Optimal. Leaf size=27 \[ -\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0681749, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{2} \sqrt{1-x^2} (2-x)-\frac{1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(x*Sqrt[1 - x^2])/(1 + x),x]
[Out]
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Rubi in Sympy [A] time = 5.2659, size = 29, normalized size = 1.07 \[ - \frac{\sqrt{- x^{2} + 1}}{2} - \frac{\operatorname{asin}{\left (x \right )}}{2} - \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{2 \left (x + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(-x**2+1)**(1/2)/(1+x),x)
[Out]
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Mathematica [A] time = 0.0261381, size = 26, normalized size = 0.96 \[ \left (\frac{x}{2}-1\right ) \sqrt{1-x^2}-\frac{1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[(x*Sqrt[1 - x^2])/(1 + x),x]
[Out]
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Maple [A] time = 0.009, size = 34, normalized size = 1.3 \[{\frac{x}{2}\sqrt{-{x}^{2}+1}}-{\frac{\arcsin \left ( x \right ) }{2}}-\sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(-x^2+1)^(1/2)/(1+x),x)
[Out]
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Maxima [A] time = 0.785224, size = 38, normalized size = 1.41 \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1} - \frac{1}{2} \, \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)*x/(x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285531, size = 123, normalized size = 4.56 \[ -\frac{2 \, x^{3} - 2 \, x^{2} - 2 \,{\left (x^{2} + 2 \, \sqrt{-x^{2} + 1} - 2\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (x^{3} - 2 \, x^{2} - 2 \, x\right )} \sqrt{-x^{2} + 1} - 2 \, x}{2 \,{\left (x^{2} + 2 \, \sqrt{-x^{2} + 1} - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)*x/(x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.61977, size = 29, normalized size = 1.07 \[ \begin{cases} \frac{x \sqrt{- x^{2} + 1}}{2} - \sqrt{- x^{2} + 1} - \frac{\operatorname{asin}{\left (x \right )}}{2} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(-x**2+1)**(1/2)/(1+x),x)
[Out]
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GIAC/XCAS [A] time = 0.283352, size = 26, normalized size = 0.96 \[ \frac{1}{2} \, \sqrt{-x^{2} + 1}{\left (x - 2\right )} - \frac{1}{2} \, \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1)*x/(x + 1),x, algorithm="giac")
[Out]