Optimal. Leaf size=12 \[ -\frac{1}{2} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]
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Rubi [A] time = 0.0056575, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {619, 216} \[ -\frac{1}{2} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3 x-4 x^2}} \, dx &=-\left (\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,3-8 x\right )\right )\\ &=-\frac{1}{2} \sin ^{-1}\left (1-\frac{8 x}{3}\right )\\ \end{align*}
Mathematica [B] time = 0.0138426, size = 40, normalized size = 3.33 \[ -\frac{\sqrt{-x (4 x-3)} \sin ^{-1}\left (\sqrt{1-\frac{4 x}{3}}\right )}{\sqrt{3-4 x} \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 9, normalized size = 0.8 \begin{align*}{\frac{1}{2}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78698, size = 11, normalized size = 0.92 \begin{align*} -\frac{1}{2} \, \arcsin \left (-\frac{8}{3} \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.33111, size = 47, normalized size = 3.92 \begin{align*} -\arctan \left (\frac{\sqrt{-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 4 x^{2} + 3 x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2208, size = 11, normalized size = 0.92 \begin{align*} \frac{1}{2} \, \arcsin \left (\frac{8}{3} \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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