3.575 \(\int \frac{1+x}{\sqrt{4-x^2}} \, dx\)

Optimal. Leaf size=20 \[ \sin ^{-1}\left (\frac{x}{2}\right )-\sqrt{4-x^2} \]

[Out]

-Sqrt[4 - x^2] + ArcSin[x/2]

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Rubi [A]  time = 0.022315, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \sin ^{-1}\left (\frac{x}{2}\right )-\sqrt{4-x^2} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/Sqrt[4 - x^2],x]

[Out]

-Sqrt[4 - x^2] + ArcSin[x/2]

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Rubi in Sympy [A]  time = 3.03159, size = 12, normalized size = 0.6 \[ - \sqrt{- x^{2} + 4} + \operatorname{asin}{\left (\frac{x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)/(-x**2+4)**(1/2),x)

[Out]

-sqrt(-x**2 + 4) + asin(x/2)

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Mathematica [A]  time = 0.0164561, size = 20, normalized size = 1. \[ \sin ^{-1}\left (\frac{x}{2}\right )-\sqrt{4-x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)/Sqrt[4 - x^2],x]

[Out]

-Sqrt[4 - x^2] + ArcSin[x/2]

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Maple [A]  time = 0.009, size = 17, normalized size = 0.9 \[ \arcsin \left ({\frac{x}{2}} \right ) -\sqrt{-{x}^{2}+4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(-x^2+4)^(1/2),x)

[Out]

arcsin(1/2*x)-(-x^2+4)^(1/2)

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Maxima [A]  time = 0.796813, size = 22, normalized size = 1.1 \[ -\sqrt{-x^{2} + 4} + \arcsin \left (\frac{1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/sqrt(-x^2 + 4),x, algorithm="maxima")

[Out]

-sqrt(-x^2 + 4) + arcsin(1/2*x)

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Fricas [A]  time = 0.216395, size = 63, normalized size = 3.15 \[ \frac{x^{2} - 2 \,{\left (\sqrt{-x^{2} + 4} - 2\right )} \arctan \left (\frac{\sqrt{-x^{2} + 4} - 2}{x}\right )}{\sqrt{-x^{2} + 4} - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/sqrt(-x^2 + 4),x, algorithm="fricas")

[Out]

(x^2 - 2*(sqrt(-x^2 + 4) - 2)*arctan((sqrt(-x^2 + 4) - 2)/x))/(sqrt(-x^2 + 4) -
2)

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Sympy [A]  time = 0.333511, size = 12, normalized size = 0.6 \[ - \sqrt{- x^{2} + 4} + \operatorname{asin}{\left (\frac{x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)/(-x**2+4)**(1/2),x)

[Out]

-sqrt(-x**2 + 4) + asin(x/2)

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GIAC/XCAS [A]  time = 0.217151, size = 22, normalized size = 1.1 \[ -\sqrt{-x^{2} + 4} + \arcsin \left (\frac{1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/sqrt(-x^2 + 4),x, algorithm="giac")

[Out]

-sqrt(-x^2 + 4) + arcsin(1/2*x)