Optimal. Leaf size=43 \[ \frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{20 x^5}-\frac{9}{4 x}-\frac{9}{8} \tan ^{-1}(x)+\frac{9}{8} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.0122302, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {28, 290, 325, 298, 203, 206} \[ \frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{20 x^5}-\frac{9}{4 x}-\frac{9}{8} \tan ^{-1}(x)+\frac{9}{8} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 290
Rule 325
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^6 \left (-1+x^4\right )^2} \, dx\\ &=\frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{4} \int \frac{1}{x^6 \left (-1+x^4\right )} \, dx\\ &=-\frac{9}{20 x^5}+\frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{4} \int \frac{1}{x^2 \left (-1+x^4\right )} \, dx\\ &=-\frac{9}{20 x^5}-\frac{9}{4 x}+\frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{4} \int \frac{x^2}{-1+x^4} \, dx\\ &=-\frac{9}{20 x^5}-\frac{9}{4 x}+\frac{1}{4 x^5 \left (1-x^4\right )}+\frac{9}{8} \int \frac{1}{1-x^2} \, dx-\frac{9}{8} \int \frac{1}{1+x^2} \, dx\\ &=-\frac{9}{20 x^5}-\frac{9}{4 x}+\frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{8} \tan ^{-1}(x)+\frac{9}{8} \tanh ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0209894, size = 51, normalized size = 1.19 \[ -\frac{x^3}{4 \left (x^4-1\right )}-\frac{1}{5 x^5}-\frac{2}{x}-\frac{9}{16} \log (1-x)+\frac{9}{16} \log (x+1)-\frac{9}{8} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 52, normalized size = 1.2 \begin{align*} -{\frac{x}{8\,{x}^{2}+8}}-{\frac{9\,\arctan \left ( x \right ) }{8}}-{\frac{1}{5\,{x}^{5}}}-2\,{x}^{-1}-{\frac{1}{16+16\,x}}+{\frac{9\,\ln \left ( 1+x \right ) }{16}}-{\frac{1}{16\,x-16}}-{\frac{9\,\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 = 1.51062, size = 57, normalized size = 1.33 \begin{align*} -\frac{45 \, x^{8} - 36 \, x^{4} - 4}{20 \,{\left (x^{9} - x^{5}\right )}} - \frac{9}{8} \, \arctan \left (x\right ) + \frac{9}{16} \, \log \left (x + 1\right ) - \frac{9}{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 = 1.47535, size = 171, normalized size = 3.98 \begin{align*} -\frac{180 \, x^{8} - 144 \, x^{4} + 90 \,{\left (x^{9} - x^{5}\right )} \arctan \left (x\right ) - 45 \,{\left (x^{9} - x^{5}\right )} \log \left (x + 1\right ) + 45 \,{\left (x^{9} - x^{5}\right )} \log \left (x - 1\right ) - 16}{80 \,{\left (x^{9} - x^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.223132, size = 44, normalized size = 1.02 \begin{align*} - \frac{9 \log{\left (x - 1 \right )}}{16} + \frac{9 \log{\left (x + 1 \right )}}{16} - \frac{9 \operatorname{atan}{\left (x \right )}}{8} - \frac{45 x^{8} - 36 x^{4} - 4}{20 x^{9} - 20 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1035, size = 58, normalized size = 1.35 \begin{align*} -\frac{x^{3}}{4 \,{\left (x^{4} - 1\right )}} - \frac{10 \, x^{4} + 1}{5 \, x^{5}} - \frac{9}{8} \, \arctan \left (x\right ) + \frac{9}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{9}{16} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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