Optimal. Leaf size=127 \[ \frac {2 x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};\frac {2 x^4}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+i\right ) (m+1)}-\frac {2 x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};\frac {2 x^4}{1+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+i\right ) (m+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1375, 364} \[ \frac {2 x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};\frac {2 x^4}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+i\right ) (m+1)}-\frac {2 x^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};\frac {2 x^4}{1+i \sqrt {3}}\right )}{\sqrt {3} \left (-\sqrt {3}+i\right ) (m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1375
Rubi steps
\begin {align*} \int \frac {x^m}{1-x^4+x^8} \, dx &=-\frac {i \int \frac {x^m}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}+x^4} \, dx}{\sqrt {3}}+\frac {i \int \frac {x^m}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}+x^4} \, dx}{\sqrt {3}}\\ &=\frac {2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};\frac {2 x^4}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (i+\sqrt {3}\right ) (1+m)}-\frac {2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};\frac {2 x^4}{1+i \sqrt {3}}\right )}{\sqrt {3} \left (i-\sqrt {3}\right ) (1+m)}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 79, normalized size = 0.62 \[ \frac {x^m \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {\left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right )}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]}{4 m} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{x^{8} - x^{4} + 1}, x\right ) \]
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 \[ \int \frac {x^{m}}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{x^{8}-x^{4}+1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{x^{8} - x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m}{x^8-x^4+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{x^{8} - x^{4} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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