3.6.75 \(\int \frac {32+e^{-18 x-34 x^2-11 x^3-x^4} (576+2176 x+1056 x^2+128 x^3)}{e^{-54 x-102 x^2-33 x^3-3 x^4}-3 e^{-36 x-68 x^2-22 x^3-2 x^4} x+3 e^{-18 x-34 x^2-11 x^3-x^4} x^2-x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac {16}{\left (e^{2 x-x (5+x) (4+x (6+x))}-x\right )^2} \]

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Rubi [F]  time = 4.40, 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 {32+e^{-18 x-34 x^2-11 x^3-x^4} \left (576+2176 x+1056 x^2+128 x^3\right )}{e^{-54 x-102 x^2-33 x^3-3 x^4}-3 e^{-36 x-68 x^2-22 x^3-2 x^4} x+3 e^{-18 x-34 x^2-11 x^3-x^4} x^2-x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(32 + E^(-18*x - 34*x^2 - 11*x^3 - x^4)*(576 + 2176*x + 1056*x^2 + 128*x^3))/(E^(-54*x - 102*x^2 - 33*x^3
- 3*x^4) - 3*E^(-36*x - 68*x^2 - 22*x^3 - 2*x^4)*x + 3*E^(-18*x - 34*x^2 - 11*x^3 - x^4)*x^2 - x^3),x]

[Out]

-576*Defer[Int][E^(2*x*(18 + 34*x + 11*x^2 + x^3))/(-1 + E^(x*(18 + 34*x + 11*x^2 + x^3))*x)^3, x] - 32*Defer[
Int][E^(2*x*(18 + 34*x + 11*x^2 + x^3))/(x*(-1 + E^(x*(18 + 34*x + 11*x^2 + x^3))*x)^3), x] - 2176*Defer[Int][
(E^(2*x*(18 + 34*x + 11*x^2 + x^3))*x)/(-1 + E^(x*(18 + 34*x + 11*x^2 + x^3))*x)^3, x] - 1056*Defer[Int][(E^(2
*x*(18 + 34*x + 11*x^2 + x^3))*x^2)/(-1 + E^(x*(18 + 34*x + 11*x^2 + x^3))*x)^3, x] - 128*Defer[Int][(E^(2*x*(
18 + 34*x + 11*x^2 + x^3))*x^3)/(-1 + E^(x*(18 + 34*x + 11*x^2 + x^3))*x)^3, x] - 32*Defer[Int][E^(2*x*(18 + 3
4*x + 11*x^2 + x^3))/(x*(-1 + E^(x*(18 + 34*x + 11*x^2 + x^3))*x)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32 e^{2 x \left (18+34 x+11 x^2+x^3\right )} \left (18+e^{x \left (18+34 x+11 x^2+x^3\right )}+68 x+33 x^2+4 x^3\right )}{\left (1-e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3} \, dx\\ &=32 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )} \left (18+e^{x \left (18+34 x+11 x^2+x^3\right )}+68 x+33 x^2+4 x^3\right )}{\left (1-e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3} \, dx\\ &=32 \int \left (-\frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{x \left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^2}-\frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )} \left (1+18 x+68 x^2+33 x^3+4 x^4\right )}{x \left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3}\right ) \, dx\\ &=-\left (32 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{x \left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^2} \, dx\right )-32 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )} \left (1+18 x+68 x^2+33 x^3+4 x^4\right )}{x \left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3} \, dx\\ &=-\left (32 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{x \left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^2} \, dx\right )-32 \int \left (\frac {18 e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3}+\frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{x \left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3}+\frac {68 e^{2 x \left (18+34 x+11 x^2+x^3\right )} x}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3}+\frac {33 e^{2 x \left (18+34 x+11 x^2+x^3\right )} x^2}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3}+\frac {4 e^{2 x \left (18+34 x+11 x^2+x^3\right )} x^3}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3}\right ) \, dx\\ &=-\left (32 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{x \left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3} \, dx\right )-32 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{x \left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^2} \, dx-128 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )} x^3}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3} \, dx-576 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3} \, dx-1056 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )} x^2}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3} \, dx-2176 \int \frac {e^{2 x \left (18+34 x+11 x^2+x^3\right )} x}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.80, size = 43, normalized size = 1.59 \begin {gather*} \frac {16 e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 + E^(-18*x - 34*x^2 - 11*x^3 - x^4)*(576 + 2176*x + 1056*x^2 + 128*x^3))/(E^(-54*x - 102*x^2 - 3
3*x^3 - 3*x^4) - 3*E^(-36*x - 68*x^2 - 22*x^3 - 2*x^4)*x + 3*E^(-18*x - 34*x^2 - 11*x^3 - x^4)*x^2 - x^3),x]

[Out]

(16*E^(2*x*(18 + 34*x + 11*x^2 + x^3)))/(-1 + E^(x*(18 + 34*x + 11*x^2 + x^3))*x)^2

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fricas [A]  time = 0.65, size = 51, normalized size = 1.89 \begin {gather*} \frac {16}{x^{2} - 2 \, x e^{\left (-x^{4} - 11 \, x^{3} - 34 \, x^{2} - 18 \, x\right )} + e^{\left (-2 \, x^{4} - 22 \, x^{3} - 68 \, x^{2} - 36 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^3+1056*x^2+2176*x+576)*exp(-x^4-11*x^3-34*x^2-18*x)+32)/(exp(-x^4-11*x^3-34*x^2-18*x)^3-3*x*
exp(-x^4-11*x^3-34*x^2-18*x)^2+3*x^2*exp(-x^4-11*x^3-34*x^2-18*x)-x^3),x, algorithm="fricas")

