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 = 0.0083581, antiderivative size = 29, normalized size of antiderivative = 1., 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 290
Rule 298
Rule 203
Rule 206
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*}
Mathematica [A] time = 0.0120117, 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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Maple [A] time = 0.01, size = 42, normalized size = 1.5 \begin{align*} -{\frac{x}{8\,{x}^{2}+8}}-{\frac{\arctan \left ( x \right ) }{8}}-{\frac{1}{16+16\,x}}+{\frac{\ln \left ( 1+x \right ) }{16}}-{\frac{1}{16\,x-16}}-{\frac{\ln \left ( x-1 \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50077, size = 39, normalized size = 1.34 \begin{align*} -\frac{x^{3}}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{8} \, \arctan \left (x\right ) + \frac{1}{16} \, \log \left (x + 1\right ) - \frac{1}{16} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.51697, size = 128, normalized size = 4.41 \begin{align*} -\frac{4 \, x^{3} + 2 \,{\left (x^{4} - 1\right )} \arctan \left (x\right ) -{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.148697, size = 27, normalized size = 0.93 \begin{align*} - \frac{x^{3}}{4 x^{4} - 4} - \frac{\log{\left (x - 1 \right )}}{16} + \frac{\log{\left (x + 1 \right )}}{16} - \frac{\operatorname{atan}{\left (x \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11053, size = 42, normalized size = 1.45 \begin{align*} -\frac{x^{3}}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{8} \, \arctan \left (x\right ) + \frac{1}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{16} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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