Optimal. Leaf size=39 \[ -\frac {\sqrt {16-x^4}}{64 x^4}-\frac {1}{256} \tanh ^{-1}\left (\frac {\sqrt {16-x^4}}{4}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 206} \[ -\frac {\sqrt {16-x^4}}{64 x^4}-\frac {1}{256} \tanh ^{-1}\left (\frac {\sqrt {16-x^4}}{4}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {16-x^4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {16-x} x^2} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {16-x^4}}{64 x^4}+\frac {1}{128} \operatorname {Subst}\left (\int \frac {1}{\sqrt {16-x} x} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {16-x^4}}{64 x^4}-\frac {1}{64} \operatorname {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\sqrt {16-x^4}\right )\\ &=-\frac {\sqrt {16-x^4}}{64 x^4}-\frac {1}{256} \tanh ^{-1}\left (\frac {\sqrt {16-x^4}}{4}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.95 \[ \frac {1}{256} \left (-\frac {4 \sqrt {16-x^4}}{x^4}-\tanh ^{-1}\left (\sqrt {1-\frac {x^4}{16}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 50, normalized size = 1.28 \[ -\frac {x^{4} \log \left (\sqrt {-x^{4} + 16} + 4\right ) - x^{4} \log \left (\sqrt {-x^{4} + 16} - 4\right ) + 8 \, \sqrt {-x^{4} + 16}}{512 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 45, normalized size = 1.15 \[ -\frac {\sqrt {-x^{4} + 16}}{64 \, x^{4}} - \frac {1}{512} \, \log \left (\sqrt {-x^{4} + 16} + 4\right ) + \frac {1}{512} \, \log \left (-\sqrt {-x^{4} + 16} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.77 \[ -\frac {\arctanh \left (\frac {4}{\sqrt {-x^{4}+16}}\right )}{256}-\frac {\sqrt {-x^{4}+16}}{64 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 43, normalized size = 1.10 \[ -\frac {\sqrt {-x^{4} + 16}}{64 \, x^{4}} - \frac {1}{512} \, \log \left (\sqrt {-x^{4} + 16} + 4\right ) + \frac {1}{512} \, \log \left (\sqrt {-x^{4} + 16} - 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 29, normalized size = 0.74 \[ -\frac {\mathrm {atanh}\left (\frac {\sqrt {16-x^4}}{4}\right )}{256}-\frac {\sqrt {16-x^4}}{64\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.01, size = 73, normalized size = 1.87 \[ \begin {cases} - \frac {\operatorname {acosh}{\left (\frac {4}{x^{2}} \right )}}{256} - \frac {\sqrt {-1 + \frac {16}{x^{4}}}}{64 x^{2}} & \text {for}\: \frac {16}{\left |{x^{4}}\right |} > 1 \\\frac {i \operatorname {asin}{\left (\frac {4}{x^{2}} \right )}}{256} - \frac {i}{64 x^{2} \sqrt {1 - \frac {16}{x^{4}}}} + \frac {i}{4 x^{6} \sqrt {1 - \frac {16}{x^{4}}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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