Optimal. Leaf size=26 \[ \frac {9 \left (x+\left (-2+e^{e^x}\right ) \left (-5+x^2\right )\right )}{2-x^2} \]
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Rubi [F] time = 0.59, 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 {18+108 x+9 x^2+e^{e^x} \left (-54 x+e^x \left (-90+63 x^2-9 x^4\right )\right )}{4-4 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {18+108 x+9 x^2+e^{e^x} \left (-54 x+e^x \left (-90+63 x^2-9 x^4\right )\right )}{\left (-2+x^2\right )^2} \, dx\\ &=\int \left (\frac {18}{\left (-2+x^2\right )^2}+\frac {108 x}{\left (-2+x^2\right )^2}-\frac {54 e^{e^x} x}{\left (-2+x^2\right )^2}+\frac {9 x^2}{\left (-2+x^2\right )^2}-\frac {9 e^{e^x+x} \left (-5+x^2\right )}{-2+x^2}\right ) \, dx\\ &=9 \int \frac {x^2}{\left (-2+x^2\right )^2} \, dx-9 \int \frac {e^{e^x+x} \left (-5+x^2\right )}{-2+x^2} \, dx+18 \int \frac {1}{\left (-2+x^2\right )^2} \, dx-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx+108 \int \frac {x}{\left (-2+x^2\right )^2} \, dx\\ &=\frac {54}{2-x^2}+\frac {9 x}{2-x^2}-9 \int \left (e^{e^x+x}-\frac {3 e^{e^x+x}}{-2+x^2}\right ) \, dx-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx\\ &=\frac {54}{2-x^2}+\frac {9 x}{2-x^2}-9 \int e^{e^x+x} \, dx+27 \int \frac {e^{e^x+x}}{-2+x^2} \, dx-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx\\ &=\frac {54}{2-x^2}+\frac {9 x}{2-x^2}-9 \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )+27 \int \left (-\frac {e^{e^x+x}}{2 \sqrt {2} \left (\sqrt {2}-x\right )}-\frac {e^{e^x+x}}{2 \sqrt {2} \left (\sqrt {2}+x\right )}\right ) \, dx-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx\\ &=-9 e^{e^x}+\frac {54}{2-x^2}+\frac {9 x}{2-x^2}-54 \int \frac {e^{e^x} x}{\left (-2+x^2\right )^2} \, dx-\frac {27 \int \frac {e^{e^x+x}}{\sqrt {2}-x} \, dx}{2 \sqrt {2}}-\frac {27 \int \frac {e^{e^x+x}}{\sqrt {2}+x} \, dx}{2 \sqrt {2}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 23, normalized size = 0.88 \begin {gather*} -\frac {9 \left (6+x+e^{e^x} \left (-5+x^2\right )\right )}{-2+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 21, normalized size = 0.81 \begin {gather*} -\frac {9 \, {\left ({\left (x^{2} - 5\right )} e^{\left (e^{x}\right )} + x + 6\right )}}{x^{2} - 2} \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 {9 \, {\left (x^{2} - {\left ({\left (x^{4} - 7 \, x^{2} + 10\right )} e^{x} + 6 \, x\right )} e^{\left (e^{x}\right )} + 12 \, x + 2\right )}}{x^{4} - 4 \, x^{2} + 4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 27, normalized size = 1.04
method | result | size |
norman | \(\frac {-9 x -9 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2}+45 \,{\mathrm e}^{{\mathrm e}^{x}}-54}{x^{2}-2}\) | \(27\) |
risch | \(\frac {-9 x -54}{x^{2}-2}-\frac {9 \left (x^{2}-5\right ) {\mathrm e}^{{\mathrm e}^{x}}}{x^{2}-2}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 37, normalized size = 1.42 \begin {gather*} -\frac {9 \, {\left (x^{2} - 5\right )} e^{\left (e^{x}\right )}}{x^{2} - 2} - \frac {9 \, x}{x^{2} - 2} - \frac {54}{x^{2} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 34, normalized size = 1.31 \begin {gather*} -\frac {9\,x+54}{x^2-2}-\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (9\,x^2-45\right )}{x^2-2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 27, normalized size = 1.04 \begin {gather*} \frac {\left (45 - 9 x^{2}\right ) e^{e^{x}}}{x^{2} - 2} + \frac {- 9 x - 54}{x^{2} - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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