3.9.68 \(\int \frac {1}{\sqrt {1-x^2}} \, dx\) [868]

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

[Out]

arcsin(x)

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Rubi [A]
time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {222} \begin {gather*} \text {ArcSin}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - x^2],x]

[Out]

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]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(14\) vs. \(2(2)=4\).
time = 0.00, size = 14, normalized size = 7.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - x^2],x]

[Out]

ArcTan[x/Sqrt[1 - x^2]]

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Maple [A]
time = 0.61, size = 3, normalized size = 1.50

method	result	size
default	\(\arcsin \left (x \right )\)	\(3\)
meijerg	\(\arcsin \left (x \right )\)	\(3\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(x)

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Maxima [A]
time = 0.49, size = 2, normalized size = 1.00 \begin {gather*} \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

arcsin(x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (2) = 4\).
time = 0.34, size = 18, normalized size = 9.00 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-2*arctan((sqrt(-x^2 + 1) - 1)/x)

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Sympy [A]
time = 0.05, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {asin}{\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+1)**(1/2),x)

[Out]

asin(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (2) = 4\).
time = 3.74, size = 17, normalized size = 8.50 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(-x^2 + 1)*x + 1/2*arcsin(x)

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Mupad [B]
time = 0.01, size = 2, normalized size = 1.00 \begin {gather*} \mathrm {asin}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1 - x^2)^(1/2),x)

[Out]

asin(x)

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