Optimal. Leaf size=39 \[ -\frac {\sqrt {9+4 x^2}}{2 x^2}-\frac {2}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right ) \]
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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, 43, 65,
213} \begin {gather*} -\frac {\sqrt {4 x^2+9}}{2 x^2}-\frac {2}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {4 x^2+9}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt {9+4 x^2}}{x^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {9+4 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9+4 x^2}}{2 x^2}+\text {Subst}\left (\int \frac {1}{x \sqrt {9+4 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {9+4 x^2}}{2 x^2}+\frac {1}{2} \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}}{2 x^2}-\frac {2}{3} \tanh ^{-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}}{2 x^2}-\frac {2}{3} \tanh ^{-1}\left (\frac {1}{3} \sqrt {9+4 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 41, normalized size = 1.05
method | result | size |
risch | \(-\frac {\sqrt {4 x^{2}+9}}{2 x^{2}}-\frac {2 \arctanh \left (\frac {3}{\sqrt {4 x^{2}+9}}\right )}{3}\) | \(30\) |
trager | \(-\frac {\sqrt {4 x^{2}+9}}{2 x^{2}}-\frac {2 \ln \left (\frac {\sqrt {4 x^{2}+9}+3}{x}\right )}{3}\) | \(34\) |
default | \(-\frac {\left (4 x^{2}+9\right )^{\frac {3}{2}}}{18 x^{2}}+\frac {2 \sqrt {4 x^{2}+9}}{9}-\frac {2 \arctanh \left (\frac {3}{\sqrt {4 x^{2}+9}}\right )}{3}\) | \(41\) |
meijerg | \(-\frac {-\frac {9 \sqrt {\pi }\, \left (8+\frac {16 x^{2}}{9}\right )}{16 x^{2}}+\frac {9 \sqrt {\pi }\, \sqrt {1+\frac {4 x^{2}}{9}}}{2 x^{2}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {4 x^{2}}{9}}}{2}\right )-\left (-1+2 \ln \left (x \right )-2 \ln \left (3\right )\right ) \sqrt {\pi }+\frac {9 \sqrt {\pi }}{2 x^{2}}}{3 \sqrt {\pi }}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 35, normalized size = 0.90 \begin {gather*} \frac {2}{9} \, \sqrt {4 \, x^{2} + 9} - \frac {{\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{18 \, x^{2}} - \frac {2}{3} \, \operatorname {arsinh}\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 = 1.63, size = 57, normalized size = 1.46 \begin {gather*} -\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}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.80, size = 24, normalized size = 0.62 \begin {gather*} - \frac {2 \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{3} - \frac {\sqrt {1 + \frac {9}{4 x^{2}}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 43, normalized size = 1.10 \begin {gather*} -\frac {\sqrt {4 \, x^{2} + 9}}{2 \, x^{2}} - \frac {1}{3} \, \log \left (\sqrt {4 \, x^{2} + 9} + 3\right ) + \frac {1}{3} \, \log \left (\sqrt {4 \, x^{2} + 9} - 3\right ) \end {gather*}
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 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {2\,\sqrt {x^2+\frac {9}{4}}}{3}\right )}{3}-\frac {\sqrt {x^2+\frac {9}{4}}}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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