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

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 = 0.0050743, 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, Rules used = {288, 216} \[ \sin ^{-1}(x)-\frac{x}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

Int[-(x^2/(1 - x^2)^(3/2)),x]

[Out]

-(x/Sqrt[1 - x^2]) + ArcSin[x]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int -\frac{x^2}{\left (1-x^2\right )^{3/2}} \, dx &=-\frac{x}{\sqrt{1-x^2}}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{x}{\sqrt{1-x^2}}+\sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0068797, size = 32, normalized size = 1.88 \[ \frac{\sqrt{1-x^2} x+x^2 \sin ^{-1}(x)-\sin ^{-1}(x)}{x^2-1} \]

Antiderivative was successfully verified.

[In]

Integrate[-(x^2/(1 - x^2)^(3/2)),x]

[Out]

(x*Sqrt[1 - x^2] - ArcSin[x] + x^2*ArcSin[x])/(-1 + x^2)

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Maple [A]  time = 0.004, size = 16, normalized size = 0.9 \begin{align*} \arcsin \left ( x \right ) -{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

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 = 1.41049, size = 20, normalized size = 1.18 \begin{align*} -\frac{x}{\sqrt{-x^{2} + 1}} + \arcsin \left (x\right ) \end{align*}

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 [B]  time = 2.16875, size = 103, normalized size = 6.06 \begin{align*} -\frac{2 \,{\left (x^{2} - 1\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) - \sqrt{-x^{2} + 1} x}{x^{2} - 1} \end{align*}

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

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Sympy [B]  time = 0.5255, size = 34, normalized size = 2. \begin{align*} \frac{x^{2} \operatorname{asin}{\left (x \right )}}{x^{2} - 1} + \frac{x \sqrt{1 - x^{2}}}{x^{2} - 1} - \frac{\operatorname{asin}{\left (x \right )}}{x^{2} - 1} \end{align*}

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

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Giac [A]  time = 1.07129, size = 28, normalized size = 1.65 \begin{align*} \frac{\sqrt{-x^{2} + 1} x}{x^{2} - 1} + \arcsin \left (x\right ) \end{align*}

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)