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

Optimal. Leaf size=17 \[ \frac{\sinh ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]

[Out]

ArcSinh[(Sqrt[b]*x)/3]/Sqrt[b]

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Rubi [A]  time = 0.0021277, antiderivative size = 17, 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} \[ \frac{\sinh ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 + b*x^2],x]

[Out]

ArcSinh[(Sqrt[b]*x)/3]/Sqrt[b]

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

Mathematica [A]  time = 0.007021, size = 17, normalized size = 1. \[ \frac{\sinh ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 + b*x^2],x]

[Out]

ArcSinh[(Sqrt[b]*x)/3]/Sqrt[b]

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Maple [A]  time = 0.002, size = 21, normalized size = 1.2 \begin{align*}{\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+9} \right ){\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x*b^(1/2)+(b*x^2+9)^(1/2))/b^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.26105, size = 157, normalized size = 9.24 \begin{align*} \left [\frac{\log \left (-\sqrt{b} x - \sqrt{b x^{2} + 9}\right )}{\sqrt{b}}, -\frac{2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + 9} \sqrt{-b} - 3 \, \sqrt{-b}}{b x}\right )}{b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[log(-sqrt(b)*x - sqrt(b*x^2 + 9))/sqrt(b), -2*sqrt(-b)*arctan((sqrt(b*x^2 + 9)*sqrt(-b) - 3*sqrt(-b))/(b*x))/
b]

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Sympy [A]  time = 0.943127, size = 14, normalized size = 0.82 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(sqrt(b)*x/3)/sqrt(b)

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Giac [A]  time = 2.49941, size = 30, normalized size = 1.76 \begin{align*} -\frac{\log \left (-\sqrt{b} x + \sqrt{b x^{2} + 9}\right )}{\sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-sqrt(b)*x + sqrt(b*x^2 + 9))/sqrt(b)