Integrand size = 16, antiderivative size = 117 \[ \int \frac {x^m}{1-3 x^4+x^8} \, dx=\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},\frac {2 x^4}{3-\sqrt {5}}\right )}{\sqrt {5} \left (3-\sqrt {5}\right ) (1+m)}-\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},\frac {2 x^4}{3+\sqrt {5}}\right )}{\sqrt {5} \left (3+\sqrt {5}\right ) (1+m)} \]
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Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1389, 371} \[ \int \frac {x^m}{1-3 x^4+x^8} \, dx=\frac {2 x^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{4},\frac {m+5}{4},\frac {2 x^4}{3-\sqrt {5}}\right )}{\sqrt {5} \left (3-\sqrt {5}\right ) (m+1)}-\frac {2 x^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{4},\frac {m+5}{4},\frac {2 x^4}{3+\sqrt {5}}\right )}{\sqrt {5} \left (3+\sqrt {5}\right ) (m+1)} \]
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Rule 371
Rule 1389
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {x^m}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}}-\frac {\int \frac {x^m}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}} \\ & = \frac {2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};\frac {2 x^4}{3-\sqrt {5}}\right )}{\sqrt {5} \left (3-\sqrt {5}\right ) (1+m)}-\frac {2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};\frac {2 x^4}{3+\sqrt {5}}\right )}{\sqrt {5} \left (3+\sqrt {5}\right ) (1+m)} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.60 (sec) , antiderivative size = 575, normalized size of antiderivative = 4.91 \[ \int \frac {x^m}{1-3 x^4+x^8} \, dx=\frac {x^m \left (-\text {RootSum}\left [-1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m}}{-\text {$\#$1}+2 \text {$\#$1}^3}\&\right ]+\frac {\text {RootSum}\left [-1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {m x^2+m^2 x^2+2 m x \text {$\#$1}+m^2 x \text {$\#$1}+2 \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \text {$\#$1}^2+3 m \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \text {$\#$1}^2+m^2 \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \text {$\#$1}^2+m \left (\frac {x}{\text {$\#$1}}\right )^{-m} \text {$\#$1}^2}{-\text {$\#$1}+2 \text {$\#$1}^3}\&\right ]-\left (2+3 m+m^2\right ) \text {RootSum}\left [-1+\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m}}{\text {$\#$1}+2 \text {$\#$1}^3}\&\right ]-\text {RootSum}\left [-1+\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {m x^2+m^2 x^2+2 m x \text {$\#$1}+m^2 x \text {$\#$1}+2 \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \text {$\#$1}^2+3 m \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \text {$\#$1}^2+m^2 \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \text {$\#$1}^2+m \left (\frac {x}{\text {$\#$1}}\right )^{-m} \text {$\#$1}^2}{\text {$\#$1}+2 \text {$\#$1}^3}\&\right ]}{2+3 m+m^2}\right )}{4 m} \]
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\[\int \frac {x^{m}}{x^{8}-3 x^{4}+1}d x\]
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\[ \int \frac {x^m}{1-3 x^4+x^8} \, dx=\int { \frac {x^{m}}{x^{8} - 3 \, x^{4} + 1} \,d x } \]
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\[ \int \frac {x^m}{1-3 x^4+x^8} \, dx=\int \frac {x^{m}}{\left (x^{4} - x^{2} - 1\right ) \left (x^{4} + x^{2} - 1\right )}\, dx \]
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\[ \int \frac {x^m}{1-3 x^4+x^8} \, dx=\int { \frac {x^{m}}{x^{8} - 3 \, x^{4} + 1} \,d x } \]
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\[ \int \frac {x^m}{1-3 x^4+x^8} \, dx=\int { \frac {x^{m}}{x^{8} - 3 \, x^{4} + 1} \,d x } \]
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Timed out. \[ \int \frac {x^m}{1-3 x^4+x^8} \, dx=\int \frac {x^m}{x^8-3\,x^4+1} \,d x \]
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