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

Optimal. Leaf size=16 \[ \sin ^{-1}(x)-\sqrt{1-x^2} \]

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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Rubi [A]  time = 0.0209544, antiderivative size = 16, 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 \[ \sin ^{-1}(x)-\sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/Sqrt[1 - x^2],x]

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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Rubi in Sympy [A]  time = 3.08412, size = 10, normalized size = 0.62 \[ - \sqrt{- x^{2} + 1} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 1) + asin(x)

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Mathematica [A]  time = 0.0143807, size = 16, normalized size = 1. \[ \sin ^{-1}(x)-\sqrt{1-x^2} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 - x^2] + ArcSin[x]

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Maple [A]  time = 0.006, size = 15, normalized size = 0.9 \[ \arcsin \left ( x \right ) -\sqrt{-{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.794237, size = 19, normalized size = 1.19 \[ -\sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1) + arcsin(x)

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Fricas [A]  time = 0.222216, size = 63, normalized size = 3.94 \[ \frac{x^{2} - 2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}{\sqrt{-x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^2 - 2*(sqrt(-x^2 + 1) - 1)*arctan((sqrt(-x^2 + 1) - 1)/x))/(sqrt(-x^2 + 1) -
1)

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Sympy [A]  time = 0.388194, size = 10, normalized size = 0.62 \[ - \sqrt{- x^{2} + 1} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 1) + asin(x)

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GIAC/XCAS [A]  time = 0.215208, size = 19, normalized size = 1.19 \[ -\sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1) + arcsin(x)