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

Optimal. Leaf size=4 \[ x+\sin ^{-1}(x) \]

[Out]

x + ArcSin[x]

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Rubi [A]  time = 0.0717603, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ x+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x + ArcSin[x]

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Rubi in Sympy [A]  time = 4.15297, size = 3, normalized size = 0.75 \[ x + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + asin(x)

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Mathematica [A]  time = 0.0195708, size = 4, normalized size = 1. \[ x+\sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

x + ArcSin[x]

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Maple [B]  time = 0.012, size = 51, normalized size = 12.8 \[ x+{\frac{\ln \left ( -1+x \right ) }{2}}-{\frac{\ln \left ( 1+x \right ) }{2}}+{\it Artanh} \left ( x \right ) +{\frac{1}{2}\sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x}}+\arcsin \left ( x \right ) -{\frac{1}{2}\sqrt{- \left ( -1+x \right ) ^{2}-2\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x+1/2*ln(-1+x)-1/2*ln(1+x)+arctanh(x)+1/2*(-(1+x)^2+2+2*x)^(1/2)+arcsin(x)-1/2*(
-(-1+x)^2-2*x+2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{x^{2} + \sqrt{-x^{2} + 1} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(x^2 + sqrt(-x^2 + 1) - 1), x)

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Fricas [A]  time = 0.266583, size = 27, normalized size = 6.75 \[ x - 2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - 2*arctan((sqrt(-x^2 + 1) - 1)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{x^{2} + \sqrt{- x^{2} + 1} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/(x**2 + sqrt(-x**2 + 1) - 1), x)

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GIAC/XCAS [A]  time = 0.26808, size = 5, normalized size = 1.25 \[ x + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + arcsin(x)