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 = 0.0113191, 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 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[9 - b*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.97757, size = 14, normalized size = 0.82 \[ \frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-b*x**2+9)**(1/2),x)

[Out]

asin(sqrt(b)*x/3)/sqrt(b)

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Mathematica [A]  time = 0.0127152, 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 = 0.004, size = 21, normalized size = 1.2 \[{1\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}+9}}}} \right ){\frac{1}{\sqrt{b}}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244151, size = 1, normalized size = 0.06 \[ \left [\frac{\log \left (\frac{\sqrt{-b x^{2} + 9} b x - 3 \, b x +{\left (b x^{2} + 3 \, \sqrt{-b x^{2} + 9} - 9\right )} \sqrt{-b}}{\sqrt{-b x^{2} + 9} - 3}\right )}{\sqrt{-b}}, -\frac{2 \, \arctan \left (\frac{\sqrt{-b x^{2} + 9} - 3}{\sqrt{b} x}\right )}{\sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.52015, size = 39, normalized size = 2.29 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x^{2}}\right |}{9} > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.217633, size = 36, normalized size = 2.12 \[ -\frac{{\rm ln}\left (-\sqrt{-b} x + \sqrt{-b x^{2} + 9}\right )}{\sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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