Optimal. Leaf size=37 \[ \frac{1}{4 \left (1-x^4\right )}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^4\right )+2 \log (x) \]
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Rubi [A] time = 0.0185109, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {28, 266, 44} \[ \frac{1}{4 \left (1-x^4\right )}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^4\right )+2 \log (x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^5 \left (-1+x^4\right )^2} \, dx\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{(-1+x)^2 x^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^2}-\frac{2}{-1+x}+\frac{1}{x^2}+\frac{2}{x}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}+\frac{1}{4 \left (1-x^4\right )}+2 \log (x)-\frac{1}{2} \log \left (1-x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0126081, size = 35, normalized size = 0.95 \[ -\frac{1}{4 \left (x^4-1\right )}-\frac{1}{4 x^4}-\frac{1}{2} \log \left (1-x^4\right )+2 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 54, normalized size = 1.5 \begin{align*}{\frac{1}{8\,{x}^{2}+8}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{2}}-{\frac{1}{4\,{x}^{4}}}+2\,\ln \left ( x \right ) +{\frac{1}{16+16\,x}}-{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{1}{16\,x-16}}-{\frac{\ln \left ( x-1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974259, size = 47, normalized size = 1.27 \begin{align*} -\frac{2 \, x^{4} - 1}{4 \,{\left (x^{8} - x^{4}\right )}} - \frac{1}{2} \, \log \left (x^{4} - 1\right ) + \frac{1}{2} \, \log \left (x^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45873, size = 111, normalized size = 3. \begin{align*} -\frac{2 \, x^{4} + 2 \,{\left (x^{8} - x^{4}\right )} \log \left (x^{4} - 1\right ) - 8 \,{\left (x^{8} - x^{4}\right )} \log \left (x\right ) - 1}{4 \,{\left (x^{8} - x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.160786, size = 29, normalized size = 0.78 \begin{align*} - \frac{2 x^{4} - 1}{4 x^{8} - 4 x^{4}} + 2 \log{\left (x \right )} - \frac{\log{\left (x^{4} - 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11053, size = 49, normalized size = 1.32 \begin{align*} -\frac{2 \, x^{4} - 1}{4 \,{\left (x^{8} - x^{4}\right )}} + \frac{1}{2} \, \log \left (x^{4}\right ) - \frac{1}{2} \, \log \left ({\left | x^{4} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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