3.538 \(\int \frac{1}{x \sqrt{9+4 x^2}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right ) \]

[Out]

-ArcTanh[Sqrt[9 + 4*x^2]/3]/3

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Rubi [A]  time = 0.0100864, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 207} \[ -\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[9 + 4*x^2]),x]

[Out]

-ArcTanh[Sqrt[9 + 4*x^2]/3]/3

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{9+4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{9+4 x}} \, dx,x,x^2\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-\frac{9}{4}+\frac{x^2}{4}} \, dx,x,\sqrt{9+4 x^2}\right )\\ &=-\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9+4 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0024164, size = 20, normalized size = 1. \[ -\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[9 + 4*x^2]),x]

[Out]

-ArcTanh[Sqrt[9 + 4*x^2]/3]/3

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Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{3}{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{4\,{x}^{2}+9}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*arctanh(3/(4*x^2+9)^(1/2))

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Maxima [A]  time = 2.99021, size = 12, normalized size = 0.6 \begin{align*} -\frac{1}{3} \, \operatorname{arsinh}\left (\frac{3}{2 \,{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.24206, size = 103, normalized size = 5.15 \begin{align*} -\frac{1}{3} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} + 3\right ) + \frac{1}{3} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(-2*x + sqrt(4*x^2 + 9) + 3) + 1/3*log(-2*x + sqrt(4*x^2 + 9) - 3)

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Sympy [A]  time = 0.997542, size = 8, normalized size = 0.4 \begin{align*} - \frac{\operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.20068, size = 39, normalized size = 1.95 \begin{align*} -\frac{1}{6} \, \log \left (\sqrt{4 \, x^{2} + 9} + 3\right ) + \frac{1}{6} \, \log \left (\sqrt{4 \, x^{2} + 9} - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*log(sqrt(4*x^2 + 9) + 3) + 1/6*log(sqrt(4*x^2 + 9) - 3)