Optimal. Leaf size=14 \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.0042074, 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 = {641, 216} \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{1-x}{\sqrt{1-x^2}} \, dx &=\sqrt{1-x^2}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\sqrt{1-x^2}+\sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0056058, size = 14, normalized size = 1. \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 13, normalized size = 0.9 \begin{align*} \arcsin \left ( x \right ) +\sqrt{-{x}^{2}+1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52499, 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 = 2.04453, 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 = 0.131368, size = 10, normalized size = 0.71 \begin{align*} \sqrt{1 - x^{2}} + \operatorname{asin}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19809, 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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