Optimal. Leaf size=16 \[ -\frac{1}{4 x^4}-\frac{1}{4} \tan ^{-1}\left (x^4\right ) \]
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Rubi [A] time = 0.0059629, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {275, 325, 203} \[ -\frac{1}{4 x^4}-\frac{1}{4} \tan ^{-1}\left (x^4\right ) \]
Antiderivative was successfully verified.
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Rule 275
Rule 325
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (1+x^8\right )} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+x^2\right )} \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}-\frac{1}{4} \tan ^{-1}\left (x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0045518, size = 16, normalized size = 1. \[ \frac{1}{4} \tan ^{-1}\left (\frac{1}{x^4}\right )-\frac{1}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{4}}}-{\frac{\arctan \left ({x}^{4} \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43088, size = 16, normalized size = 1. \begin{align*} -\frac{1}{4 \, x^{4}} - \frac{1}{4} \, \arctan \left (x^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2209, size = 43, normalized size = 2.69 \begin{align*} -\frac{x^{4} \arctan \left (x^{4}\right ) + 1}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.133271, size = 14, normalized size = 0.88 \begin{align*} - \frac{\operatorname{atan}{\left (x^{4} \right )}}{4} - \frac{1}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16312, size = 16, normalized size = 1. \begin{align*} -\frac{1}{4 \, x^{4}} - \frac{1}{4} \, \arctan \left (x^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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