Optimal. Leaf size=25 \[ \frac{1}{2} \sqrt{1-4 x^2} x+\frac{1}{4} \sin ^{-1}(2 x) \]
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Rubi [A] time = 0.0034196, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {195, 216} \[ \frac{1}{2} \sqrt{1-4 x^2} x+\frac{1}{4} \sin ^{-1}(2 x) \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-4 x^2} \, dx &=\frac{1}{2} x \sqrt{1-4 x^2}+\frac{1}{2} \int \frac{1}{\sqrt{1-4 x^2}} \, dx\\ &=\frac{1}{2} x \sqrt{1-4 x^2}+\frac{1}{4} \sin ^{-1}(2 x)\\ \end{align*}
Mathematica [A] time = 0.0070851, size = 25, normalized size = 1. \[ \frac{1}{2} \sqrt{1-4 x^2} x+\frac{1}{4} \sin ^{-1}(2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 20, normalized size = 0.8 \begin{align*}{\frac{\arcsin \left ( 2\,x \right ) }{4}}+{\frac{x}{2}\sqrt{-4\,{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40291, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} + 1} x + \frac{1}{4} \, \arcsin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98043, size = 90, normalized size = 3.6 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} + 1} x - \frac{1}{2} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 1} - 1}{2 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.186725, size = 19, normalized size = 0.76 \begin{align*} \frac{x \sqrt{1 - 4 x^{2}}}{2} + \frac{\operatorname{asin}{\left (2 x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05781, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} + 1} x + \frac{1}{4} \, \arcsin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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