3.562 \(\int \frac{1}{x^3 \sqrt{-9+4 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{\sqrt{4 x^2-9}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right ) \]

[Out]

Sqrt[-9 + 4*x^2]/(18*x^2) + (2*ArcTan[Sqrt[-9 + 4*x^2]/3])/27

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Rubi [A]  time = 0.0158369, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 203} \[ \frac{\sqrt{4 x^2-9}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[-9 + 4*x^2]),x]

[Out]

Sqrt[-9 + 4*x^2]/(18*x^2) + (2*ArcTan[Sqrt[-9 + 4*x^2]/3])/27

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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^3 \sqrt{-9+4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{-9+4 x}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9+4 x^2}}{18 x^2}+\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-9+4 x}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9+4 x^2}}{18 x^2}+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{\frac{9}{4}+\frac{x^2}{4}} \, dx,x,\sqrt{-9+4 x^2}\right )\\ &=\frac{\sqrt{-9+4 x^2}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{-9+4 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0207119, size = 54, normalized size = 1.38 \[ \frac{4}{81} \sqrt{4 x^2-9} \left (\frac{9}{8 x^2}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{4 x^2}{9}}\right )}{2 \sqrt{1-\frac{4 x^2}{9}}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[-9 + 4*x^2]),x]

[Out]

(4*Sqrt[-9 + 4*x^2]*(9/(8*x^2) + ArcTanh[Sqrt[1 - (4*x^2)/9]]/(2*Sqrt[1 - (4*x^2)/9])))/81

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Maple [A]  time = 0.003, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{18\,{x}^{2}}\sqrt{4\,{x}^{2}-9}}-{\frac{2}{27}\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^3/(4*x^2-9)^(1/2),x)

[Out]

1/18*(4*x^2-9)^(1/2)/x^2-2/27*arctan(3/(4*x^2-9)^(1/2))

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Maxima [A]  time = 1.93946, size = 32, normalized size = 0.82 \begin{align*} \frac{\sqrt{4 \, x^{2} - 9}}{18 \, x^{2}} - \frac{2}{27} \, \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^3/(4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

1/18*sqrt(4*x^2 - 9)/x^2 - 2/27*arcsin(3/2/abs(x))

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Fricas [A]  time = 1.34103, size = 101, normalized size = 2.59 \begin{align*} \frac{8 \, x^{2} \arctan \left (-\frac{2}{3} \, x + \frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) + 3 \, \sqrt{4 \, x^{2} - 9}}{54 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/54*(8*x^2*arctan(-2/3*x + 1/3*sqrt(4*x^2 - 9)) + 3*sqrt(4*x^2 - 9))/x^2

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Sympy [A]  time = 2.23335, size = 99, normalized size = 2.54 \begin{align*} \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{27} - \frac{i}{9 x \sqrt{-1 + \frac{9}{4 x^{2}}}} + \frac{i}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{27} + \frac{1}{9 x \sqrt{1 - \frac{9}{4 x^{2}}}} - \frac{1}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(4*x**2-9)**(1/2),x)

[Out]

Piecewise((2*I*acosh(3/(2*x))/27 - I/(9*x*sqrt(-1 + 9/(4*x**2))) + I/(4*x**3*sqrt(-1 + 9/(4*x**2))), 9/(4*Abs(
x**2)) > 1), (-2*asin(3/(2*x))/27 + 1/(9*x*sqrt(1 - 9/(4*x**2))) - 1/(4*x**3*sqrt(1 - 9/(4*x**2))), True))

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Giac [A]  time = 2.46387, size = 39, normalized size = 1. \begin{align*} \frac{\sqrt{4 \, x^{2} - 9}}{18 \, x^{2}} + \frac{2}{27} \, \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^3/(4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

1/18*sqrt(4*x^2 - 9)/x^2 + 2/27*arctan(1/3*sqrt(4*x^2 - 9))