Optimal. Leaf size=12 \[ \frac{1}{2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-1\right ) \]
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Rubi [A] time = 0.0016934, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {221} \[ \frac{1}{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{16-x^4}} \, dx &=\frac{1}{2} F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )\\ \end{align*}
Mathematica [A] time = 0.0120847, size = 12, normalized size = 1. \[ \frac{1}{2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-1\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 34, normalized size = 2.8 \begin{align*}{\frac{1}{2}\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4}{\it EllipticF} \left ({\frac{x}{2}},i \right ){\frac{1}{\sqrt{-{x}^{4}+16}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{4} + 16}}{x^{4} - 16}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.535752, size = 31, normalized size = 2.58 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{4} + 16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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