Optimal. Leaf size=30 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+5}{4};x^4\right )}{m+1} \]
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Rubi [A] time = 0.0056888, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 364} \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+5}{4};x^4\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 28
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m}{1-2 x^4+x^8} \, dx &=\int \frac{x^m}{\left (-1+x^4\right )^2} \, dx\\ &=\frac{x^{1+m} \, _2F_1\left (2,\frac{1+m}{4};\frac{5+m}{4};x^4\right )}{1+m}\\ \end{align*}
Mathematica [A] time = 0.006415, size = 32, normalized size = 1.07 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+1}{4}+1;x^4\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{{x}^{8}-2\,{x}^{4}+1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{x^{8} - 2 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m}}{x^{8} - 2 \, x^{4} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{x^{8} - 2 \, x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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