x))/e**((4*x**2)/(x**4 + 6*x**2 + 9))