Optimal. Leaf size=18 \[ -\frac{\sqrt{1-x^4}}{2 x^2} \]
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Rubi [A] time = 0.003119, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\sqrt{1-x^4}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{1-x^4}} \, dx &=-\frac{\sqrt{1-x^4}}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0024418, size = 18, normalized size = 1. \[ -\frac{\sqrt{1-x^4}}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 26, normalized size = 1.4 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) }{2\,{x}^{2}}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0105, size = 19, normalized size = 1.06 \begin{align*} -\frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58737, size = 34, normalized size = 1.89 \begin{align*} -\frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.785154, size = 34, normalized size = 1.89 \begin{align*} \begin{cases} - \frac{i \sqrt{x^{4} - 1}}{2 x^{2}} & \text{for}\: \left |{x^{4}}\right | > 1 \\- \frac{\sqrt{1 - x^{4}}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15157, size = 12, normalized size = 0.67 \begin{align*} -\frac{1}{2} \, \sqrt{\frac{1}{x^{4}} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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