Optimal. Leaf size=11 \[ \frac{2}{3} (x+1)^{3/2} \]
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Rubi [A] time = 0.0010932, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {26, 32} \[ \frac{2}{3} (x+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 26
Rule 32
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^2}}{\sqrt{1-x}} \, dx &=\int \sqrt{1+x} \, dx\\ &=\frac{2}{3} (1+x)^{3/2}\\ \end{align*}
Mathematica [B] time = 0.0216299, size = 27, normalized size = 2.45 \[ \frac{2 (x+1) \sqrt{1-x^2}}{3 \sqrt{1-x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 22, normalized size = 2. \begin{align*}{\frac{2+2\,x}{3}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{1-x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986706, size = 9, normalized size = 0.82 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46537, size = 68, normalized size = 6.18 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}{\left (x + 1\right )} \sqrt{-x + 1}}{3 \,{\left (x - 1\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{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt{1 - x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10436, size = 18, normalized size = 1.64 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{4}{3} \, \sqrt{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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