Optimal. Leaf size=28 \[ \frac{1}{2} \sqrt{1-x} \sqrt{x+1} x+\frac{1}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0035754, antiderivative size = 28, 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 = {38, 41, 216} \[ \frac{1}{2} \sqrt{1-x} \sqrt{x+1} x+\frac{1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-x} \sqrt{1+x} \, dx &=\frac{1}{2} \sqrt{1-x} x \sqrt{1+x}+\frac{1}{2} \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=\frac{1}{2} \sqrt{1-x} x \sqrt{1+x}+\frac{1}{2} \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{2} \sqrt{1-x} x \sqrt{1+x}+\frac{1}{2} \sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0064911, size = 20, normalized size = 0.71 \[ \frac{1}{2} \left (\sqrt{1-x^2} x+\sin ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 57, normalized size = 2. \begin{align*} -{\frac{1}{2} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{1}{2}\sqrt{1-x}\sqrt{1+x}}+{\frac{\arcsin \left ( x \right ) }{2}\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 = 1.54687, size = 23, normalized size = 0.82 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48498, size = 101, normalized size = 3.61 \begin{align*} \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} - \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 = 2.67175, size = 133, normalized size = 4.75 \begin{align*} \begin{cases} - i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x - 1}} - \frac{3 i \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{x - 1}} + \frac{i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{1 - x}} + \frac{3 \left (x + 1\right )^{\frac{3}{2}}}{2 \sqrt{1 - x}} - \frac{\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 = 1.09921, size = 36, normalized size = 1.29 \begin{align*} \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \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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