Optimal. Leaf size=35 \[ -\frac{\sqrt{1-x^4}}{4 x^4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]
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Rubi [A] time = 0.0144941, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 206} \[ -\frac{\sqrt{1-x^4}}{4 x^4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^5 \sqrt{1-x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^2} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{1-x^4}}{4 x^4}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{1-x^4}}{4 x^4}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^4}\right )\\ &=-\frac{\sqrt{1-x^4}}{4 x^4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right )\\ \end{align*}
Mathematica [A] time = 0.0077894, size = 35, normalized size = 1. \[ -\frac{\sqrt{1-x^4}}{4 x^4}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 28, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{4}}\sqrt{-{x}^{4}+1}}-{\frac{1}{4}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{4}+1}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02626, size = 58, normalized size = 1.66 \begin{align*} -\frac{\sqrt{-x^{4} + 1}}{4 \, x^{4}} - \frac{1}{8} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{-x^{4} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70179, size = 120, normalized size = 3.43 \begin{align*} -\frac{x^{4} \log \left (\sqrt{-x^{4} + 1} + 1\right ) - x^{4} \log \left (\sqrt{-x^{4} + 1} - 1\right ) + 2 \, \sqrt{-x^{4} + 1}}{8 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.04632, size = 73, normalized size = 2.09 \begin{align*} \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{\sqrt{-1 + \frac{1}{x^{4}}}}{4 x^{2}} & \text{for}\: \frac{1}{\left |{x^{4}}\right |} > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{i}{4 x^{2} \sqrt{1 - \frac{1}{x^{4}}}} + \frac{i}{4 x^{6} \sqrt{1 - \frac{1}{x^{4}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16814, size = 61, normalized size = 1.74 \begin{align*} -\frac{\sqrt{-x^{4} + 1}}{4 \, x^{4}} - \frac{1}{8} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{8} \, \log \left (-\sqrt{-x^{4} + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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