Optimal. Leaf size=2 \[ \sin ^{-1}(x) \]
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Rubi [A] time = 0.0014997, antiderivative size = 2, 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 = {41, 216} \[ \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx &=\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0042368, size = 2, normalized size = 1. \[ \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 27, normalized size = 13.5 \begin{align*}{\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 = 1.53142, size = 3, normalized size = 1.5 \begin{align*} \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81425, size = 61, normalized size = 30.5 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.09657, size = 41, normalized size = 20.5 \begin{align*} \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05936, size = 18, normalized size = 9. \begin{align*} 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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