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

Optimal. Leaf size=6 \[ \sinh ^{-1}\left (\frac{x}{3}\right ) \]

[Out]

ArcSinh[x/3]

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Rubi [A]  time = 0.0011798, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {215} \[ \sinh ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 + x^2],x]

[Out]

ArcSinh[x/3]

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

Mathematica [A]  time = 0.0034179, size = 6, normalized size = 1. \[ \sinh ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 + x^2],x]

[Out]

ArcSinh[x/3]

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Maple [A]  time = 0.001, size = 5, normalized size = 0.8 \begin{align*}{\it Arcsinh} \left ({\frac{x}{3}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+9)^(1/2),x)

[Out]

arcsinh(1/3*x)

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Maxima [A]  time = 1.40911, size = 5, normalized size = 0.83 \begin{align*} \operatorname{arsinh}\left (\frac{1}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(1/3*x)

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Fricas [B]  time = 1.92847, size = 35, normalized size = 5.83 \begin{align*} -\log \left (-x + \sqrt{x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(x^2 + 9))

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Sympy [A]  time = 0.12484, size = 3, normalized size = 0.5 \begin{align*} \operatorname{asinh}{\left (\frac{x}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(x/3)

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Giac [B]  time = 1.05858, size = 19, normalized size = 3.17 \begin{align*} -\log \left (-x + \sqrt{x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x + sqrt(x^2 + 9))