Optimal. Leaf size=20 \[ \sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0209621, 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 \[ \sqrt{1-x} \sqrt{x+1}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - x]/Sqrt[1 + x],x]
[Out]
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Rubi in Sympy [A] time = 3.70917, size = 15, normalized size = 0.75 \[ \sqrt{- x + 1} \sqrt{x + 1} + \operatorname{asin}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)**(1/2)/(1+x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.00930959, size = 28, normalized size = 1.4 \[ \sqrt{1-x^2}+2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - x]/Sqrt[1 + x],x]
[Out]
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Maple [B] time = 0.006, size = 41, normalized size = 2.1 \[ \sqrt{1-x}\sqrt{1+x}+{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)^(1/2)/(1+x)^(1/2),x)
[Out]
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Maxima [A] time = 1.48377, size = 16, normalized size = 0.8 \[ \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x + 1)/sqrt(x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209621, size = 81, normalized size = 4.05 \[ -\frac{x^{2} + 2 \,{\left (\sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right )}{\sqrt{x + 1} \sqrt{-x + 1} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x + 1)/sqrt(x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.51074, size = 100, normalized size = 5. \[ \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{i \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{x - 1}} - \frac{2 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{\left (x + 1\right )^{\frac{3}{2}}}{\sqrt{- x + 1}} + \frac{2 \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x)**(1/2)/(1+x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.208422, size = 36, normalized size = 1.8 \[ \sqrt{x + 1} \sqrt{-x + 1} + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x + 1)/sqrt(x + 1),x, algorithm="giac")
[Out]