3.3.68 \(\int \frac {1}{\sqrt {9+4 x^2}} \, dx\) [268]

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

[Out]

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

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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 = {221} \begin {gather*} \frac {1}{2} \sinh ^{-1}\left (\frac {2 x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 + 4*x^2],x]

[Out]

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

Rule 221

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*}

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Mathematica [A]
time = 0.01, size = 20, normalized size = 2.00 \begin {gather*} -\frac {1}{2} \log \left (-2 x+\sqrt {9+4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 + 4*x^2],x]

[Out]

-1/2*Log[-2*x + Sqrt[9 + 4*x^2]]

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Mathics [A]
time = 1.65, size = 6, normalized size = 0.60 \begin {gather*} \frac {\text {ArcSinh}\left [\frac {2 x}{3}\right ]}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(4*x^2+9)^(1/2),x]')

[Out]

ArcSinh[2 x / 3] / 2

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Maple [A]
time = 0.05, size = 7, normalized size = 0.70

method	result	size
default	\(\frac {\arcsinh \left (\frac {2 x}{3}\right )}{2}\)	\(7\)
meijerg	\(\frac {\arcsinh \left (\frac {2 x}{3}\right )}{2}\)	\(7\)
trager	\(\frac {\ln \left (2 x +\sqrt {4 x^{2}+9}\right )}{2}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2+9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.35, size = 6, normalized size = 0.60 \begin {gather*} \frac {1}{2} \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (6) = 12\).
time = 0.32, size = 16, normalized size = 1.60 \begin {gather*} -\frac {1}{2} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 9}\right ) \end {gather*}

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 = 0.07, size = 7, normalized size = 0.70 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {2 x}{3} \right )}}{2} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (6) = 12\).
time = 0.00, size = 21, normalized size = 2.10 \begin {gather*} -\frac {\ln \left (\sqrt {4 x^{2}+9}-2 x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x^2+9)^(1/2),x)

[Out]

-1/2*log(-2*x + sqrt(4*x^2 + 9))

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

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

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