Optimal. Leaf size=21 \[ 2 E\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )-2 F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {313, 227, 1195,
21, 435} \begin {gather*} 2 E\left (\left .\text {ArcSin}\left (\frac {x}{2}\right )\right |-1\right )-2 F\left (\left .\text {ArcSin}\left (\frac {x}{2}\right )\right |-1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 227
Rule 313
Rule 435
Rule 1195
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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 24, normalized size = 1.14 \begin {gather*} \frac {1}{12} x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {x^4}{16}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 42 vs. \(2 (17 ) = 34\).
time = 0.16, size = 43, normalized size = 2.05
method | result | size |
meijerg | \(\frac {x^{3} \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], \frac {x^{4}}{16}\right )}{12}\) | \(17\) |
default | \(-\frac {2 \sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \left (\EllipticF \left (\frac {x}{2}, i\right )-\EllipticE \left (\frac {x}{2}, i\right )\right )}{\sqrt {-x^{4}+16}}\) | \(43\) |
elliptic | \(-\frac {2 \sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \left (\EllipticF \left (\frac {x}{2}, i\right )-\EllipticE \left (\frac {x}{2}, i\right )\right )}{\sqrt {-x^{4}+16}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 14, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {-x^{4} + 16}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 32 vs. \(2 (12) = 24\).
time = 0.32, size = 32, normalized size = 1.52 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x^2}{\sqrt {16-x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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