3.6.99 \(\int \frac {e^{-\frac {e^3 (-3 x+x^3)}{1+e^3 (36 x-24 x^2+4 x^3)}} (e^3 (3-3 x^2)+e^6 (72 x^2-96 x^3+24 x^4))}{1+e^3 (72 x-48 x^2+8 x^3)+e^6 (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6)} \, dx\)

Optimal. Leaf size=28 \[ 4+e^{-\frac {-3+x^2}{(6-2 x)^2+\frac {1}{e^3 x}}} \]

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Rubi [F]  time = 2.73, 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 {e^3 \left (-3 x+x^3\right )}{1+e^3 \left (36 x-24 x^2+4 x^3\right )}\right ) \left (e^3 \left (3-3 x^2\right )+e^6 \left (72 x^2-96 x^3+24 x^4\right )\right )}{1+e^3 \left (72 x-48 x^2+8 x^3\right )+e^6 \left (1296 x^2-1728 x^3+864 x^4-192 x^5+16 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^3*(3 - 3*x^2) + E^6*(72*x^2 - 96*x^3 + 24*x^4))/(E^((E^3*(-3*x + x^3))/(1 + E^3*(36*x - 24*x^2 + 4*x^3)
))*(1 + E^3*(72*x - 48*x^2 + 8*x^3) + E^6*(1296*x^2 - 1728*x^3 + 864*x^4 - 192*x^5 + 16*x^6))),x]

[Out]

-9*Defer[Int][E^(3 - (E^3*x*(-3 + x^2))/(1 + 36*E^3*x - 24*E^3*x^2 + 4*E^3*x^3))/(1 + 36*E^3*x - 24*E^3*x^2 +
4*E^3*x^3)^2, x] - 6*(1 + 72*E^3)*Defer[Int][(E^(3 - (E^3*x*(-3 + x^2))/(1 + 36*E^3*x - 24*E^3*x^2 + 4*E^3*x^3
))*x)/(1 + 36*E^3*x - 24*E^3*x^2 + 4*E^3*x^3)^2, x] - 3*(1 - 48*E^3)*Defer[Int][(E^(3 - (E^3*x*(-3 + x^2))/(1
+ 36*E^3*x - 24*E^3*x^2 + 4*E^3*x^3))*x^2)/(1 + 36*E^3*x - 24*E^3*x^2 + 4*E^3*x^3)^2, x] + 12*Defer[Int][E^(3
- (E^3*x*(-3 + x^2))/(1 + 36*E^3*x - 24*E^3*x^2 + 4*E^3*x^3))/(1 + 36*E^3*x - 24*E^3*x^2 + 4*E^3*x^3), x] + 6*
Defer[Int][(E^(3 - (E^3*x*(-3 + x^2))/(1 + 36*E^3*x - 24*E^3*x^2 + 4*E^3*x^3))*x)/(1 + 36*E^3*x - 24*E^3*x^2 +
4*E^3*x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) \left (3 e^3-3 e^3 \left (1-24 e^3\right ) x^2-96 e^6 x^3+24 e^6 x^4\right )}{\left (1+36 e^3 x-24 e^3 x^2+4 e^3 x^3\right )^2} \, dx\\ &=\int \left (\frac {3 \exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) \left (-3-2 \left (1+72 e^3\right ) x-\left (1-48 e^3\right ) x^2\right )}{\left (1+36 e^3 x-24 e^3 x^2+4 e^3 x^3\right )^2}+\frac {6 \exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) (2+x)}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) \, dx\\ &=3 \int \frac {\exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) \left (-3-2 \left (1+72 e^3\right ) x-\left (1-48 e^3\right ) x^2\right )}{\left (1+36 e^3 x-24 e^3 x^2+4 e^3 x^3\right )^2} \, dx+6 \int \frac {\exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) (2+x)}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3} \, dx\\ &=3 \int \left (-\frac {3 \exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right )}{\left (1+36 e^3 x-24 e^3 x^2+4 e^3 x^3\right )^2}-\frac {2 \exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) \left (1+72 e^3\right ) x}{\left (1+36 e^3 x-24 e^3 x^2+4 e^3 x^3\right )^2}+\frac {\exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) \left (-1+48 e^3\right ) x^2}{\left (1+36 e^3 x-24 e^3 x^2+4 e^3 x^3\right )^2}\right ) \, dx+6 \int \left (\frac {2 \exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}+\frac {\exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) x}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) \, dx\\ &=6 \int \frac {\exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) x}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3} \, dx-9 \int \frac {\exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right )}{\left (1+36 e^3 x-24 e^3 x^2+4 e^3 x^3\right )^2} \, dx+12 \int \frac {\exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3} \, dx-\left (3 \left (1-48 e^3\right )\right ) \int \frac {\exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) x^2}{\left (1+36 e^3 x-24 e^3 x^2+4 e^3 x^3\right )^2} \, dx-\left (6 \left (1+72 e^3\right )\right ) \int \frac {\exp \left (3-\frac {e^3 x \left (-3+x^2\right )}{1+36 e^3 x-24 e^3 x^2+4 e^3 x^3}\right ) x}{\left (1+36 e^3 x-24 e^3 x^2+4 e^3 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.75, size = 28, normalized size = 1.00 \begin {gather*} e^{-\frac {e^3 x \left (-3+x^2\right )}{1+4 e^3 (-3+x)^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^3*(3 - 3*x^2) + E^6*(72*x^2 - 96*x^3 + 24*x^4))/(E^((E^3*(-3*x + x^3))/(1 + E^3*(36*x - 24*x^2 +
4*x^3)))*(1 + E^3*(72*x - 48*x^2 + 8*x^3) + E^6*(1296*x^2 - 1728*x^3 + 864*x^4 - 192*x^5 + 16*x^6))),x]

