Optimal. Leaf size=32 \[ \frac {x^{m+1} \, _2F_1\left (2,\frac {m+1}{4};\frac {m+5}{4};-x^4\right )}{m+1} \]
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Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 364} \[ \frac {x^{m+1} \, _2F_1\left (2,\frac {m+1}{4};\frac {m+5}{4};-x^4\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 28
Rule 364
Rubi steps
\begin {align*} \int \frac {x^m}{1+2 x^4+x^8} \, dx &=\int \frac {x^m}{\left (1+x^4\right )^2} \, dx\\ &=\frac {x^{1+m} \, _2F_1\left (2,\frac {1+m}{4};\frac {5+m}{4};-x^4\right )}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.06 \[ \frac {x^{m+1} \, _2F_1\left (2,\frac {m+1}{4};\frac {m+1}{4}+1;-x^4\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{x^{8} + 2 \, x^{4} + 1}, x\right ) \]
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 \[ \int \frac {x^{m}}{x^{8} + 2 \, x^{4} + 1}\,{d x} \]
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 \[ \int \frac {x^{m}}{x^{8}+2 x^{4}+1}\, dx \]
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 \[ \int \frac {x^{m}}{x^{8} + 2 \, x^{4} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m}{x^8+2\,x^4+1} \,d x \]
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 \[ \int \frac {x^{m}}{\left (x^{4} + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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