Optimal. Leaf size=31 \[ \frac {1}{96} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )-\frac {\sqrt {16-x^4}}{48 x^3} \]
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Rubi [A] time = 0.01, 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 = {325, 221} \[ \frac {1}{96} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )-\frac {\sqrt {16-x^4}}{48 x^3} \]
Antiderivative was successfully verified.
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Rule 221
Rule 325
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] time = 0.00, size = 24, normalized size = 0.77 \[ -\frac {\, _2F_1\left (-\frac {3}{4},\frac {1}{2};\frac {1}{4};\frac {x^4}{16}\right )}{12 x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 16}}{x^{8} - 16 \, x^{4}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-x^{4} + 16} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 49, normalized size = 1.58 \[ \frac {\sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \EllipticF \left (\frac {x}{2}, i\right )}{96 \sqrt {-x^{4}+16}}-\frac {\sqrt {-x^{4}+16}}{48 x^{3}} \]
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 \[ \int \frac {1}{\sqrt {-x^{4} + 16} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,\sqrt {16-x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.11, size = 36, normalized size = 1.16 \[ \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{16}} \right )}}{16 x^{3} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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