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 = 0.0486989, antiderivative size = 127, normalized size of antiderivative = 1., 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 1375
Rule 364
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*}
Mathematica [C] time = 0.0374923, size = 52, normalized size = 0.41 \[ -\frac{x^{m+1} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\, _2F_1\left (1,m+1;m+2;\frac{x}{\text{$\#$1}}\right )}{\text{$\#$1}^4-2}\& \right ]}{4 (m+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{{x}^{8}-{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} - 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} - x^{4} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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} - x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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