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

Optimal. Leaf size=34 \[ -\frac {\sqrt {-4 x^2-9}}{x}-2 \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {277, 217, 203} \[ -\frac {\sqrt {-4 x^2-9}}{x}-2 \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 - 4*x^2]/x^2,x]

[Out]

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

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])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 49, normalized size = 1.44 \[ \frac {\sqrt {-4 x^2-9} \left (2 x \sinh ^{-1}\left (\frac {2 x}{3}\right )-\sqrt {4 x^2+9}\right )}{x \sqrt {4 x^2+9}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 - 4*x^2]/x^2,x]

[Out]

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

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fricas [C]  time = 0.91, size = 64, normalized size = 1.88 \[ \frac {-i \, x \log \left (-\frac {8 \, x + 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) + i \, x \log \left (-\frac {8 \, x - 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) - \sqrt {-4 \, x^{2} - 9}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-I*x*log(-(8*x + 4*I*sqrt(-4*x^2 - 9))/x) + I*x*log(-(8*x - 4*I*sqrt(-4*x^2 - 9))/x) - sqrt(-4*x^2 - 9))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-4 \, x^{2} - 9}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-4*x^2 - 9)/x^2, x)

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maple [A]  time = 0.00, size = 43, normalized size = 1.26 \[ \frac {4 \sqrt {-4 x^{2}-9}\, x}{9}-2 \arctan \left (\frac {2 x}{\sqrt {-4 x^{2}-9}}\right )+\frac {\left (-4 x^{2}-9\right )^{\frac {3}{2}}}{9 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 2.92, size = 21, normalized size = 0.62 \[ -\frac {\sqrt {-4 \, x^{2} - 9}}{x} + 2 i \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.77, size = 41, normalized size = 1.21 \[ -\frac {\sqrt {-4\,x^2-9}}{x}-\frac {\mathrm {asin}\left (\frac {x\,2{}\mathrm {i}}{3}\right )\,\sqrt {-4\,x^2-9}\,2{}\mathrm {i}}{3\,\sqrt {\frac {4\,x^2}{9}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- 4*x^2 - 9)^(1/2)/x^2,x)

[Out]

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

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sympy [A]  time = 0.42, size = 32, normalized size = 0.94 \[ - 2 \operatorname {atan}{\left (\frac {2 x}{\sqrt {- 4 x^{2} - 9}} \right )} - \frac {\sqrt {- 4 x^{2} - 9}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*atan(2*x/sqrt(-4*x**2 - 9)) - sqrt(-4*x**2 - 9)/x

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