Optimal. Leaf size=29 \[ \frac {x^3}{4 \left (1-x^4\right )}-\frac {1}{8} \tan ^{-1}(x)+\frac {1}{8} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 29, 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, 290, 298, 203, 206} \[ \frac {x^3}{4 \left (1-x^4\right )}-\frac {1}{8} \tan ^{-1}(x)+\frac {1}{8} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 203
Rule 206
Rule 290
Rule 298
Rubi steps
\begin {align*} \int \frac {x^2}{1-2 x^4+x^8} \, dx &=\int \frac {x^2}{\left (-1+x^4\right )^2} \, dx\\ &=\frac {x^3}{4 \left (1-x^4\right )}-\frac {1}{4} \int \frac {x^2}{-1+x^4} \, dx\\ &=\frac {x^3}{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^3}{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 = 33, normalized size = 1.14 \[ \frac {1}{16} \left (-\frac {4 x^3}{x^4-1}-\log (1-x)+\log (x+1)-2 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 45, normalized size = 1.55 \[ -\frac {4 \, x^{3} + 2 \, {\left (x^{4} - 1\right )} \arctan \relax (x) - {\left (x^{4} - 1\right )} \log \left (x + 1\right ) + {\left (x^{4} - 1\right )} \log \left (x - 1\right )}{16 \, {\left (x^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 31, normalized size = 1.07 \[ -\frac {x^{3}}{4 \, {\left (x^{4} - 1\right )}} - \frac {1}{8} \, \arctan \relax (x) + \frac {1}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{16} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 1.45 \[ -\frac {x}{8 \left (x^{2}+1\right )}-\frac {\arctan \relax (x )}{8}-\frac {\ln \left (x -1\right )}{16}+\frac {\ln \left (x +1\right )}{16}-\frac {1}{16 \left (x +1\right )}-\frac {1}{16 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.71, size = 29, normalized size = 1.00 \[ -\frac {x^{3}}{4 \, {\left (x^{4} - 1\right )}} - \frac {1}{8} \, \arctan \relax (x) + \frac {1}{16} \, \log \left (x + 1\right ) - \frac {1}{16} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 23, normalized size = 0.79 \[ \frac {\mathrm {atanh}\relax (x)}{8}-\frac {\mathrm {atan}\relax (x)}{8}-\frac {x^3}{4\,\left (x^4-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 27, normalized size = 0.93 \[ - \frac {x^{3}}{4 x^{4} - 4} - \frac {\log {\left (x - 1 \right )}}{16} + \frac {\log {\left (x + 1 \right )}}{16} - \frac {\operatorname {atan}{\relax (x )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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