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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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