Optimal. Leaf size=39 \[ \frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right )-\frac {\sqrt {4 x^2+9}}{18 x^2} \]
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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, 207} \[ \frac {2}{27} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right )-\frac {\sqrt {4 x^2+9}}{18 x^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {9+4 x^2}} \, dx &=\frac {1}{2} \operatorname {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} \operatorname {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} \operatorname {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} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.95 \[ \frac {1}{54} \left (4 \tanh ^{-1}\left (\sqrt {\frac {4 x^2}{9}+1}\right )-\frac {3 \sqrt {4 x^2+9}}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 57, normalized size = 1.46 \[ \frac {4 \, x^{2} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} + 3\right ) - 4 \, x^{2} \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9} - 3\right ) - 3 \, \sqrt {4 \, x^{2} + 9}}{54 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 43, normalized size = 1.10 \[ -\frac {\sqrt {4 \, x^{2} + 9}}{18 \, x^{2}} + \frac {1}{27} \, \log \left (\sqrt {4 \, x^{2} + 9} + 3\right ) - \frac {1}{27} \, \log \left (\sqrt {4 \, x^{2} + 9} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.77 \[ \frac {2 \arctanh \left (\frac {3}{\sqrt {4 x^{2}+9}}\right )}{27}-\frac {\sqrt {4 x^{2}+9}}{18 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 24, normalized size = 0.62 \[ -\frac {\sqrt {4 \, x^{2} + 9}}{18 \, x^{2}} + \frac {2}{27} \, \operatorname {arsinh}\left (\frac {3}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 25, normalized size = 0.64 \[ \frac {2\,\mathrm {atanh}\left (\frac {2\,\sqrt {x^2+\frac {9}{4}}}{3}\right )}{27}-\frac {\sqrt {x^2+\frac {9}{4}}}{9\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.10, size = 44, normalized size = 1.13 \[ \frac {2 \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {1}{9 x \sqrt {1 + \frac {9}{4 x^{2}}}} - \frac {1}{4 x^{3} \sqrt {1 + \frac {9}{4 x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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