Optimal. Leaf size=12 \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right ) \]
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Rubi [A] time = 0.0040121, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {110} \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 110
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x}}{\sqrt{-x} \sqrt{1+x}} \, dx &=-2 E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0261283, size = 66, normalized size = 5.5 \[ -\frac{2 x \sqrt{1-x^2} \left (x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};x^2\right )-3 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right )\right )}{3 \sqrt{1-x} \sqrt{-x (x+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.097, size = 17, normalized size = 1.4 \begin{align*} 2\,\sqrt{2}{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x + 1}}{\sqrt{-x} \sqrt{x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x} \sqrt{x + 1} \sqrt{-x + 1}}{x^{2} + x}, 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{\sqrt{1 - x}}{\sqrt{- x} \sqrt{x + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x + 1}}{\sqrt{-x} \sqrt{x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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