Optimal. Leaf size=31 \[ -\frac {\sqrt {16-x^4}}{48 x^3}+\frac {1}{96} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {331, 227}
\begin {gather*} \frac {1}{96} F\left (\left .\text {ArcSin}\left (\frac {x}{2}\right )\right |-1\right )-\frac {\sqrt {16-x^4}}{48 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {16-x^4}} \, dx &=-\frac {\sqrt {16-x^4}}{48 x^3}+\frac {1}{48} \int \frac {1}{\sqrt {16-x^4}} \, dx\\ &=-\frac {\sqrt {16-x^4}}{48 x^3}+\frac {1}{96} 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 = 10.01, size = 24, normalized size = 0.77 \begin {gather*} -\frac {\, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};\frac {x^4}{16}\right )}{12 x^3} \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. 48 vs. \(2 (23 ) = 46\).
time = 0.16, size = 49, normalized size = 1.58
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], \frac {x^{4}}{16}\right )}{12 x^{3}}\) | \(17\) |
default | \(-\frac {\sqrt {-x^{4}+16}}{48 x^{3}}+\frac {\sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{96 \sqrt {-x^{4}+16}}\) | \(49\) |
elliptic | \(-\frac {\sqrt {-x^{4}+16}}{48 x^{3}}+\frac {\sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{96 \sqrt {-x^{4}+16}}\) | \(49\) |
risch | \(\frac {x^{4}-16}{48 x^{3} \sqrt {-x^{4}+16}}+\frac {\sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{96 \sqrt {-x^{4}+16}}\) | \(54\) |
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 = 27, normalized size = 0.87 \begin {gather*} \frac {x^{3} F(\arcsin \left (\frac {1}{2} \, x\right )\,|\,-1) - 2 \, \sqrt {-x^{4} + 16}}{96 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 36, normalized size = 1.16 \begin {gather*} \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{16}} \right )}}{16 x^{3} \Gamma \left (\frac {1}{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 = -0.03 \begin {gather*} \int \frac {1}{x^4\,\sqrt {16-x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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