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 = 0.0014627, 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, Rules used = {215} \[ \frac{1}{2} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 + 4*x^2],x]

[Out]

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

Rule 215

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{9+4 x^2}} \, dx &=\frac{1}{2} \sinh ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0040377, 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 = 0.003, size = 7, normalized size = 0.7 \begin{align*}{\frac{1}{2}{\it Arcsinh} \left ({\frac{2\,x}{3}} \right ) } \end{align*}

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 = 1.40079, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{2} \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.61173, size = 46, normalized size = 4.6 \begin{align*} -\frac{1}{2} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(-2*x + sqrt(4*x^2 + 9))

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Sympy [A]  time = 0.140751, size = 7, normalized size = 0.7 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{2} \end{align*}

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 [B]  time = 1.07223, size = 22, normalized size = 2.2 \begin{align*} -\frac{1}{2} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(-2*x + sqrt(4*x^2 + 9))