[Out]

16/(x^2 - 2*x*e^(-x^4 - 11*x^3 - 34*x^2 - 18*x) + e^(-2*x^4 - 22*x^3 - 68*x^2 - 36*x))

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giac [A]  time = 0.92, size = 51, normalized size = 1.89 \begin {gather*} \frac {16}{x^{2} - 2 \, x e^{\left (-x^{4} - 11 \, x^{3} - 34 \, x^{2} - 18 \, x\right )} + e^{\left (-2 \, x^{4} - 22 \, x^{3} - 68 \, x^{2} - 36 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^3+1056*x^2+2176*x+576)*exp(-x^4-11*x^3-34*x^2-18*x)+32)/(exp(-x^4-11*x^3-34*x^2-18*x)^3-3*x*
exp(-x^4-11*x^3-34*x^2-18*x)^2+3*x^2*exp(-x^4-11*x^3-34*x^2-18*x)-x^3),x, algorithm="giac")

[Out]

16/(x^2 - 2*x*e^(-x^4 - 11*x^3 - 34*x^2 - 18*x) + e^(-2*x^4 - 22*x^3 - 68*x^2 - 36*x))

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maple [A]  time = 0.13, size = 26, normalized size = 0.96

method	result	size
risch	\(\frac {16}{\left (x -{\mathrm e}^{-x \left (x^{3}+11 x^{2}+34 x +18\right )}\right )^{2}}\)	\(26\)
norman	\(\frac {16}{\left (x -{\mathrm e}^{-x^{4}-11 x^{3}-34 x^{2}-18 x}\right )^{2}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((128*x^3+1056*x^2+2176*x+576)*exp(-x^4-11*x^3-34*x^2-18*x)+32)/(exp(-x^4-11*x^3-34*x^2-18*x)^3-3*x*exp(-x
^4-11*x^3-34*x^2-18*x)^2+3*x^2*exp(-x^4-11*x^3-34*x^2-18*x)-x^3),x,method=_RETURNVERBOSE)

[Out]

16/(x-exp(-x*(x^3+11*x^2+34*x+18)))^2

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maxima [B]  time = 0.63, size = 71, normalized size = 2.63 \begin {gather*} \frac {16 \, e^{\left (2 \, x^{4} + 22 \, x^{3} + 68 \, x^{2} + 36 \, x\right )}}{x^{2} e^{\left (2 \, x^{4} + 22 \, x^{3} + 68 \, x^{2} + 36 \, x\right )} - 2 \, x e^{\left (x^{4} + 11 \, x^{3} + 34 \, x^{2} + 18 \, x\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^3+1056*x^2+2176*x+576)*exp(-x^4-11*x^3-34*x^2-18*x)+32)/(exp(-x^4-11*x^3-34*x^2-18*x)^3-3*x*
exp(-x^4-11*x^3-34*x^2-18*x)^2+3*x^2*exp(-x^4-11*x^3-34*x^2-18*x)-x^3),x, algorithm="maxima")

[Out]

16*e^(2*x^4 + 22*x^3 + 68*x^2 + 36*x)/(x^2*e^(2*x^4 + 22*x^3 + 68*x^2 + 36*x) - 2*x*e^(x^4 + 11*x^3 + 34*x^2 +
18*x) + 1)

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mupad [B]  time = 0.60, size = 56, normalized size = 2.07 \begin {gather*} \frac {16}{x^2+{\mathrm {e}}^{-36\,x}\,{\mathrm {e}}^{-2\,x^4}\,{\mathrm {e}}^{-22\,x^3}\,{\mathrm {e}}^{-68\,x^2}-2\,x\,{\mathrm {e}}^{-18\,x}\,{\mathrm {e}}^{-x^4}\,{\mathrm {e}}^{-11\,x^3}\,{\mathrm {e}}^{-34\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- 18*x - 34*x^2 - 11*x^3 - x^4)*(2176*x + 1056*x^2 + 128*x^3 + 576) + 32)/(exp(- 54*x - 102*x^2 - 33*
x^3 - 3*x^4) - 3*x*exp(- 36*x - 68*x^2 - 22*x^3 - 2*x^4) + 3*x^2*exp(- 18*x - 34*x^2 - 11*x^3 - x^4) - x^3),x)

[Out]

16/(x^2 + exp(-36*x)*exp(-2*x^4)*exp(-22*x^3)*exp(-68*x^2) - 2*x*exp(-18*x)*exp(-x^4)*exp(-11*x^3)*exp(-34*x^2
))

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sympy [B]  time = 0.23, size = 49, normalized size = 1.81 \begin {gather*} \frac {16}{x^{2} - 2 x e^{- x^{4} - 11 x^{3} - 34 x^{2} - 18 x} + e^{- 2 x^{4} - 22 x^{3} - 68 x^{2} - 36 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x**3+1056*x**2+2176*x+576)*exp(-x**4-11*x**3-34*x**2-18*x)+32)/(exp(-x**4-11*x**3-34*x**2-18*x
)**3-3*x*exp(-x**4-11*x**3-34*x**2-18*x)**2+3*x**2*exp(-x**4-11*x**3-34*x**2-18*x)-x**3),x)

[Out]

16/(x**2 - 2*x*exp(-x**4 - 11*x**3 - 34*x**2 - 18*x) + exp(-2*x**4 - 22*x**3 - 68*x**2 - 36*x))

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