3.6.76 \(\int \frac {1}{\sqrt {9+b x^2}} \, dx\) [576]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, 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 {\sinh ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 + b*x^2],x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.65 \begin {gather*} -\frac {\log \left (-\sqrt {b} x+\sqrt {9+b x^2}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 + b*x^2],x]

[Out]

-(Log[-(Sqrt[b]*x) + Sqrt[9 + b*x^2]]/Sqrt[b])

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Maple [A]
time = 0.04, size = 21, normalized size = 1.24

method	result	size
meijerg	\(\frac {\arcsinh \left (\frac {x \sqrt {b}}{3}\right )}{\sqrt {b}}\)	\(12\)
default	\(\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+9}\right )}{\sqrt {b}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 11, normalized size = 0.65 \begin {gather*} \frac {\operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {b} x\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(1/3*sqrt(b)*x)/sqrt(b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (11) = 22\).
time = 1.71, size = 65, normalized size = 3.82 \begin {gather*} \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 {gather*}

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 = 0.40, size = 14, normalized size = 0.82 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{3} \right )}}{\sqrt {b}} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (11) = 22\).
time = 0.78, size = 35, normalized size = 2.06 \begin {gather*} \frac {1}{2} \, \sqrt {b x^{2} + 9} x - \frac {9 \, \log \left (-\sqrt {b} x + \sqrt {b x^{2} + 9}\right )}{2 \, \sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(b*x^2 + 9)*x - 9/2*log(-sqrt(b)*x + sqrt(b*x^2 + 9))/sqrt(b)

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Mupad [B]
time = 0.04, size = 11, normalized size = 0.65 \begin {gather*} \frac {\mathrm {asinh}\left (\frac {\sqrt {b}\,x}{3}\right )}{\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2 + 9)^(1/2),x)

[Out]

asinh((b^(1/2)*x)/3)/b^(1/2)

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