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]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 0.43, size = 45, normalized size = 2.65 \[ -\frac {2 \, {\left (x^{2} - 1\right )} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - \sqrt {-x^{2} + 1} x}{x^{2} - 1} \]

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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giac [A]  time = 1.07, size = 21, normalized size = 1.24 \[ \frac {\sqrt {-x^{2} + 1} x}{x^{2} - 1} + \arcsin \relax (x) \]

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)

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maple [A]  time = 0.00, size = 16, normalized size = 0.94 \[ -\frac {x}{\sqrt {-x^{2}+1}}+\arcsin \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^2/(-x^2+1)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.53, size = 15, normalized size = 0.88 \[ -\frac {x}{\sqrt {-x^{2} + 1}} + \arcsin \relax (x) \]

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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mupad [B]  time = 0.16, size = 37, normalized size = 2.18 \[ \mathrm {asin}\relax (x)+\frac {\sqrt {1-x^2}}{2\,\left (x-1\right )}+\frac {\sqrt {1-x^2}}{2\,\left (x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^2/(1 - x^2)^(3/2),x)

[Out]

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

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sympy [B]  time = 0.51, size = 34, normalized size = 2.00 \[ \frac {x^{2} \operatorname {asin}{\relax (x )}}{x^{2} - 1} + \frac {x \sqrt {1 - x^{2}}}{x^{2} - 1} - \frac {\operatorname {asin}{\relax (x )}}{x^{2} - 1} \]

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

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