[Out]

E^(-((E^3*x*(-3 + x^2))/(1 + 4*E^3*(-3 + x)^2*x)))

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fricas [A]  time = 0.79, size = 32, normalized size = 1.14 \begin {gather*} e^{\left (-\frac {{\left (x^{3} - 3 \, x\right )} e^{3}}{4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{3} + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192*x^5+864*x^4-1728*x^3+1296*x^2)*exp(
3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x, algorithm="fricas")

[Out]

e^(-(x^3 - 3*x)*e^3/(4*(x^3 - 6*x^2 + 9*x)*e^3 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, {\left (8 \, {\left (x^{4} - 4 \, x^{3} + 3 \, x^{2}\right )} e^{6} - {\left (x^{2} - 1\right )} e^{3}\right )} e^{\left (-\frac {{\left (x^{3} - 3 \, x\right )} e^{3}}{4 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{3} + 1}\right )}}{16 \, {\left (x^{6} - 12 \, x^{5} + 54 \, x^{4} - 108 \, x^{3} + 81 \, x^{2}\right )} e^{6} + 8 \, {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} e^{3} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192*x^5+864*x^4-1728*x^3+1296*x^2)*exp(
3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x, algorithm="giac")

[Out]

integrate(3*(8*(x^4 - 4*x^3 + 3*x^2)*e^6 - (x^2 - 1)*e^3)*e^(-(x^3 - 3*x)*e^3/(4*(x^3 - 6*x^2 + 9*x)*e^3 + 1))
/(16*(x^6 - 12*x^5 + 54*x^4 - 108*x^3 + 81*x^2)*e^6 + 8*(x^3 - 6*x^2 + 9*x)*e^3 + 1), x)

