Optimal. Leaf size=30 \[ \sqrt {4 x^2-9}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {4 x^2-9}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 30, 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, 50, 63, 203} \[ \sqrt {4 x^2-9}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {-9+4 x^2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {-9+4 x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {-9+4 x^2}-\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-9+4 x}} \, dx,x,x^2\right )\\ &=\sqrt {-9+4 x^2}-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{\frac {9}{4}+\frac {x^2}{4}} \, dx,x,\sqrt {-9+4 x^2}\right )\\ &=\sqrt {-9+4 x^2}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {-9+4 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.00 \[ \sqrt {4 x^2-9}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 28, normalized size = 0.93 \[ \sqrt {4 \, x^{2} - 9} - 6 \, \arctan \left (-\frac {2}{3} \, x + \frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 24, normalized size = 0.80 \[ \sqrt {4 \, x^{2} - 9} - 3 \, \arctan \left (\frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.83 \[ 3 \arctan \left (\frac {3}{\sqrt {4 x^{2}-9}}\right )+\sqrt {4 x^{2}-9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 19, normalized size = 0.63 \[ \sqrt {4 \, x^{2} - 9} + 3 \, \arcsin \left (\frac {3}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.34, size = 24, normalized size = 0.80 \[ \sqrt {4\,x^2-9}-3\,\mathrm {atan}\left (\frac {\sqrt {4\,x^2-9}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.37, size = 82, normalized size = 2.73 \[ \begin {cases} \sqrt {4 x^{2} - 9} - 3 i \log {\relax (x )} + \frac {3 i \log {\left (x^{2} \right )}}{2} + 3 \operatorname {asin}{\left (\frac {3}{2 x} \right )} & \text {for}\: \frac {4 \left |{x^{2}}\right |}{9} > 1 \\i \sqrt {9 - 4 x^{2}} + \frac {3 i \log {\left (x^{2} \right )}}{2} - 3 i \log {\left (\sqrt {1 - \frac {4 x^{2}}{9}} + 1 \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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