Optimal. Leaf size=39 \[ \frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9+4 x^2}\right ) \]
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Rubi [A]
time = 0.01, 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 = {272, 44, 65,
209} \begin {gather*} \frac {2}{27} \text {ArcTan}\left (\frac {1}{3} \sqrt {4 x^2-9}\right )+\frac {\sqrt {4 x^2-9}}{18 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 {-9+4 x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {-9+4 x}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {1}{9} \text {Subst}\left (\int \frac {1}{x \sqrt {-9+4 x}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {1}{18} \text {Subst}\left (\int \frac {1}{\frac {9}{4}+\frac {x^2}{4}} \, dx,x,\sqrt {-9+4 x^2}\right )\\ &=\frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9+4 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 39, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9+4 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 30, normalized size = 0.77
method | result | size |
default | \(\frac {\sqrt {4 x^{2}-9}}{18 x^{2}}-\frac {2 \arctan \left (\frac {3}{\sqrt {4 x^{2}-9}}\right )}{27}\) | \(30\) |
risch | \(\frac {\sqrt {4 x^{2}-9}}{18 x^{2}}-\frac {2 \arctan \left (\frac {3}{\sqrt {4 x^{2}-9}}\right )}{27}\) | \(30\) |
trager | \(\frac {\sqrt {4 x^{2}-9}}{18 x^{2}}-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {4 x^{2}-9}-3 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x}\right )}{27}\) | \(47\) |
meijerg | \(-\frac {2 \sqrt {-\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}\, \left (-\frac {9 \sqrt {\pi }\, \left (-\frac {16 x^{2}}{9}+8\right )}{32 x^{2}}+\frac {9 \sqrt {\pi }\, \sqrt {1-\frac {4 x^{2}}{9}}}{4 x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {4 x^{2}}{9}}}{2}\right )-\frac {\left (1+2 \ln \left (x \right )-2 \ln \left (3\right )+i \pi \right ) \sqrt {\pi }}{2}+\frac {9 \sqrt {\pi }}{4 x^{2}}\right )}{27 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 24, normalized size = 0.62 \begin {gather*} \frac {\sqrt {4 \, x^{2} - 9}}{18 \, x^{2}} - \frac {2}{27} \, \arcsin \left (\frac {3}{2 \, {\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.35, size = 38, normalized size = 0.97 \begin {gather*} \frac {8 \, x^{2} \arctan \left (-\frac {2}{3} \, x + \frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) + 3 \, \sqrt {4 \, x^{2} - 9}}{54 \, 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 = 1.08, size = 99, normalized size = 2.54 \begin {gather*} \begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {i}{9 x \sqrt {-1 + \frac {9}{4 x^{2}}}} + \frac {i}{4 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {4}{9} \\- \frac {2 \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{27} + \frac {1}{9 x \sqrt {1 - \frac {9}{4 x^{2}}}} - \frac {1}{4 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.91, size = 29, normalized size = 0.74 \begin {gather*} \frac {\sqrt {4 \, x^{2} - 9}}{18 \, x^{2}} + \frac {2}{27} \, \arctan \left (\frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.88, size = 29, normalized size = 0.74 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {\sqrt {4\,x^2-9}}{3}\right )}{27}+\frac {\sqrt {4\,x^2-9}}{18\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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