3.84 \(\int -\frac{x^2}{\left (1-x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=17 \[ \sin ^{-1}(x)-\frac{x}{\sqrt{1-x^2}} \]

[Out]

-(x/Sqrt[1 - x^2]) + ArcSin[x]

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Rubi [A]  time = 0.0171994, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \sin ^{-1}(x)-\frac{x}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]  Int[-(x^2/(1 - x^2)^(3/2)),x]

[Out]

-(x/Sqrt[1 - x^2]) + ArcSin[x]

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Rubi in Sympy [A]  time = 1.66846, size = 12, normalized size = 0.71 \[ - \frac{x}{\sqrt{- x^{2} + 1}} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(-x**2/(-x**2+1)**(3/2),x)

[Out]

-x/sqrt(-x**2 + 1) + asin(x)

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Mathematica [A]  time = 0.0236906, size = 17, normalized size = 1. \[ \sin ^{-1}(x)-\frac{x}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[-(x^2/(1 - x^2)^(3/2)),x]

[Out]

-(x/Sqrt[1 - x^2]) + ArcSin[x]

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Maple [A]  time = 0.007, size = 16, normalized size = 0.9 \[ \arcsin \left ( x \right ) -{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(-x^2/(-x^2+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.48503, size = 20, normalized size = 1.18 \[ -\frac{x}{\sqrt{-x^{2} + 1}} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/(-x^2 + 1)^(3/2),x, algorithm="maxima")

[Out]

-x/sqrt(-x^2 + 1) + arcsin(x)

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Fricas [A]  time = 0.242648, size = 86, normalized size = 5.06 \[ -\frac{2 \,{\left (x^{2} + \sqrt{-x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) - \sqrt{-x^{2} + 1} x + x}{x^{2} + \sqrt{-x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/(-x^2 + 1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.97049, size = 34, normalized size = 2. \[ \frac{x^{2} \operatorname{asin}{\left (x \right )}}{x^{2} - 1} + \frac{x \sqrt{- x^{2} + 1}}{x^{2} - 1} - \frac{\operatorname{asin}{\left (x \right )}}{x^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x**2/(-x**2+1)**(3/2),x)

[Out]

x**2*asin(x)/(x**2 - 1) + x*sqrt(-x**2 + 1)/(x**2 - 1) - asin(x)/(x**2 - 1)

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GIAC/XCAS [A]  time = 0.208514, size = 28, normalized size = 1.65 \[ \frac{\sqrt{-x^{2} + 1} x}{x^{2} - 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/(-x^2 + 1)^(3/2),x, algorithm="giac")

[Out]

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