3.6.57 \(\int \frac {e^{3 e} (8-2 x)+e^{6 e+x} (-8-2 x+2 x^2)}{x^3-2 e^{3 e+x} x^3+e^{6 e+2 x} x^3} \, dx\)

Optimal. Leaf size=27 \[ -3+e+\frac {2 (-2+x)}{x \left (e^{-3 e} x-e^x x\right )} \]

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Rubi [F]  time = 1.18, 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 {e^{3 e} (8-2 x)+e^{6 e+x} \left (-8-2 x+2 x^2\right )}{x^3-2 e^{3 e+x} x^3+e^{6 e+2 x} x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(3*E)*(8 - 2*x) + E^(6*E + x)*(-8 - 2*x + 2*x^2))/(x^3 - 2*E^(3*E + x)*x^3 + E^(6*E + 2*x)*x^3),x]

[Out]

-8*E^(3*E)*Defer[Int][1/((-1 + E^(3*E + x))*x^3), x] - 4*E^(3*E)*Defer[Int][1/((-1 + E^(3*E + x))^2*x^2), x] -
2*E^(3*E)*Defer[Int][1/((-1 + E^(3*E + x))*x^2), x] + 2*E^(3*E)*Defer[Int][1/((-1 + E^(3*E + x))^2*x), x] + 2
*E^(3*E)*Defer[Int][1/((-1 + E^(3*E + x))*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{3 e} \left (4-x+e^{3 e+x} \left (-4-x+x^2\right )\right )}{\left (1-e^{3 e+x}\right )^2 x^3} \, dx\\ &=\left (2 e^{3 e}\right ) \int \frac {4-x+e^{3 e+x} \left (-4-x+x^2\right )}{\left (1-e^{3 e+x}\right )^2 x^3} \, dx\\ &=\left (2 e^{3 e}\right ) \int \left (\frac {-2+x}{\left (-1+e^{3 e+x}\right )^2 x^2}+\frac {-4-x+x^2}{\left (-1+e^{3 e+x}\right ) x^3}\right ) \, dx\\ &=\left (2 e^{3 e}\right ) \int \frac {-2+x}{\left (-1+e^{3 e+x}\right )^2 x^2} \, dx+\left (2 e^{3 e}\right ) \int \frac {-4-x+x^2}{\left (-1+e^{3 e+x}\right ) x^3} \, dx\\ &=\left (2 e^{3 e}\right ) \int \left (-\frac {2}{\left (-1+e^{3 e+x}\right )^2 x^2}+\frac {1}{\left (-1+e^{3 e+x}\right )^2 x}\right ) \, dx+\left (2 e^{3 e}\right ) \int \left (-\frac {4}{\left (-1+e^{3 e+x}\right ) x^3}-\frac {1}{\left (-1+e^{3 e+x}\right ) x^2}+\frac {1}{\left (-1+e^{3 e+x}\right ) x}\right ) \, dx\\ &=-\left (\left (2 e^{3 e}\right ) \int \frac {1}{\left (-1+e^{3 e+x}\right ) x^2} \, dx\right )+\left (2 e^{3 e}\right ) \int \frac {1}{\left (-1+e^{3 e+x}\right )^2 x} \, dx+\left (2 e^{3 e}\right ) \int \frac {1}{\left (-1+e^{3 e+x}\right ) x} \, dx-\left (4 e^{3 e}\right ) \int \frac {1}{\left (-1+e^{3 e+x}\right )^2 x^2} \, dx-\left (8 e^{3 e}\right ) \int \frac {1}{\left (-1+e^{3 e+x}\right ) x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 26, normalized size = 0.96 \begin {gather*} \frac {2 e^{3 e} (2-x)}{\left (-1+e^{3 e+x}\right ) x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*E)*(8 - 2*x) + E^(6*E + x)*(-8 - 2*x + 2*x^2))/(x^3 - 2*E^(3*E + x)*x^3 + E^(6*E + 2*x)*x^3),x
]

[Out]

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

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fricas [A]  time = 0.63, size = 34, normalized size = 1.26 \begin {gather*} -\frac {2 \, {\left (x - 2\right )} e^{\left (6 \, e\right )}}{x^{2} e^{\left (x + 6 \, e\right )} - x^{2} e^{\left (3 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-2*x-8)*exp(x)*exp(3*exp(1))^2+(-2*x+8)*exp(3*exp(1)))/(x^3*exp(x)^2*exp(3*exp(1))^2-2*x^3*ex
p(x)*exp(3*exp(1))+x^3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.32, size = 36, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (x e^{\left (3 \, e\right )} - 2 \, e^{\left (3 \, e\right )}\right )}}{x^{2} e^{\left (x + 3 \, e\right )} - x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-2*x-8)*exp(x)*exp(3*exp(1))^2+(-2*x+8)*exp(3*exp(1)))/(x^3*exp(x)^2*exp(3*exp(1))^2-2*x^3*ex
p(x)*exp(3*exp(1))+x^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.25, size = 25, normalized size = 0.93

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{3 \,{\mathrm e}} \left (x -2\right )}{x^{2} \left ({\mathrm e}^{x +3 \,{\mathrm e}}-1\right )}\)	\(25\)
norman	\(\frac {4 \,{\mathrm e}^{3 \,{\mathrm e}}-2 \,{\mathrm e}^{3 \,{\mathrm e}} x}{x^{2} \left ({\mathrm e}^{x} {\mathrm e}^{3 \,{\mathrm e}}-1\right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-2*x-8)*exp(x)*exp(3*exp(1))^2+(-2*x+8)*exp(3*exp(1)))/(x^3*exp(x)^2*exp(3*exp(1))^2-2*x^3*exp(x)*e
xp(3*exp(1))+x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 1.05, size = 36, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (x e^{\left (3 \, e\right )} - 2 \, e^{\left (3 \, e\right )}\right )}}{x^{2} e^{\left (x + 3 \, e\right )} - x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-2*x-8)*exp(x)*exp(3*exp(1))^2+(-2*x+8)*exp(3*exp(1)))/(x^3*exp(x)^2*exp(3*exp(1))^2-2*x^3*ex
p(x)*exp(3*exp(1))+x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.60, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2\,{\mathrm {e}}^{3\,\mathrm {e}}\,\left (x-2\right )}{x^2\,\left ({\mathrm {e}}^{x+3\,\mathrm {e}}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(3*exp(1))*(2*x - 8) + exp(6*exp(1))*exp(x)*(2*x - 2*x^2 + 8))/(x^3 - 2*x^3*exp(3*exp(1))*exp(x) + x^
3*exp(6*exp(1))*exp(2*x)),x)

[Out]

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

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sympy [A]  time = 0.13, size = 34, normalized size = 1.26 \begin {gather*} \frac {- 2 x e^{3 e} + 4 e^{3 e}}{x^{2} e^{3 e} e^{x} - x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-2*x-8)*exp(x)*exp(3*exp(1))**2+(-2*x+8)*exp(3*exp(1)))/(x**3*exp(x)**2*exp(3*exp(1))**2-2*x
**3*exp(x)*exp(3*exp(1))+x**3),x)

[Out]

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

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