Optimal. Leaf size=32 \[ \frac{1}{4 x^2 \left (1-x^4\right )}-\frac{3}{4 x^2}+\frac{3}{4} \tanh ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.0126115, antiderivative size = 32, 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, 275, 290, 325, 207} \[ \frac{1}{4 x^2 \left (1-x^4\right )}-\frac{3}{4 x^2}+\frac{3}{4} \tanh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 275
Rule 290
Rule 325
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^3 \left (-1+x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (-1+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{4 x^2 \left (1-x^4\right )}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (-1+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{3}{4 x^2}+\frac{1}{4 x^2 \left (1-x^4\right )}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,x^2\right )\\ &=-\frac{3}{4 x^2}+\frac{1}{4 x^2 \left (1-x^4\right )}+\frac{3}{4} \tanh ^{-1}\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.017206, size = 41, normalized size = 1.28 \[ \frac{1}{8} \left (\frac{4-6 x^4}{x^2 \left (x^4-1\right )}-3 \log \left (1-x^2\right )+3 \log \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 50, normalized size = 1.6 \begin{align*} -{\frac{1}{8\,{x}^{2}+8}}+{\frac{3\,\ln \left ({x}^{2}+1 \right ) }{8}}-{\frac{1}{2\,{x}^{2}}}+{\frac{1}{16+16\,x}}-{\frac{3\,\ln \left ( 1+x \right ) }{8}}-{\frac{1}{16\,x-16}}-{\frac{3\,\ln \left ( x-1 \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01926, size = 50, normalized size = 1.56 \begin{align*} -\frac{3 \, x^{4} - 2}{4 \,{\left (x^{6} - x^{2}\right )}} + \frac{3}{8} \, \log \left (x^{2} + 1\right ) - \frac{3}{8} \, \log \left (x^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.42239, size = 119, normalized size = 3.72 \begin{align*} -\frac{6 \, x^{4} - 3 \,{\left (x^{6} - x^{2}\right )} \log \left (x^{2} + 1\right ) + 3 \,{\left (x^{6} - x^{2}\right )} \log \left (x^{2} - 1\right ) - 4}{8 \,{\left (x^{6} - x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.155546, size = 36, normalized size = 1.12 \begin{align*} - \frac{3 x^{4} - 2}{4 x^{6} - 4 x^{2}} - \frac{3 \log{\left (x^{2} - 1 \right )}}{8} + \frac{3 \log{\left (x^{2} + 1 \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12248, size = 51, normalized size = 1.59 \begin{align*} -\frac{3 \, x^{4} - 2}{4 \,{\left (x^{6} - x^{2}\right )}} + \frac{3}{8} \, \log \left (x^{2} + 1\right ) - \frac{3}{8} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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