Optimal. Leaf size=32 \[ \frac {x^{1+m} \, _2F_1\left (2,\frac {1+m}{4};\frac {5+m}{4};-x^4\right )}{1+m} \]
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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, 371}
\begin {gather*} \frac {x^{m+1} \, _2F_1\left (2,\frac {m+1}{4};\frac {m+5}{4};-x^4\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 371
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 = 0.02, size = 34, normalized size = 1.06 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (2,\frac {1+m}{4};1+\frac {1+m}{4};-x^4\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \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^{4} + 1\right )^{2}}\, 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 = -0.03 \begin {gather*} \int \frac {x^m}{x^8+2\,x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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