3.577 \(\int \frac{1}{\sqrt{9-b x^2}} \, dx\)

Optimal. Leaf size=17 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]

[Out]

ArcSin[(Sqrt[b]*x)/3]/Sqrt[b]

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Rubi [A]  time = 0.0025019, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {216} \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 - b*x^2],x]

[Out]

ArcSin[(Sqrt[b]*x)/3]/Sqrt[b]

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{9-b x^2}} \, dx &=\frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0072171, size = 17, normalized size = 1. \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 - b*x^2],x]

[Out]

ArcSin[(Sqrt[b]*x)/3]/Sqrt[b]

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Maple [A]  time = 0.003, size = 21, normalized size = 1.2 \begin{align*}{\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}+9}}}} \right ){\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x^2+9)^(1/2),x)

[Out]

1/b^(1/2)*arctan(b^(1/2)*x/(-b*x^2+9)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.27607, size = 146, normalized size = 8.59 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-\sqrt{-b} x - \sqrt{-b x^{2} + 9}\right )}{b}, -\frac{2 \, \arctan \left (\frac{\sqrt{-b x^{2} + 9} - 3}{\sqrt{b} x}\right )}{\sqrt{b}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

[-sqrt(-b)*log(-sqrt(-b)*x - sqrt(-b*x^2 + 9))/b, -2*arctan((sqrt(-b*x^2 + 9) - 3)/(sqrt(b)*x))/sqrt(b)]

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Sympy [A]  time = 1.00037, size = 39, normalized size = 2.29 \begin{align*} \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x^{2}}\right |}{9} > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x**2+9)**(1/2),x)

[Out]

Piecewise((-I*acosh(sqrt(b)*x/3)/sqrt(b), Abs(b*x**2)/9 > 1), (asin(sqrt(b)*x/3)/sqrt(b), True))

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Giac [B]  time = 2.40067, size = 36, normalized size = 2.12 \begin{align*} -\frac{\log \left (-\sqrt{-b} x + \sqrt{-b x^{2} + 9}\right )}{\sqrt{-b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+9)^(1/2),x, algorithm="giac")

[Out]

-log(-sqrt(-b)*x + sqrt(-b*x^2 + 9))/sqrt(-b)