3.1.84 \(\int -\frac {x^2}{(1-x^2)^{3/2}} \, dx\) [84]

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

[Out]

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

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Rubi [A]
time = 0.00, 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 = {294, 222} \begin {gather*} \sin ^{-1}(x)-\frac {x}{\sqrt {1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-(x^2/(1 - x^2)^(3/2)),x]

[Out]

-(x/Sqrt[1 - x^2]) + ArcSin[x]

Rule 222

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 294

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 [B] Leaf count is larger than twice the leaf count of optimal. \(35\) vs. \(2(17)=34\).
time = 0.05, size = 35, normalized size = 2.06 \begin {gather*} -\frac {x}{\sqrt {1-x^2}}+2 \tan ^{-1}\left (\frac {x}{-1+\sqrt {1-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-(x^2/(1 - x^2)^(3/2)),x]

[Out]

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

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Mathics [A]
time = 2.01, size = 15, normalized size = 0.88 \begin {gather*} -\frac {x}{\sqrt {1-x^2}}+\text {ArcSin}\left [x\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x / Sqrt[1 - x ^ 2] + ArcSin[x]

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Maple [A]
time = 0.07, size = 16, normalized size = 0.94

method	result	size
default	\(\arcsin \left (x \right )-\frac {x}{\sqrt {-x^{2}+1}}\)	\(16\)
risch	\(\arcsin \left (x \right )-\frac {x}{\sqrt {-x^{2}+1}}\)	\(16\)
meijerg	\(-\frac {i \left (-\frac {i \sqrt {\pi }\, x}{\sqrt {-x^{2}+1}}+i \sqrt {\pi }\, \arcsin \left (x \right )\right )}{\sqrt {\pi }}\)	\(32\)
trager	\(\frac {x \sqrt {-x^{2}+1}}{x^{2}-1}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.34, size = 15, normalized size = 0.88 \begin {gather*} -\frac {x}{\sqrt {-x^{2} + 1}} + \arcsin \left (x\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (15) = 30\).
time = 0.32, size = 45, normalized size = 2.65 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\)
time = 0.20, size = 34, normalized size = 2.00 \begin {gather*} \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 {gather*}

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 = 0.00, size = 26, normalized size = 1.53 \begin {gather*} -\frac {2 x \sqrt {-x^{2}+1}}{2 \left (-x^{2}+1\right )}+\arcsin x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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