Optimal. Leaf size=35 \[ \frac {\sqrt {-16+x^2}}{32 x^2}+\frac {1}{128} \tan ^{-1}\left (\frac {1}{4} \sqrt {-16+x^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 = {272, 44, 65,
209} \begin {gather*} \frac {1}{128} \text {ArcTan}\left (\frac {\sqrt {x^2-16}}{4}\right )+\frac {\sqrt {x^2-16}}{32 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 209
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {-16+x^2}} \, dx &=\frac {1}{2} \text {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} \text {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} \text {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*}
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Mathematica [A]
time = 0.02, size = 35, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-16+x^2}}{32 x^2}+\frac {1}{128} \tan ^{-1}\left (\frac {1}{4} \sqrt {-16+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 26, normalized size = 0.74
method | result | size |
default | \(\frac {\sqrt {x^{2}-16}}{32 x^{2}}-\frac {\arctan \left (\frac {4}{\sqrt {x^{2}-16}}\right )}{128}\) | \(26\) |
risch | \(\frac {\sqrt {x^{2}-16}}{32 x^{2}}-\frac {\arctan \left (\frac {4}{\sqrt {x^{2}-16}}\right )}{128}\) | \(26\) |
trager | \(\frac {\sqrt {x^{2}-16}}{32 x^{2}}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {x^{2}-16}-4 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x}\right )}{128}\) | \(43\) |
meijerg | \(-\frac {\sqrt {-\mathrm {signum}\left (-1+\frac {x^{2}}{16}\right )}\, \left (\frac {16 \sqrt {\pi }}{x^{2}}-\frac {\left (1-6 \ln \left (2\right )+2 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}-\frac {2 \sqrt {\pi }\, \left (-\frac {x^{2}}{4}+8\right )}{x^{2}}+\frac {16 \sqrt {\pi }\, \sqrt {1-\frac {x^{2}}{16}}}{x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {x^{2}}{16}}}{2}\right )\right )}{128 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {x^{2}}{16}\right )}}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.84, size = 22, normalized size = 0.63 \begin {gather*} \frac {\sqrt {x^{2} - 16}}{32 \, x^{2}} - \frac {1}{128} \, \arcsin \left (\frac {4}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.74, size = 33, normalized size = 0.94 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.10, size = 66, normalized size = 1.89 \begin {gather*} \begin {cases} \frac {i \operatorname {acosh}{\left (\frac {4}{x} \right )}}{128} - \frac {i}{32 x \sqrt {-1 + \frac {16}{x^{2}}}} + \frac {i}{2 x^{3} \sqrt {-1 + \frac {16}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {1}{16} \\- \frac {\operatorname {asin}{\left (\frac {4}{x} \right )}}{128} + \frac {\sqrt {1 - \frac {16}{x^{2}}}}{32 x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 25, normalized size = 0.71 \begin {gather*} \frac {\sqrt {x^{2} - 16}}{32 \, x^{2}} + \frac {1}{128} \, \arctan \left (\frac {1}{4} \, \sqrt {x^{2} - 16}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 25, normalized size = 0.71 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {x^2-16}}{4}\right )}{128}+\frac {\sqrt {x^2-16}}{32\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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