3.87.92 \(\int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} (8 x^3-8 x^5+2 x^7+e^{-x} (4+2 x-4 x^2-x^3))}{4 x^3-4 x^5+x^7} \, dx\)

Optimal. Leaf size=26 \[ e^{2 x-\frac {e^{-x}}{2 x^2-x^4}} \]

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Rubi [F]  time = 3.35, 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^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x + 1/(E^x*(-2*x^2 + x^4)))*(8*x^3 - 8*x^5 + 2*x^7 + (4 + 2*x - 4*x^2 - x^3)/E^x))/(4*x^3 - 4*x^5 +
x^7),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{x^3 \left (4-4 x^2+x^4\right )} \, dx\\ &=\int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{x^3 \left (-2+x^2\right )^2} \, dx\\ &=\int \left (2 e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}}-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} \left (-4-2 x+4 x^2+x^3\right )}{x^3 \left (-2+x^2\right )^2}\right ) \, dx\\ &=2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx-\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} \left (-4-2 x+4 x^2+x^3\right )}{x^3 \left (-2+x^2\right )^2} \, dx\\ &=2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx-\int \left (-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^3}-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{2 x^2}+\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} x}{\left (-2+x^2\right )^2}+\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{2 \left (-2+x^2\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^2} \, dx-\frac {1}{2} \int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{-2+x^2} \, dx+2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx+\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^3} \, dx-\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} x}{\left (-2+x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^2} \, dx-\frac {1}{2} \int \left (-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{2 \sqrt {2} \left (\sqrt {2}-x\right )}-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{2 \sqrt {2} \left (\sqrt {2}+x\right )}\right ) \, dx+2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx+\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^3} \, dx-\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} x}{\left (-2+x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^2} \, dx+2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx+\frac {\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{\sqrt {2}-x} \, dx}{4 \sqrt {2}}+\frac {\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{\sqrt {2}+x} \, dx}{4 \sqrt {2}}+\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^3} \, dx-\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} x}{\left (-2+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.99, size = 22, normalized size = 0.85 \begin {gather*} e^{2 x+\frac {e^{-x}}{x^2 \left (-2+x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x + 1/(E^x*(-2*x^2 + x^4)))*(8*x^3 - 8*x^5 + 2*x^7 + (4 + 2*x - 4*x^2 - x^3)/E^x))/(4*x^3 - 4*
x^5 + x^7),x]

[Out]

E^(2*x + 1/(E^x*x^2*(-2 + x^2)))

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fricas [A]  time = 0.48, size = 28, normalized size = 1.08 \begin {gather*} e^{\left (\frac {2 \, x^{5} - 4 \, x^{3} + e^{\left (-x\right )}}{x^{4} - 2 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-4*x^2+2*x+4)*exp(-x)+2*x^7-8*x^5+8*x^3)*exp(2*x)/(x^7-4*x^5+4*x^3)/exp(-exp(-x)/(x^4-2*x^2)),
x, algorithm="fricas")

[Out]

e^((2*x^5 - 4*x^3 + e^(-x))/(x^4 - 2*x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{7} - 8 \, x^{5} + 8 \, x^{3} - {\left (x^{3} + 4 \, x^{2} - 2 \, x - 4\right )} e^{\left (-x\right )}\right )} e^{\left (2 \, x + \frac {e^{\left (-x\right )}}{x^{4} - 2 \, x^{2}}\right )}}{x^{7} - 4 \, x^{5} + 4 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-4*x^2+2*x+4)*exp(-x)+2*x^7-8*x^5+8*x^3)*exp(2*x)/(x^7-4*x^5+4*x^3)/exp(-exp(-x)/(x^4-2*x^2)),
x, algorithm="giac")

[Out]

integrate((2*x^7 - 8*x^5 + 8*x^3 - (x^3 + 4*x^2 - 2*x - 4)*e^(-x))*e^(2*x + e^(-x)/(x^4 - 2*x^2))/(x^7 - 4*x^5
+ 4*x^3), x)

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maple [A]  time = 0.09, size = 28, normalized size = 1.08

method	result	size
risch	\({\mathrm e}^{\frac {2 x^{5}-4 x^{3}+{\mathrm e}^{-x}}{x^{2} \left (x^{2}-2\right )}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-4*x^2+2*x+4)*exp(-x)+2*x^7-8*x^5+8*x^3)*exp(2*x)/(x^7-4*x^5+4*x^3)/exp(-exp(-x)/(x^4-2*x^2)),x,meth
od=_RETURNVERBOSE)

[Out]

exp((2*x^5-4*x^3+exp(-x))/x^2/(x^2-2))

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maxima [A]  time = 0.49, size = 27, normalized size = 1.04 \begin {gather*} e^{\left (2 \, x + \frac {e^{\left (-x\right )}}{2 \, {\left (x^{2} - 2\right )}} - \frac {e^{\left (-x\right )}}{2 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-4*x^2+2*x+4)*exp(-x)+2*x^7-8*x^5+8*x^3)*exp(2*x)/(x^7-4*x^5+4*x^3)/exp(-exp(-x)/(x^4-2*x^2)),
x, algorithm="maxima")

[Out]

e^(2*x + 1/2*e^(-x)/(x^2 - 2) - 1/2*e^(-x)/x^2)

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mupad [B]  time = 5.72, size = 25, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{-\frac {{\mathrm {e}}^{-x}}{2\,x^2-x^4}}\,{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-exp(-x)/(2*x^2 - x^4))*exp(2*x)*(exp(-x)*(2*x - 4*x^2 - x^3 + 4) + 8*x^3 - 8*x^5 + 2*x^7))/(4*x^3 -
4*x^5 + x^7),x)

[Out]

exp(-exp(-x)/(2*x^2 - x^4))*exp(2*x)

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sympy [A]  time = 0.27, size = 19, normalized size = 0.73 \begin {gather*} e^{2 x} e^{\frac {e^{- x}}{x^{4} - 2 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-4*x**2+2*x+4)*exp(-x)+2*x**7-8*x**5+8*x**3)*exp(2*x)/(x**7-4*x**5+4*x**3)/exp(-exp(-x)/(x**4
-2*x**2)),x)

[Out]

exp(2*x)*exp(exp(-x)/(x**4 - 2*x**2))

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