Optimal. Leaf size=27 \[ \frac {x}{4 \left (1-x^4\right )}-\frac {1}{8} \tan ^{-1}(x)-\frac {1}{8} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {28, 288, 212, 206, 203} \[ \frac {x}{4 \left (1-x^4\right )}-\frac {1}{8} \tan ^{-1}(x)-\frac {1}{8} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 203
Rule 206
Rule 212
Rule 288
Rubi steps
\begin {align*} \int \frac {x^4}{1-2 x^4+x^8} \, dx &=\int \frac {x^4}{\left (-1+x^4\right )^2} \, dx\\ &=\frac {x}{4 \left (1-x^4\right )}+\frac {1}{4} \int \frac {1}{-1+x^4} \, dx\\ &=\frac {x}{4 \left (1-x^4\right )}-\frac {1}{8} \int \frac {1}{1-x^2} \, dx-\frac {1}{8} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{4 \left (1-x^4\right )}-\frac {1}{8} \tan ^{-1}(x)-\frac {1}{8} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.15 \[ \frac {1}{16} \left (-\frac {4 x}{x^4-1}+\log (1-x)-\log (x+1)-2 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 43, normalized size = 1.59 \[ -\frac {2 \, {\left (x^{4} - 1\right )} \arctan \relax (x) + {\left (x^{4} - 1\right )} \log \left (x + 1\right ) - {\left (x^{4} - 1\right )} \log \left (x - 1\right ) + 4 \, x}{16 \, {\left (x^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 29, normalized size = 1.07 \[ -\frac {x}{4 \, {\left (x^{4} - 1\right )}} - \frac {1}{8} \, \arctan \relax (x) - \frac {1}{16} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{16} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 1.56 \[ \frac {x}{8 x^{2}+8}-\frac {\arctan \relax (x )}{8}+\frac {\ln \left (x -1\right )}{16}-\frac {\ln \left (x +1\right )}{16}-\frac {1}{16 \left (x +1\right )}-\frac {1}{16 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.92, size = 27, normalized size = 1.00 \[ -\frac {x}{4 \, {\left (x^{4} - 1\right )}} - \frac {1}{8} \, \arctan \relax (x) - \frac {1}{16} \, \log \left (x + 1\right ) + \frac {1}{16} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 21, normalized size = 0.78 \[ -\frac {\mathrm {atan}\relax (x)}{8}-\frac {\mathrm {atanh}\relax (x)}{8}-\frac {x}{4\,\left (x^4-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 26, normalized size = 0.96 \[ - \frac {x}{4 x^{4} - 4} + \frac {\log {\left (x - 1 \right )}}{16} - \frac {\log {\left (x + 1 \right )}}{16} - \frac {\operatorname {atan}{\relax (x )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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