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

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

[Out]

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

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Rubi [A]  time = 0.0161465, 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, 47, 63, 204} \[ -\frac{\sqrt{-4 x^2-9}}{2 x^2}-\frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 - 4*x^2]/x^3,x]

[Out]

-Sqrt[-9 - 4*x^2]/(2*x^2) - (2*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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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 204

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

Rubi steps

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

Mathematica [A]  time = 0.0136417, size = 55, normalized size = 1.41 \[ \frac{12 x^2+4 \sqrt{4 x^2+9} x^2 \tanh ^{-1}\left (\sqrt{\frac{4 x^2}{9}+1}\right )+27}{6 x^2 \sqrt{-4 x^2-9}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 - 4*x^2]/x^3,x]

[Out]

(27 + 12*x^2 + 4*x^2*Sqrt[9 + 4*x^2]*ArcTanh[Sqrt[1 + (4*x^2)/9]])/(6*x^2*Sqrt[-9 - 4*x^2])

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Maple [A]  time = 0.003, size = 41, normalized size = 1.1 \begin{align*}{\frac{1}{18\,{x}^{2}} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{2}{9}\sqrt{-4\,{x}^{2}-9}}+{\frac{2}{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((-4*x^2-9)^(1/2)/x^3,x)

[Out]

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

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Maxima [C]  time = 3.57064, size = 69, normalized size = 1.77 \begin{align*} \frac{2}{9} \, \sqrt{-4 \, x^{2} - 9} + \frac{{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{18 \, x^{2}} + \frac{2}{3} i \, \log \left (\frac{6 \, \sqrt{4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*sqrt(-4*x^2 - 9) + 1/18*(-4*x^2 - 9)^(3/2)/x^2 + 2/3*I*log(6*sqrt(4*x^2 + 9)/abs(x) + 18/abs(x))

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Fricas [C]  time = 1.25728, size = 170, normalized size = 4.36 \begin{align*} \frac{-2 i \, x^{2} \log \left (-\frac{4 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{3 \, x}\right ) + 2 i \, x^{2} \log \left (-\frac{4 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{3 \, x}\right ) - 3 \, \sqrt{-4 \, x^{2} - 9}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.74531, size = 27, normalized size = 0.69 \begin{align*} - \frac{2 i \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{3} - \frac{i \sqrt{1 + \frac{9}{4 x^{2}}}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*asinh(3/(2*x))/3 - I*sqrt(1 + 9/(4*x**2))/x

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Giac [C]  time = 2.00082, size = 39, normalized size = 1. \begin{align*} -\frac{i \, \sqrt{4 \, x^{2} + 9}}{2 \, x^{2}} - \frac{2}{3} \, \arctan \left (\frac{1}{3} i \, \sqrt{4 \, x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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