Integrand size = 13, antiderivative size = 35 \[ \int \frac {1}{x^3 \sqrt {-16+x^2}} \, dx=\frac {\sqrt {-16+x^2}}{32 x^2}+\frac {1}{128} \arctan \left (\frac {1}{4} \sqrt {-16+x^2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 44, 65, 209} \[ \int \frac {1}{x^3 \sqrt {-16+x^2}} \, dx=\frac {1}{128} \arctan \left (\frac {\sqrt {x^2-16}}{4}\right )+\frac {\sqrt {x^2-16}}{32 x^2} \]
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Rule 44
Rule 65
Rule 209
Rule 272
Rubi steps \begin{align*} \text {integral}& = \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} \arctan \left (\frac {1}{4} \sqrt {-16+x^2}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt {-16+x^2}} \, dx=\frac {\sqrt {-16+x^2}}{32 x^2}+\frac {1}{128} \arctan \left (\frac {1}{4} \sqrt {-16+x^2}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 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 | \(\frac {\sqrt {x^{2}-16}}{32 x^{2}}-\frac {\arctan \left (\frac {4}{\sqrt {x^{2}-16}}\right )}{128}\) | \(26\) |
pseudoelliptic | \(\frac {\arctan \left (\frac {\sqrt {x^{2}-16}}{4}\right ) x^{2}+4 \sqrt {x^{2}-16}}{128 x^{2}}\) | \(30\) |
trager | \(\frac {\sqrt {x^{2}-16}}{32 x^{2}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-\sqrt {x^{2}-16}}{x}\right )}{128}\) | \(46\) |
meijerg | \(-\frac {\sqrt {-\operatorname {signum}\left (-1+\frac {x^{2}}{16}\right )}\, \left (\frac {16 \sqrt {\pi }}{x^{2}}-\frac {\left (1-6 \ln \left (2\right )+2 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}-\frac {2 \sqrt {\pi }\, \left (-\frac {x^{2}}{4}+8\right )}{x^{2}}+\frac {16 \sqrt {\pi }\, \sqrt {1-\frac {x^{2}}{16}}}{x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {x^{2}}{16}}}{2}\right )\right )}{128 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (-1+\frac {x^{2}}{16}\right )}}\) | \(106\) |
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Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^3 \sqrt {-16+x^2}} \, dx=\frac {x^{2} \arctan \left (-\frac {1}{4} \, x + \frac {1}{4} \, \sqrt {x^{2} - 16}\right ) + 2 \, \sqrt {x^{2} - 16}}{64 \, x^{2}} \]
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Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.89 \[ \int \frac {1}{x^3 \sqrt {-16+x^2}} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {4}{x} \right )}}{128} - \frac {i}{32 x \sqrt {-1 + \frac {16}{x^{2}}}} + \frac {i}{2 x^{3} \sqrt {-1 + \frac {16}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {1}{16} \\- \frac {\operatorname {asin}{\left (\frac {4}{x} \right )}}{128} + \frac {\sqrt {1 - \frac {16}{x^{2}}}}{32 x} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^3 \sqrt {-16+x^2}} \, dx=\frac {\sqrt {x^{2} - 16}}{32 \, x^{2}} - \frac {1}{128} \, \arcsin \left (\frac {4}{{\left | x \right |}}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^3 \sqrt {-16+x^2}} \, dx=\frac {\sqrt {x^{2} - 16}}{32 \, x^{2}} + \frac {1}{128} \, \arctan \left (\frac {1}{4} \, \sqrt {x^{2} - 16}\right ) \]
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Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^3 \sqrt {-16+x^2}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {x^2-16}}{4}\right )}{128}+\frac {\sqrt {x^2-16}}{32\,x^2} \]
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