3.87 \(\int \frac{1}{\sqrt{a^2-x^2}} \, dx\)

Optimal. Leaf size=16 \[ \tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right ) \]

[Out]

ArcTan[x/Sqrt[a^2 - x^2]]

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Rubi [A]  time = 0.0022237, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {217, 203} \[ \tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a^2 - x^2],x]

[Out]

ArcTan[x/Sqrt[a^2 - x^2]]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a^2-x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{x}{\sqrt{a^2-x^2}}\right )\\ &=\tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0024595, size = 16, normalized size = 1. \[ \tan ^{-1}\left (\frac{x}{\sqrt{a^2-x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a^2 - x^2],x]

[Out]

ArcTan[x/Sqrt[a^2 - x^2]]

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Maple [A]  time = 0.003, size = 15, normalized size = 0.9 \begin{align*} \arctan \left ({x{\frac{1}{\sqrt{{a}^{2}-{x}^{2}}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2-x^2)^(1/2),x)

[Out]

arctan(x/(a^2-x^2)^(1/2))

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Maxima [A]  time = 1.42585, size = 11, normalized size = 0.69 \begin{align*} \arcsin \left (\frac{x}{\sqrt{a^{2}}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(x/sqrt(a^2))

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Fricas [A]  time = 0.508721, size = 50, normalized size = 3.12 \begin{align*} -2 \, \arctan \left (-\frac{a - \sqrt{a^{2} - x^{2}}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(-(a - sqrt(a^2 - x^2))/x)

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Sympy [A]  time = 1.01036, size = 20, normalized size = 1.25 \begin{align*} \begin{cases} - i \operatorname{acosh}{\left (\frac{x}{a} \right )} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{a^{2}}\right |} > 1 \\\operatorname{asin}{\left (\frac{x}{a} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*acosh(x/a), Abs(x**2)/Abs(a**2) > 1), (asin(x/a), True))

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Giac [A]  time = 1.11871, size = 12, normalized size = 0.75 \begin{align*} \arcsin \left (\frac{x}{a}\right ) \mathrm{sgn}\left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(x/a)*sgn(a)