Optimal. Leaf size=23 \[ -\frac{2 \sqrt{1-x}}{\sqrt{x+1}}-\sin ^{-1}(x) \]
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Rubi [A] time = 0.0033808, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {47, 41, 216} \[ -\frac{2 \sqrt{1-x}}{\sqrt{x+1}}-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 47
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x}}{(1+x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-x}}{\sqrt{1+x}}-\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{2 \sqrt{1-x}}{\sqrt{1+x}}-\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2 \sqrt{1-x}}{\sqrt{1+x}}-\sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0368303, size = 34, normalized size = 1.48 \[ 2 \left (\frac{x-1}{\sqrt{1-x^2}}+\sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 67, normalized size = 2.9 \begin{align*} 2\,{\frac{ \left ( -1+x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }\sqrt{1-x}\sqrt{1+x}}}-{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53654, size = 28, normalized size = 1.22 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{x + 1} - \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76957, size = 131, normalized size = 5.7 \begin{align*} \frac{2 \,{\left ({\left (x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - x - \sqrt{x + 1} \sqrt{-x + 1} - 1\right )}}{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.46917, size = 104, normalized size = 4.52 \begin{align*} \begin{cases} 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{2 i \sqrt{x + 1}}{\sqrt{x - 1}} + \frac{4 i}{\sqrt{x - 1} \sqrt{x + 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{2 \sqrt{x + 1}}{\sqrt{1 - x}} - \frac{4}{\sqrt{1 - x} \sqrt{x + 1}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.06955, size = 74, normalized size = 3.22 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{\sqrt{x + 1}} - \frac{\sqrt{x + 1}}{\sqrt{2} - \sqrt{-x + 1}} - 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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