Optimal. Leaf size=21 \[ \sin ^{-1}(x)-\sqrt{1-x} \sqrt{x+1} \]
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Rubi [A] time = 0.00379, antiderivative size = 21, 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} \[ \sin ^{-1}(x)-\sqrt{1-x} \sqrt{x+1} \]
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 = 0.0066771, size = 32, normalized size = 1.52 \[ -\sqrt{1-x^2}-2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 42, normalized size = 2. \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 = 1.525, size = 19, normalized size = 0.9 \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 = 1.63437, size = 97, normalized size = 4.62 \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 = 1.84416, size = 100, normalized size = 4.76 \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 = 1.06617, size = 38, normalized size = 1.81 \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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