Optimal. Leaf size=127 \[ \frac {2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};-\frac {2 x^4}{1-i \sqrt {3}}\right )}{\sqrt {3} \left (i+\sqrt {3}\right ) (1+m)}-\frac {2 x^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};-\frac {2 x^4}{1+i \sqrt {3}}\right )}{\sqrt {3} \left (i-\sqrt {3}\right ) (1+m)} \]
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Rubi [A]
time = 0.06, antiderivative size = 127, normalized size of antiderivative = 1.00, 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 = {1389, 371}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 1389
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] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 0.94, size = 488, normalized size = 3.84 \begin {gather*} \frac {x^m \left (-\frac {i \left (\left (\frac {x}{-\sqrt [3]{-1}+x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {\sqrt [3]{-1}}{\sqrt [3]{-1}-x}\right )+\left (\frac {x}{-(-1)^{2/3}+x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {(-1)^{2/3}}{(-1)^{2/3}-x}\right )-\left (\frac {x}{\sqrt [3]{-1}+x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {\sqrt [3]{-1}}{\sqrt [3]{-1}+x}\right )-\left (\frac {x}{(-1)^{2/3}+x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {(-1)^{2/3}}{(-1)^{2/3}+x}\right )\right )}{\sqrt {3}}+\text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\, _2F_1\left (-m,-m;1-m;-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m}}{-\text {$\#$1}+2 \text {$\#$1}^3}\&\right ]-\frac {\text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {m x^2+m^2 x^2+2 m x \text {$\#$1}+m^2 x \text {$\#$1}+2 \, _2F_1\left (-m,-m;1-m;-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \text {$\#$1}^2+3 m \, _2F_1\left (-m,-m;1-m;-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \text {$\#$1}^2+m^2 \, _2F_1\left (-m,-m;1-m;-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m} \text {$\#$1}^2+m \left (\frac {x}{\text {$\#$1}}\right )^{-m} \text {$\#$1}^2}{-\text {$\#$1}+2 \text {$\#$1}^3}\&\right ]}{2+3 m+m^2}\right )}{4 m} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^m}{x^8+x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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