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 = 0.0779896, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, 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 = 0.40294, size = 212, normalized size = 1.67 \[ \frac{1}{4} x^{m+1} \left (\frac{x^2 \text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2+1\& ,\frac{\, _2F_1\left (1,m+3;m+4;\frac{x}{\text{$\#$1}}\right )}{\text{$\#$1}^2-2}\& \right ]}{m+3}-\frac{\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2+1\& ,\frac{\, _2F_1\left (1,m+1;m+2;\frac{x}{\text{$\#$1}}\right )}{\text{$\#$1}^2-2}\& \right ]}{m+1}+\frac{3 i \left (\sqrt{3}+i\right ) \, _2F_1\left (1,m+1;m+2;-\sqrt [3]{-1} x\right )+3 i \left (\sqrt{3}+i\right ) \, _2F_1\left (1,m+1;m+2;\sqrt [3]{-1} x\right )-6 \left (\, _2F_1\left (1,m+1;m+2;-(-1)^{2/3} x\right )+\, _2F_1\left (1,m+1;m+2;(-1)^{2/3} x\right )\right )}{6 \left (\sqrt [3]{-1}-2\right ) (m+1)}\right ) \]
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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, 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} + x^{4} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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