Optimal. Leaf size=13 \[ \frac{\sin ^{-1}(2 x)}{2 \sqrt{6}} \]
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Rubi [A] time = 0.0025095, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {41, 216} \[ \frac{\sin ^{-1}(2 x)}{2 \sqrt{6}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-6 x} \sqrt{2+4 x}} \, dx &=\int \frac{1}{\sqrt{6-24 x^2}} \, dx\\ &=\frac{\sin ^{-1}(2 x)}{2 \sqrt{6}}\\ \end{align*}
Mathematica [A] time = 0.0151583, size = 13, normalized size = 1. \[ \frac{\sin ^{-1}(2 x)}{2 \sqrt{6}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 37, normalized size = 2.9 \begin{align*}{\frac{\arcsin \left ( 2\,x \right ) \sqrt{6}}{12}\sqrt{ \left ( 2+4\,x \right ) \left ( 3-6\,x \right ) }{\frac{1}{\sqrt{3-6\,x}}}{\frac{1}{\sqrt{2+4\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43274, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{12} \, \sqrt{6} \arcsin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46493, size = 90, normalized size = 6.92 \begin{align*} -\frac{1}{12} \, \sqrt{6} \arctan \left (\frac{\sqrt{6} \sqrt{4 \, x + 2} \sqrt{-6 \, x + 3}}{12 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.28419, size = 41, normalized size = 3.15 \begin{align*} \begin{cases} - \frac{\sqrt{6} i \operatorname{acosh}{\left (\sqrt{x + \frac{1}{2}} \right )}}{6} & \text{for}\: \left |{x + \frac{1}{2}}\right | > 1 \\\frac{\sqrt{6} \operatorname{asin}{\left (\sqrt{x + \frac{1}{2}} \right )}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07293, size = 20, normalized size = 1.54 \begin{align*} \frac{1}{6} \, \sqrt{6} \arcsin \left (\frac{1}{2} \, \sqrt{4 \, x + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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