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

Optimal. Leaf size=31 \[ (3-x) \left (-x+e^{2+x-\frac {4 x^2}{\left (3+x^2\right )^2}} x^2\right ) \]

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Rubi [F]  time = 68.71, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (-\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}\right ) \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+\exp \left (\frac {-18-9 x-8 x^2-6 x^3-2 x^4-x^5}{9+6 x^2+x^4}\right ) \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

-3*x + x^2 - 8*Defer[Int][E^((18 + 9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5)/(3 + x^2)^2), x] - (6*I)*Sqrt[3]*Defer[I
nt][E^((18 + 9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5)/(3 + x^2)^2)/(I*Sqrt[3] - x)^3, x] + 21*Defer[Int][E^((18 + 9*
x + 8*x^2 + 6*x^3 + 2*x^4 + x^5)/(3 + x^2)^2)/(I*Sqrt[3] - x)^2, x] - 12*Defer[Int][E^((18 + 9*x + 8*x^2 + 6*x
^3 + 2*x^4 + x^5)/(3 + x^2)^2)/(I*Sqrt[3] - x), x] + (9*I)*Sqrt[3]*Defer[Int][E^((18 + 9*x + 8*x^2 + 6*x^3 + 2
*x^4 + x^5)/(3 + x^2)^2)/(I*Sqrt[3] - x), x] + 6*Defer[Int][E^((18 + 9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5)/(3 + x
^2)^2)*x, x] - Defer[Int][E^((18 + 9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5)/(3 + x^2)^2)*x^3, x] - (6*I)*Sqrt[3]*Def
er[Int][E^((18 + 9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5)/(3 + x^2)^2)/(I*Sqrt[3] + x)^3, x] + 21*Defer[Int][E^((18
+ 9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5)/(3 + x^2)^2)/(I*Sqrt[3] + x)^2, x] + 12*Defer[Int][E^((18 + 9*x + 8*x^2 +
6*x^3 + 2*x^4 + x^5)/(3 + x^2)^2)/(I*Sqrt[3] + x), x] + (9*I)*Sqrt[3]*Defer[Int][E^((18 + 9*x + 8*x^2 + 6*x^3
+ 2*x^4 + x^5)/(3 + x^2)^2)/(I*Sqrt[3] + x), x] + 432*Defer[Int][(E^((18 + 9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5)
/(3 + x^2)^2)*x)/(3 + x^2)^3, x] - 216*Defer[Int][(E^((18 + 9*x + 8*x^2 + 6*x^3 + 2*x^4 + x^5)/(3 + x^2)^2)*x)
/(3 + x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) \left (162 x+63 x^3+24 x^4+51 x^5-8 x^6-3 x^7-x^9+\exp \left (-\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) (-3+2 x) \left (3+x^2\right )^3\right )}{\left (3+x^2\right )^3} \, dx\\ &=\int \left (-3+2 x+\frac {162 \exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x}{\left (3+x^2\right )^3}+\frac {63 \exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^3}{\left (3+x^2\right )^3}+\frac {24 \exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^4}{\left (3+x^2\right )^3}+\frac {51 \exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^5}{\left (3+x^2\right )^3}-\frac {8 \exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^6}{\left (3+x^2\right )^3}-\frac {3 \exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^7}{\left (3+x^2\right )^3}-\frac {\exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^9}{\left (3+x^2\right )^3}\right ) \, dx\\ &=-3 x+x^2-3 \int \frac {\exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^7}{\left (3+x^2\right )^3} \, dx-8 \int \frac {\exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^6}{\left (3+x^2\right )^3} \, dx+24 \int \frac {\exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^4}{\left (3+x^2\right )^3} \, dx+51 \int \frac {\exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^5}{\left (3+x^2\right )^3} \, dx+63 \int \frac {\exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^3}{\left (3+x^2\right )^3} \, dx+162 \int \frac {\exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x}{\left (3+x^2\right )^3} \, dx-\int \frac {\exp \left (\frac {18+9 x+8 x^2+6 x^3+2 x^4+x^5}{\left (3+x^2\right )^2}\right ) x^9}{\left (3+x^2\right )^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 47, normalized size = 1.52 \begin {gather*} -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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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]

[Out]

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

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fricas [A]  time = 0.69, size = 55, normalized size = 1.77 \begin {gather*} 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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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

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giac [B]  time = 1.04, size = 83, normalized size = 2.68 \begin {gather*} -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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

-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

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maple [A]  time = 0.31, size = 52, normalized size = 1.68

method	result	size
risch	\(x^{2}-3 x +\left (-x^{3}+3 x^{2}\right ) {\mathrm e}^{\frac {x^{5}+2 x^{4}+6 x^{3}+8 x^{2}+9 x +18}{\left (x^{2}+3\right )^{2}}}\)	\(52\)
norman	\(\frac {\left (x^{6} {\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}-54 \,{\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}-27 x^{2} {\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}+27 x^{2}-9 x^{3}+18 x^{4}-6 x^{5}+3 x^{6}-x^{7}-27 x \,{\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}-18 x^{3} {\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}-3 x^{5} {\mathrm e}^{\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}\right ) {\mathrm e}^{-\frac {-x^{5}-2 x^{4}-6 x^{3}-8 x^{2}-9 x -18}{x^{4}+6 x^{2}+9}}}{\left (x^{2}+3\right )^{2}}\)	\(339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

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maxima [B]  time = 0.61, size = 214, normalized size = 6.90 \begin {gather*} 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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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)

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mupad [B]  time = 3.44, size = 221, normalized size = 7.13 \begin {gather*} 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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))

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sympy [B]  time = 0.34, size = 51, normalized size = 1.65 \begin {gather*} 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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))

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