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

Optimal. Leaf size=26 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-9}}\right )}{\sqrt{b}} \]

[Out]

ArcTan[(Sqrt[b]*x)/Sqrt[-9 - b*x^2]]/Sqrt[b]

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Rubi [A]  time = 0.0172256, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-9}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-9 - b*x^2],x]

[Out]

ArcTan[(Sqrt[b]*x)/Sqrt[-9 - b*x^2]]/Sqrt[b]

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Rubi in Sympy [A]  time = 2.07762, size = 24, normalized size = 0.92 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{- b x^{2} - 9}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(b)*x/sqrt(-b*x**2 - 9))/sqrt(b)

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Mathematica [A]  time = 0.0116653, size = 26, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-9}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-9 - b*x^2],x]

[Out]

ArcTan[(Sqrt[b]*x)/Sqrt[-9 - b*x^2]]/Sqrt[b]

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Maple [A]  time = 0.004, size = 21, normalized size = 0.8 \[{1\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}-9}}}} \right ){\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^2-9)^(1/2),x)

[Out]

arctan(x*b^(1/2)/(-b*x^2-9)^(1/2))/b^(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 = 0.252234, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-2 \, \sqrt{-b x^{2} - 9} b x -{\left (2 \, b x^{2} + 9\right )} \sqrt{-b}\right )}{2 \, \sqrt{-b}}, \frac{\arctan \left (\frac{\sqrt{b} x}{\sqrt{-b x^{2} - 9}}\right )}{\sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*log(-2*sqrt(-b*x^2 - 9)*b*x - (2*b*x^2 + 9)*sqrt(-b))/sqrt(-b), arctan(sqrt
(b)*x/sqrt(-b*x^2 - 9))/sqrt(b)]

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Sympy [A]  time = 3.36872, size = 17, normalized size = 0.65 \[ - \frac{i \operatorname{asinh}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*asinh(sqrt(b)*x/3)/sqrt(b)

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GIAC/XCAS [A]  time = 0.216764, size = 38, normalized size = 1.46 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{-b} x + \sqrt{-b x^{2} - 9} \right |}\right )}{\sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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