3.268 \(\int \frac{1}{\sqrt{9+4 x^2}} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{2} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

ArcSinh[(2*x)/3]/2

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Rubi [A]  time = 0.00502021, antiderivative size = 10, 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 \[ \frac{1}{2} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[9 + 4*x^2],x]

[Out]

ArcSinh[(2*x)/3]/2

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Rubi in Sympy [A]  time = 0.521502, size = 7, normalized size = 0.7 \[ \frac{\operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4*x**2+9)**(1/2),x)

[Out]

asinh(2*x/3)/2

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Mathematica [A]  time = 0.00607744, size = 10, normalized size = 1. \[ \frac{1}{2} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[9 + 4*x^2],x]

[Out]

ArcSinh[(2*x)/3]/2

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Maple [A]  time = 0.001, size = 7, normalized size = 0.7 \[{\frac{1}{2}{\it Arcsinh} \left ({\frac{2\,x}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4*x^2+9)^(1/2),x)

[Out]

1/2*arcsinh(2/3*x)

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Maxima [A]  time = 1.49827, size = 8, normalized size = 0.8 \[ \frac{1}{2} \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(4*x^2 + 9),x, algorithm="maxima")

[Out]

1/2*arcsinh(2/3*x)

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Fricas [A]  time = 0.206729, size = 22, normalized size = 2.2 \[ -\frac{1}{2} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(4*x^2 + 9),x, algorithm="fricas")

[Out]

-1/2*log(-2*x + sqrt(4*x^2 + 9))

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Sympy [A]  time = 0.149618, size = 7, normalized size = 0.7 \[ \frac{\operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*x**2+9)**(1/2),x)

[Out]

asinh(2*x/3)/2

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GIAC/XCAS [A]  time = 0.201141, size = 22, normalized size = 2.2 \[ -\frac{1}{2} \,{\rm ln}\left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(4*x^2 + 9),x, algorithm="giac")

[Out]

-1/2*ln(-2*x + sqrt(4*x^2 + 9))