Optimal. Leaf size=14 \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0247177, antiderivative size = 14, 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 \[ \sqrt{1-x^2}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - x^2]/(1 + x),x]
[Out]
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Rubi in Sympy [A] time = 3.4256, size = 10, normalized size = 0.71 \[ \sqrt{- x^{2} + 1} + \operatorname{asin}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+1)**(1/2)/(1+x),x)
[Out]
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Mathematica [B] time = 0.0323346, size = 46, normalized size = 3.29 \[ \sqrt{1-x^2} \left (1-\frac{2 \log \left (\sqrt{x-1}+\sqrt{x+1}\right )}{\sqrt{x-1} \sqrt{x+1}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - x^2]/(1 + x),x]
[Out]
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Maple [A] time = 0.006, size = 18, normalized size = 1.3 \[ \sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x}+\arcsin \left ( x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+1)^(1/2)/(1+x),x)
[Out]
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Maxima [A] time = 0.784969, size = 16, normalized size = 1.14 \[ \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)/(x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213382, size = 65, normalized size = 4.64 \[ -\frac{x^{2} + 2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}{\sqrt{-x^{2} + 1} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)/(x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.37713, size = 15, normalized size = 1.07 \[ \begin{cases} \sqrt{- x^{2} + 1} + \operatorname{asin}{\left (x \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+1)**(1/2)/(1+x),x)
[Out]
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GIAC/XCAS [A] time = 0.211639, size = 16, normalized size = 1.14 \[ \sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)/(x + 1),x, algorithm="giac")
[Out]