Optimal. Leaf size=21 \[ 2 E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-1\right ) \]
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Rubi [A] time = 0.0152457, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {307, 221, 1181, 21, 424} \[ 2 E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-2 F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 307
Rule 221
Rule 1181
Rule 21
Rule 424
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{16-x^4}} \, dx &=-\left (4 \int \frac{1}{\sqrt{16-x^4}} \, dx\right )+4 \int \frac{1+\frac{x^2}{4}}{\sqrt{16-x^4}} \, dx\\ &=-2 F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )+4 \int \frac{1+\frac{x^2}{4}}{\sqrt{4-x^2} \sqrt{4+x^2}} \, dx\\ &=-2 F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )+\int \frac{\sqrt{4+x^2}}{\sqrt{4-x^2}} \, dx\\ &=2 E\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )-2 F\left (\left .\sin ^{-1}\left (\frac{x}{2}\right )\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.002446, size = 24, normalized size = 1.14 \[ \frac{1}{12} x^3 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};\frac{x^4}{16}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 43, normalized size = 2.1 \begin{align*} -2\,{\frac{\sqrt{-{x}^{2}+4}\sqrt{{x}^{2}+4} \left ({\it EllipticF} \left ( x/2,i \right ) -{\it EllipticE} \left ( x/2,i \right ) \right ) }{\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{x^{2}}{\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^{2}}{x^{4} - 16}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.709392, size = 32, normalized size = 1.52 \begin{align*} \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac{7}{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{x^{2}}{\sqrt{-x^{4} + 16}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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