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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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 = 0.00, size = 10, normalized size = 1.00 \[ \frac {1}{2} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 + 4*x^2],x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.41, size = 16, normalized size = 1.60 \[ -\frac {1}{2} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(-2*x + sqrt(4*x^2 + 9))

________________________________________________________________________________________

giac [B]  time = 1.07, size = 16, normalized size = 1.60 \[ -\frac {1}{2} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(-2*x + sqrt(4*x^2 + 9))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 7, normalized size = 0.70 \[ \frac {\arcsinh \left (\frac {2 x}{3}\right )}{2} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 6, normalized size = 0.60 \[ \frac {1}{2} \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 6, normalized size = 0.60 \[ \frac {\mathrm {asinh}\left (\frac {2\,x}{3}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2 + 9)^(1/2),x)

[Out]

asinh((2*x)/3)/2

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 7, normalized size = 0.70 \[ \frac {\operatorname {asinh}{\left (\frac {2 x}{3} \right )}}{2} \]

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

________________________________________________________________________________________