Optimal. Leaf size=35 \[ \frac{\sqrt{x^2-16}}{32 x^2}+\frac{1}{128} \tan ^{-1}\left (\frac{\sqrt{x^2-16}}{4}\right ) \]
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Rubi [A] time = 0.0132875, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 203} \[ \frac{\sqrt{x^2-16}}{32 x^2}+\frac{1}{128} \tan ^{-1}\left (\frac{\sqrt{x^2-16}}{4}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{-16+x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-16+x} x^2} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-16+x^2}}{32 x^2}+\frac{1}{64} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-16+x} x} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-16+x^2}}{32 x^2}+\frac{1}{32} \operatorname{Subst}\left (\int \frac{1}{16+x^2} \, dx,x,\sqrt{-16+x^2}\right )\\ &=\frac{\sqrt{-16+x^2}}{32 x^2}+\frac{1}{128} \tan ^{-1}\left (\frac{1}{4} \sqrt{-16+x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0203663, size = 46, normalized size = 1.31 \[ \frac{1}{256} \sqrt{x^2-16} \left (\frac{8}{x^2}+\frac{2 \tanh ^{-1}\left (\sqrt{1-\frac{x^2}{16}}\right )}{\sqrt{16-x^2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 26, normalized size = 0.7 \begin{align*}{\frac{1}{32\,{x}^{2}}\sqrt{{x}^{2}-16}}-{\frac{1}{128}\arctan \left ( 4\,{\frac{1}{\sqrt{{x}^{2}-16}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40181, size = 30, normalized size = 0.86 \begin{align*} \frac{\sqrt{x^{2} - 16}}{32 \, x^{2}} - \frac{1}{128} \, \arcsin \left (\frac{4}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87866, size = 96, normalized size = 2.74 \begin{align*} \frac{x^{2} \arctan \left (-\frac{1}{4} \, x + \frac{1}{4} \, \sqrt{x^{2} - 16}\right ) + 2 \, \sqrt{x^{2} - 16}}{64 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.23, size = 66, normalized size = 1.89 \begin{align*} \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{4}{x} \right )}}{128} + \frac{i \sqrt{-1 + \frac{16}{x^{2}}}}{32 x} & \text{for}\: \frac{16}{\left |{x^{2}}\right |} > 1 \\- \frac{\operatorname{asin}{\left (\frac{4}{x} \right )}}{128} + \frac{1}{32 x \sqrt{1 - \frac{16}{x^{2}}}} - \frac{1}{2 x^{3} \sqrt{1 - \frac{16}{x^{2}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05215, size = 34, normalized size = 0.97 \begin{align*} \frac{\sqrt{x^{2} - 16}}{32 \, x^{2}} + \frac{1}{128} \, \arctan \left (\frac{1}{4} \, \sqrt{x^{2} - 16}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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