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maple [A]  time = 0.35, size = 35, normalized size = 1.25

method	result	size
norman	\({\mathrm e}^{-\frac {\left (x^{3}-3 x \right ) {\mathrm e}^{3}}{\left (4 x^{3}-24 x^{2}+36 x \right ) {\mathrm e}^{3}+1}}\)	\(35\)
risch	\({\mathrm e}^{-\frac {x \left (x^{2}-3\right ) {\mathrm e}^{3}}{4 x^{3} {\mathrm e}^{3}-24 x^{2} {\mathrm e}^{3}+36 x \,{\mathrm e}^{3}+1}}\)	\(35\)
gosper	\({\mathrm e}^{-\frac {x \left (x^{2}-3\right ) {\mathrm e}^{3}}{4 x^{3} {\mathrm e}^{3}-24 x^{2} {\mathrm e}^{3}+36 x \,{\mathrm e}^{3}+1}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192*x^5+864*x^4-1728*x^3+1296*x^2)*exp(3)^2+(
8*x^3-48*x^2+72*x)*exp(3)+1)/exp((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x,method=_RETURNVERBOSE)

[Out]

1/exp((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1))

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maxima [B]  time = 1.38, size = 86, normalized size = 3.07 \begin {gather*} e^{\left (-\frac {6 \, x^{2} e^{3}}{4 \, x^{3} e^{3} - 24 \, x^{2} e^{3} + 36 \, x e^{3} + 1} + \frac {12 \, x e^{3}}{4 \, x^{3} e^{3} - 24 \, x^{2} e^{3} + 36 \, x e^{3} + 1} + \frac {1}{4 \, {\left (4 \, x^{3} e^{3} - 24 \, x^{2} e^{3} + 36 \, x e^{3} + 1\right )}} - \frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^4-96*x^3+72*x^2)*exp(3)^2+(-3*x^2+3)*exp(3))/((16*x^6-192*x^5+864*x^4-1728*x^3+1296*x^2)*exp(
3)^2+(8*x^3-48*x^2+72*x)*exp(3)+1)/exp((x^3-3*x)*exp(3)/((4*x^3-24*x^2+36*x)*exp(3)+1)),x, algorithm="maxima")

[Out]

e^(-6*x^2*e^3/(4*x^3*e^3 - 24*x^2*e^3 + 36*x*e^3 + 1) + 12*x*e^3/(4*x^3*e^3 - 24*x^2*e^3 + 36*x*e^3 + 1) + 1/4
/(4*x^3*e^3 - 24*x^2*e^3 + 36*x*e^3 + 1) - 1/4)

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mupad [B]  time = 4.75, size = 38, normalized size = 1.36 \begin {gather*} {\mathrm {e}}^{\frac {3\,x\,{\mathrm {e}}^3-x^3\,{\mathrm {e}}^3}{4\,{\mathrm {e}}^3\,x^3-24\,{\mathrm {e}}^3\,x^2+36\,{\mathrm {e}}^3\,x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(3)*(3*x - x^3))/(exp(3)*(36*x - 24*x^2 + 4*x^3) + 1))*(exp(3)*(3*x^2 - 3) - exp(6)*(72*x^2 - 96
*x^3 + 24*x^4)))/(exp(6)*(1296*x^2 - 1728*x^3 + 864*x^4 - 192*x^5 + 16*x^6) + exp(3)*(72*x - 48*x^2 + 8*x^3) +
1),x)

[Out]

exp((3*x*exp(3) - x^3*exp(3))/(36*x*exp(3) - 24*x^2*exp(3) + 4*x^3*exp(3) + 1))

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sympy [A]  time = 1.50, size = 31, normalized size = 1.11 \begin {gather*} e^{- \frac {\left (x^{3} - 3 x\right ) e^{3}}{\left (4 x^{3} - 24 x^{2} + 36 x\right ) e^{3} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x**4-96*x**3+72*x**2)*exp(3)**2+(-3*x**2+3)*exp(3))/((16*x**6-192*x**5+864*x**4-1728*x**3+1296*
x**2)*exp(3)**2+(8*x**3-48*x**2+72*x)*exp(3)+1)/exp((x**3-3*x)*exp(3)/((4*x**3-24*x**2+36*x)*exp(3)+1)),x)

[Out]

exp(-(x**3 - 3*x)*exp(3)/((4*x**3 - 24*x**2 + 36*x)*exp(3) + 1))

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