Optimal. Leaf size=14 \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.0058539, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {665, 216} \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 665
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^2}}{1+x} \, dx &=\sqrt{1-x^2}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\sqrt{1-x^2}+\sin ^{-1}(x)\\ \end{align*}
Mathematica [B] time = 0.0486737, size = 30, normalized size = 2.14 \[ \sqrt{1-x^2}-2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 18, normalized size = 1.3 \begin{align*} \sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x}+\arcsin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51349, size = 16, normalized size = 1.14 \begin{align*} \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.31635, size = 69, normalized size = 4.93 \begin{align*} \sqrt{-x^{2} + 1} - 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.68256, size = 15, normalized size = 1.07 \begin{align*} \begin{cases} \sqrt{1 - x^{2}} + \operatorname{asin}{\left (x \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20619, size = 16, normalized size = 1.14 \begin{align*} \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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