Optimal. Leaf size=13 \[ -\frac{2}{3} (1-x)^{3/2} \]
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Rubi [A] time = 0.0013626, 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 = {26, 32} \[ -\frac{2}{3} (1-x)^{3/2} \]
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 [A] time = 0.0196129, size = 25, normalized size = 1.92 \[ \frac{2 (x-1) \sqrt{1-x^2}}{3 \sqrt{x+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 20, normalized size = 1.5 \begin{align*}{\frac{2\,x-2}{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 = 1.11251, size = 16, normalized size = 1.23 \begin{align*} \frac{2}{3} \,{\left (x - 1\right )} \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46389, size = 54, normalized size = 4.15 \begin{align*} \frac{2 \, \sqrt{-x^{2} + 1}{\left (x - 1\right )}}{3 \, \sqrt{x + 1}} \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{x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1058, size = 20, normalized size = 1.54 \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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