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

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

[Out]

ArcTan[Sqrt[-9 + 4*x^2]/3]/3

________________________________________________________________________________________

Rubi [A]  time = 0.0096452, 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, 203} \[ \frac{1}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-9 + 4*x^2]),x]

[Out]

ArcTan[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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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} \tan ^{-1}\left (\frac{1}{3} \sqrt{-9+4 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0026158, size = 20, normalized size = 1. \[ \frac{1}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[-9 + 4*x^2]),x]

[Out]

ArcTan[Sqrt[-9 + 4*x^2]/3]/3

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{3}\arctan \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*arctan(3/(4*x^2-9)^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.98175, size = 12, normalized size = 0.6 \begin{align*} -\frac{1}{3} \, \arcsin \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*arcsin(3/2/abs(x))

________________________________________________________________________________________

Fricas [A]  time = 1.30312, size = 57, normalized size = 2.85 \begin{align*} \frac{2}{3} \, \arctan \left (-\frac{2}{3} \, x + \frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\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]

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

________________________________________________________________________________________

Sympy [A]  time = 1.05396, size = 26, normalized size = 1.3 \begin{align*} \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{3} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\- \frac{\operatorname{asin}{\left (\frac{3}{2 x} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*acosh(3/(2*x))/3, 9/(4*Abs(x**2)) > 1), (-asin(3/(2*x))/3, True))

________________________________________________________________________________________

Giac [A]  time = 2.14256, size = 19, normalized size = 0.95 \begin{align*} \frac{1}{3} \, \arctan \left (\frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\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]

1/3*arctan(1/3*sqrt(4*x^2 - 9))