### 3.312 $$\int \frac{1}{\sqrt{16-x^2}} \, dx$$

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

[Out]

ArcSin[x/4]

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Rubi [A]  time = 0.0012241, antiderivative size = 6, normalized size of antiderivative = 1., 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 = {216} $\sin ^{-1}\left (\frac{x}{4}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[1/Sqrt[16 - x^2],x]

[Out]

ArcSin[x/4]

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

Mathematica [A]  time = 0.0041155, size = 6, normalized size = 1. $\sin ^{-1}\left (\frac{x}{4}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/Sqrt[16 - x^2],x]

[Out]

ArcSin[x/4]

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Maple [A]  time = 0.003, size = 5, normalized size = 0.8 \begin{align*} \arcsin \left ({\frac{x}{4}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(1/4*x)

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Maxima [A]  time = 1.40092, size = 5, normalized size = 0.83 \begin{align*} \arcsin \left (\frac{1}{4} \, x\right ) \end{align*}

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]  time = 2.20828, size = 49, normalized size = 8.17 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x^{2} + 16} - 4}{x}\right ) \end{align*}

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 = 0.131268, size = 3, normalized size = 0.5 \begin{align*} \operatorname{asin}{\left (\frac{x}{4} \right )} \end{align*}

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 = 1.07072, size = 5, normalized size = 0.83 \begin{align*} \arcsin \left (\frac{1}{4} \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(1/4*x)