Optimal. Leaf size=20 \[ \sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.0034088, antiderivative size = 20, 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 = {50, 41, 216} \[ \sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx &=\sqrt{1-x} \sqrt{1+x}+\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\sqrt{1-x} \sqrt{1+x}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\sqrt{1-x} \sqrt{1+x}+\sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0135955, size = 30, normalized size = 1.5 \[ \sqrt{1-x^2}-2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 41, normalized size = 2.1 \begin{align*} \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 = 1.52339, size = 16, normalized size = 0.8 \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 = 1.89186, size = 96, normalized size = 4.8 \begin{align*} \sqrt{x + 1} \sqrt{-x + 1} - 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 = 1.60976, size = 100, normalized size = 5. \begin{align*} \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{x - 1}} - \frac{2 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{\left (x + 1\right )^{\frac{3}{2}}}{\sqrt{1 - x}} + \frac{2 \sqrt{x + 1}}{\sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05982, size = 36, normalized size = 1.8 \begin{align*} \sqrt{x + 1} \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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