3.81.80 \(\int \frac {e^{-\frac {6 x}{e^x-x^2}} (e^{2 x}-6 x^3+x^4+e^x (-6 x+4 x^2))}{e^{2 x}-2 e^x x^2+x^4} \, dx\)

Optimal. Leaf size=18 \[ e^{\frac {6}{-\frac {e^x}{x}+x}} x \]

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Rubi [F]  time = 3.52, 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^{-\frac {6 x}{e^x-x^2}} \left (e^{2 x}-6 x^3+x^4+e^x \left (-6 x+4 x^2\right )\right )}{e^{2 x}-2 e^x x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x) - 6*x^3 + x^4 + E^x*(-6*x + 4*x^2))/(E^((6*x)/(E^x - x^2))*(E^(2*x) - 2*E^x*x^2 + x^4)),x]

[Out]

Defer[Int][E^((-6*x)/(E^x - x^2)), x] - 12*Defer[Int][x^3/(E^((6*x)/(E^x - x^2))*(-E^x + x^2)^2), x] + 6*Defer
[Int][x^4/(E^((6*x)/(E^x - x^2))*(-E^x + x^2)^2), x] + 6*Defer[Int][x/(E^((6*x)/(E^x - x^2))*(-E^x + x^2)), x]
- 6*Defer[Int][x^2/(E^((6*x)/(E^x - x^2))*(-E^x + x^2)), x]

Rubi steps

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

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Mathematica [A]  time = 0.55, size = 18, normalized size = 1.00 \begin {gather*} e^{-\frac {6 x}{e^x-x^2}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x) - 6*x^3 + x^4 + E^x*(-6*x + 4*x^2))/(E^((6*x)/(E^x - x^2))*(E^(2*x) - 2*E^x*x^2 + x^4)),x]

[Out]

x/E^((6*x)/(E^x - x^2))

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fricas [A]  time = 0.58, size = 16, normalized size = 0.89 \begin {gather*} x e^{\left (\frac {6 \, x}{x^{2} - e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+(4*x^2-6*x)*exp(x)+x^4-6*x^3)*exp(-6*x/(exp(x)-x^2))/(exp(x)^2-2*exp(x)*x^2+x^4),x, algori
thm="fricas")

[Out]

x*e^(6*x/(x^2 - e^x))

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giac [A]  time = 1.88, size = 30, normalized size = 1.67 \begin {gather*} x e^{\left (-x + \frac {x^{3} - x e^{x} + 6 \, x}{x^{2} - e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+(4*x^2-6*x)*exp(x)+x^4-6*x^3)*exp(-6*x/(exp(x)-x^2))/(exp(x)^2-2*exp(x)*x^2+x^4),x, algori
thm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 17, normalized size = 0.94

method	result	size
risch	\({\mathrm e}^{-\frac {6 x}{{\mathrm e}^{x}-x^{2}}} x\)	\(17\)
norman	\(\frac {x^{3} {\mathrm e}^{-\frac {6 x}{{\mathrm e}^{x}-x^{2}}}-{\mathrm e}^{x} x \,{\mathrm e}^{-\frac {6 x}{{\mathrm e}^{x}-x^{2}}}}{-{\mathrm e}^{x}+x^{2}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)^2+(4*x^2-6*x)*exp(x)+x^4-6*x^3)*exp(-6*x/(exp(x)-x^2))/(exp(x)^2-2*exp(x)*x^2+x^4),x,method=_RETUR
NVERBOSE)

[Out]

exp(-6*x/(exp(x)-x^2))*x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 6 \, x^{3} + 2 \, {\left (2 \, x^{2} - 3 \, x\right )} e^{x} + e^{\left (2 \, x\right )}\right )} e^{\left (\frac {6 \, x}{x^{2} - e^{x}}\right )}}{x^{4} - 2 \, x^{2} e^{x} + e^{\left (2 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)^2+(4*x^2-6*x)*exp(x)+x^4-6*x^3)*exp(-6*x/(exp(x)-x^2))/(exp(x)^2-2*exp(x)*x^2+x^4),x, algori
thm="maxima")

[Out]

integrate((x^4 - 6*x^3 + 2*(2*x^2 - 3*x)*e^x + e^(2*x))*e^(6*x/(x^2 - e^x))/(x^4 - 2*x^2*e^x + e^(2*x)), x)

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mupad [B]  time = 5.33, size = 16, normalized size = 0.89 \begin {gather*} x\,{\mathrm {e}}^{-\frac {6\,x}{{\mathrm {e}}^x-x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(6*x)/(exp(x) - x^2))*(exp(2*x) - exp(x)*(6*x - 4*x^2) - 6*x^3 + x^4))/(exp(2*x) - 2*x^2*exp(x) + x^
4),x)

[Out]

x*exp(-(6*x)/(exp(x) - x^2))

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sympy [A]  time = 1.52, size = 12, normalized size = 0.67 \begin {gather*} x e^{- \frac {6 x}{- x^{2} + e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)**2+(4*x**2-6*x)*exp(x)+x**4-6*x**3)*exp(-6*x/(exp(x)-x**2))/(exp(x)**2-2*exp(x)*x**2+x**4),x
)

[Out]

x*exp(-6*x/(-x**2 + exp(x)))

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