Optimal. Leaf size=25 \[ \frac {1}{4} \tanh ^{-1}\left (x^2\right )+\frac {x^2}{4 \left (1-x^4\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {28, 275, 199, 207} \[ \frac {x^2}{4 \left (1-x^4\right )}+\frac {1}{4} \tanh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 199
Rule 207
Rule 275
Rubi steps
\begin {align*} \int \frac {x}{1-2 x^4+x^8} \, dx &=\int \frac {x}{\left (-1+x^4\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\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} \tanh ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.32 \[ \frac {1}{8} \left (-\log \left (1-x^2\right )+\log \left (x^2+1\right )-\frac {2 x^2}{x^4-1}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 40, normalized size = 1.60 \[ -\frac {2 \, x^{2} - {\left (x^{4} - 1\right )} \log \left (x^{2} + 1\right ) + {\left (x^{4} - 1\right )} \log \left (x^{2} - 1\right )}{8 \, {\left (x^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 30, normalized size = 1.20 \[ -\frac {x^{2}}{4 \, {\left (x^{4} - 1\right )}} + \frac {1}{8} \, \log \left (x^{2} + 1\right ) - \frac {1}{8} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 1.44 \[ -\frac {\ln \left (x^{2}-1\right )}{8}+\frac {\ln \left (x^{2}+1\right )}{8}-\frac {1}{8 \left (x^{2}+1\right )}-\frac {1}{8 \left (x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 29, normalized size = 1.16 \[ -\frac {x^{2}}{4 \, {\left (x^{4} - 1\right )}} + \frac {1}{8} \, \log \left (x^{2} + 1\right ) - \frac {1}{8} \, \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 21, normalized size = 0.84 \[ \frac {\mathrm {atanh}\left (x^2\right )}{4}-\frac {x^2}{4\,\left (x^4-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 26, normalized size = 1.04 \[ - \frac {x^{2}}{4 x^{4} - 4} - \frac {\log {\left (x^{2} - 1 \right )}}{8} + \frac {\log {\left (x^{2} + 1 \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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