Optimal. Leaf size=14 \[ \sqrt {1-x^2}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} \sqrt {1-x^2}+\sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 665
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*}
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Mathematica [B] time = 0.05, size = 30, normalized size = 2.14 \begin {gather*} \sqrt {1-x^2}-2 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.13, size = 32, normalized size = 2.29 \begin {gather*} \sqrt {1-x^2}-2 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 28, normalized size = 2.00 \begin {gather*} \sqrt {-x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 12, normalized size = 0.86 \begin {gather*} \sqrt {-x^{2} + 1} + \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 18, normalized size = 1.29 \begin {gather*} \arcsin \relax (x )+\sqrt {2 x -\left (x +1\right )^{2}+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 12, normalized size = 0.86 \begin {gather*} \sqrt {-x^{2} + 1} + \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 12, normalized size = 0.86 \begin {gather*} \mathrm {asin}\relax (x)+\sqrt {1-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.18, size = 15, normalized size = 1.07 \begin {gather*} \begin {cases} \sqrt {1 - x^{2}} + \operatorname {asin}{\relax (x )} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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