Optimal. Leaf size=10 \[ \frac{1}{2} \sin ^{-1}\left (x+\frac{1}{2}\right ) \]
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Rubi [A] time = 0.0067516, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {619, 216} \[ \frac{1}{2} \sin ^{-1}\left (x+\frac{1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-4 x-4 x^2}} \, dx &=-\left (\frac{1}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{64}}} \, dx,x,-4-8 x\right )\right )\\ &=\frac{1}{2} \sin ^{-1}\left (\frac{1}{2}+x\right )\\ \end{align*}
Mathematica [A] time = 0.0060364, size = 14, normalized size = 1.4 \[ -\frac{1}{2} \sin ^{-1}\left (\frac{1}{2} (-2 x-1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 7, normalized size = 0.7 \begin{align*}{\frac{1}{2}\arcsin \left ( x+{\frac{1}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6966, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{2} \, \arcsin \left (-x - \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8221, size = 88, normalized size = 8.8 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} - 4 \, x + 3}{\left (2 \, x + 1\right )}}{4 \, x^{2} + 4 \, x - 3}\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} - 4 x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15997, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{2} \, \arcsin \left (x + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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