3.4.12 \(\int \frac {1}{\sqrt {16-x^2}} \, dx\) [312]

Optimal. Leaf size=6 \[ \sin ^{-1}\left (\frac {x}{4}\right ) \]

[Out]

arcsin(1/4*x)

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Rubi [A]
time = 0.00, antiderivative size = 6, 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*} \sin ^{-1}\left (\frac {x}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[16 - x^2],x]

[Out]

ArcSin[x/4]

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 {16-x^2}} \, dx &=\sin ^{-1}\left (\frac {x}{4}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(14\) vs. \(2(6)=12\).
time = 0.02, size = 14, normalized size = 2.33 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {16-x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[16 - x^2],x]

[Out]

ArcTan[x/Sqrt[16 - x^2]]

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Mathics [A]
time = 1.67, size = 4, normalized size = 0.67 \begin {gather*} \text {ArcSin}\left [\frac {x}{4}\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/Sqrt[16 - x^2],x]')

[Out]

ArcSin[x / 4]

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Maple [A]
time = 0.08, size = 5, normalized size = 0.83

method	result	size
default	\(\arcsin \left (\frac {x}{4}\right )\)	\(5\)
meijerg	\(\arcsin \left (\frac {x}{4}\right )\)	\(5\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+16}+x \right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+16)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arcsin(1/4*x)

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Maxima [A]
time = 0.36, size = 4, normalized size = 0.67 \begin {gather*} \arcsin \left (\frac {1}{4} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+16)^(1/2),x, algorithm="maxima")

[Out]

arcsin(1/4*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (4) = 8\).
time = 0.34, size = 18, normalized size = 3.00 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x^{2} + 16} - 4}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+16)^(1/2),x, algorithm="fricas")

[Out]

-2*arctan((sqrt(-x^2 + 16) - 4)/x)

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Sympy [A]
time = 0.07, size = 3, normalized size = 0.50 \begin {gather*} \operatorname {asin}{\left (\frac {x}{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+16)**(1/2),x)

[Out]

asin(x/4)

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Giac [A]
time = 0.00, size = 5, normalized size = 0.83 \begin {gather*} \arcsin \left (\frac {x}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+16)^(1/2),x)

[Out]

arcsin(1/4*x)

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Mupad [B]
time = 0.01, size = 4, normalized size = 0.67 \begin {gather*} \mathrm {asin}\left (\frac {x}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(16 - x^2)^(1/2),x)

[Out]

asin(x/4)

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