Optimal. Leaf size=33 \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]
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Rubi [A] time = 0.0162449, antiderivative size = 33, 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} \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]
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}{x^2 (1+x)^2} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{2}{x}+\frac{1}{(1+x)^2}+\frac{2}{1+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 = 0.0118251, size = 33, normalized size = 1. \[ -\frac{1}{4 \left (x^4+1\right )}-\frac{1}{4 x^4}+\frac{1}{2} \log \left (x^4+1\right )-2 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{x}^{4}}}-{\frac{1}{4\,{x}^{4}+4}}-2\,\ln \left ( x \right ) +{\frac{\ln \left ({x}^{4}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00189, size = 45, normalized size = 1.36 \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 = 1.43895, size = 111, normalized size = 3.36 \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 = 0.158586, size = 29, normalized size = 0.88 \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 = 1.07711, size = 45, normalized size = 1.36 \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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