Optimal. Leaf size=23 \[ \frac {1}{4} \tan ^{-1}\left (x^2\right )-\frac {x^2}{4 \left (x^4+1\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {28, 275, 288, 203} \[ \frac {1}{4} \tan ^{-1}\left (x^2\right )-\frac {x^2}{4 \left (x^4+1\right )} \]
Antiderivative was successfully verified.
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Rule 28
Rule 203
Rule 275
Rule 288
Rubi steps
\begin {align*} \int \frac {x^5}{1+2 x^4+x^8} \, dx &=\int \frac {x^5}{\left (1+x^4\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {x^2}{4 \left (1+x^4\right )}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,x^2\right )\\ &=-\frac {x^2}{4 \left (1+x^4\right )}+\frac {1}{4} \tan ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ \frac {1}{4} \tan ^{-1}\left (x^2\right )-\frac {x^2}{4 \left (x^4+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 24, normalized size = 1.04 \[ -\frac {x^{2} - {\left (x^{4} + 1\right )} \arctan \left (x^{2}\right )}{4 \, {\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 19, normalized size = 0.83 \[ -\frac {x^{2}}{4 \, {\left (x^{4} + 1\right )}} + \frac {1}{4} \, \arctan \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.87 \[ -\frac {x^{2}}{4 \left (x^{4}+1\right )}+\frac {\arctan \left (x^{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.02, size = 19, normalized size = 0.83 \[ -\frac {x^{2}}{4 \, {\left (x^{4} + 1\right )}} + \frac {1}{4} \, \arctan \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 21, normalized size = 0.91 \[ \frac {\mathrm {atan}\left (x^2\right )}{4}-\frac {x^2}{4\,\left (x^4+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 15, normalized size = 0.65 \[ - \frac {x^{2}}{4 x^{4} + 4} + \frac {\operatorname {atan}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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